The Infinite Checkerboard Quandary

Before I get into the question, just want to say that last week’s Dinner Table was awesome. So awesome that it’s made us want to build something new on the site so we can see a lot more where that came from. Stay tuned.


Thanks to Andrew Snyder-Beattie for introducing me to this thought experiment.

Here’s the deal:

You’re an omnipotent god, and you’re told by your boss that you have to create an infinite checkerboard—one that extends forever in every direction in a 2-dimensional plane. You’ll then create an infinite number of new humans and put one on each square of the board, where he or she will live forever. Both the board and the lives of the humans on it will exist for an infinite eternity of future once you make them. Your boss gives you two options for the type of board you create:

Board 1: Sad with a drop of happy

The first option is a board where each and every one of the infinite number of humans you create on the squares is in a constant state of sadness and horrible suffering. But after you create the board, you’ll be able to use a medicine dropper to squeeze a single drop of happiness somewhere on the board. The drop of happiness will immediately begin to expand outward in all directions, and each time the drop’s expanding circle hits a new square, the person on that square suddenly becomes overwhelmingly happy and fulfilled and free of all suffering, and they’ll stay this way for the rest of eternity. The happiness drop will continue expanding outward for eternity.

Sad Board

Board 2: Happy with a drop of sad

The second option is the exact opposite. The board you create is full of an infinite number of blissfully happy people. But for some odd reason, your boss insists that if you choose this option, you’ll have to use that same medicine dropper to squeeze a drop of sadness onto the board, which will immediately and rapidly expand outward in all directions and continue for eternity. Each time the drop’s reach covers a new square, the person on that square becomes unbelievably sad and full of the worst suffering, and they’ll remain that way forever thereafter.

Happy Board

The question for you is two-fold:

Question 1) What is the better moral choice? As the omnipotent god, which board would you choose to create?

Question 2) If instead of being the god in control, you were told you were going to end up as one of the eternal humans on the board some other god creates, which board would you want that god to choose?


Tim’s Answer: This is a head-spinner. The issue with the happy board is that everyone on it is happy at the beginning but at some point that will end. And once it does, they will be sad forever thereafter. So each person is happy for a finite amount of time and then sad for an infinite amount of time.


The thing is, at any given point in time, the number of sad people on the happy board is finite, because they’re all contained within the expanding circle of sadness, while the number of happy people on the board is infinite, because there is an infinite number of people outside the sadness circle.

The sad board has the opposite situation. Each person on it will be sad for a finite amount of time and happy for a vastly larger, infinite amount of time—but at any given point in time there will be a far larger number of sad people on the board than happy people, because the number of sad people will be infinite and the number of happy people finite.


Let’s start by looking at Question 1. If you’re the god, and you’re trying to maximize happiness and minimize sadness, it seems like a no-brainer. No matter how far into the future you go, the happy board will have infinitely more happy people than sad people, as well as infinitely more happy people than however many happy people have amassed on the sad board. By choosing the happy board, you’re choosing far more happiness than sadness.

Now let’s look at Question 2. From a theoretical standpoint, before making your choice, the happy board seems to make more sense. Because no matter how far into the future you go, someone on the happy board is far more likely to be in the infinite happy region than the finite sad region. Think of it this way—what if instead of being placed permanently on a board, I told you that both boards existed and that you would have to choose which one on which to spend just one hour? If your hour happened to be on a sad square, it would be a terrible experience. If it were on a happy square, it would be one of the best hours of your life. But the thing is, whichever board you choose, both the square you’re placed on and the amount of time that will have passed since the drop first fell will be randomly-selected. So you can choose the happy or sad board, but at that point, you don’t know where you’ll be placed on the board, and the time that had passed since the drop fell could be seven hours, 100 quadrillion millennia, or something much larger.

If you’re being logical, it doesn’t matter how much time has passed—it’s always a much better choice to pick the happy board. At any given hour in the future, a randomly selected square on the happy board is overwhelmingly likely to leave you with a happy hour, and the opposite is true on the sad board.

Back to the original question. We’ve established that for every individual hour in the future, it would be a better choice to spend it on the happy board (if you didn’t know where you’d be on the board). And since the original question is simply choosing where to spend all future hours, isn’t the happy board an obvious choice?

Well, no. Because as soon as you’re actually on the board, the logic reverses.

The second you end up on a board, you’d much rather it be a square on the sad board. Why? Because the second you’re assigned an actual square on the sad board, the amount of time you’ll remain sad is officially finite, no matter how large that time is, and the time you’ll spend happy after the drop hits you is officially infinite. For this reason, the sad board is definitely the board you want to be standing on.

This is true even though if you had to just go onto either board for one hour, or one month—or one 100-trillion-year period—probability says you’d be insane to pick anything but the happy board.

The thing that’s so hard about this problem is that it pits infinity of time against infinity of space. And depending on which question you’re answering, one wins out over the other. In Question 1, as the god, you’re only concerned with infinity of space, because that factor says that no matter how much time passes, the happy board will always be mostly full of happy people. Time becomes irrelevant.

And if, as an individual, you were just being placed on the board for a finite period of time, you’d be thinking like the god. Infinity of space would rule the day, and you’d choose the happy board.

But as an individual permanently on the board, as in Question 2, infinity of space suddenly becomes irrelevant to you because the only space you care about—the distance between you and the outer edge of the drop’s circle—is now finite. What becomes super relevant is infinity of time, which no finite space is any match for.

If you’re concerned with infinity of space, you pick the happy board. So the logical answer to Question 1 is the happy board.

If you’re concerned with infinity of time, you pick the happy drop. So the logical answer to Question 2 is the sad board.

The troubling thing about this problem is that it makes the correct choice for the god the incorrect choice for each and every one of his subjects. And if, as the god, each individual person would rather you choose the sad board, can it still be called moral to choose the happy board? As a friend of mine asked, “How can it be morally right for god to create a world that no one would want to live in?”

On top of the mind-bendingness of this problem in general, this final question has me considering all the ways in life that collective action can be problematic. The infinite checkerboard quandary is a good thing to keep in mind when looking at how societies function—if we’re aware of this odd phenomenon, we’ll be better equipped to address it when it occurs in the real world.

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  • Alex Mac

    First comment, I’d probably do happy with a drop of sad because with infinity, no matter how fast the drop expands a majority of the people will be happy.

    • C

      I thought the same, and I think that’s the right answer for q1. But for q2 I think it’s the other option, as you will be sad for a non infinite amount of time followed for the opposite.

      If that’s correct, you at least have to address how what’s good for the individual isn’t the optimal for the whole

    • D_Aiello

      But board 1 at least gives hope.

    • Matias Frank Jensen

      But for every single individual that individual is going to be happy for a finite amount of time, but sad for eternity. So for every single person on the board, number 2 is infinitely worse than number 1.
      If you look at the board not as a collection of individuals but as an organism, then it would probably make sense to choose number 2. But since these individuals probably care most about themselves and are not terribly affected by people and infinite distance from themselves, I would argue that number 1 is most morally correct.
      That is, I would argue that it is wrong to see the board as an organism, but instead a collection of individuals.

  • Lucas Vergeest

    I would choose a drop of happiness, because I think it would be better to go from a sad to a happy state than the other way around. 🙂

    • I agree.

      More, sadness with memory of happiness is worst than only sadness.

  • Alfred

    I’d pick the happiness drop. Supposing the people lived forever, the majority of anyone’s time alive would be spent happy as no matter where you are, the happiness is expanding out and will eventually reach you. I say that even if the sadness drop makes the majority of people happy, the first one technically makes everyone happy for the infinite majority of their lifetime. My reasoning may be messed up but this is my initial answer and it is late where I am 🙂

    • Agree, exactly what I was going to say

    • v43

      wrong, there would be an infinite number of sad people outside the drop. Infinite. So the drop of happiness won’t reach everyone sooner or later. While the number of people inside the drop will slowly grow towards infinite, there will always be an infinite number of sad people outside of it.

  • Jacco

    I’d go with the sad board with a drop of happiness. It’s the more optimistic option, where as the other is very bleak if you look to the future. The people on the sad board will never know what happiness is until the drop hits them. On the other board, the people will only know happiness for a very short time until the drop of sadness gives them suffering for all time. So knowing all this as the omnipotent god that I am, I would choose the sad board to live on, knowing that one day eternal happiness will come my way.

  • René

    As the board is infinite and the drop only expands with a finite speed, there will always be an infinite amount (sounds kind of stupid, but I guess you get what I want to say) of people outside the drop and only a finite amount affected by it.
    Because of that, I would choose board 2 for both answers. For 1) because there are always more happy people (infinitely many) than sad people and for 2) because the “happy area” is way bigger than the “sad area”; thus the probability of being placed in a square that’s happy is far higher.

    Then again, the more I think about this, it doesn’t feel right. But that’s probably just because humans can’t really grasp the concept of infinity.

  • David Olsen

    Whenever I try to divide infinity by infinity I get an indeterminate value. Nevertheless, I feel that as god with utilitarian ethical system, I’d create Happy with a drop of Sad, but as an eternally living human, I’d know that sadness drop would get to me eventually, so I’d much rather live on Sad with a drop of Happy.

  • Bob Loblaw

    Q1, B2 & Q2 B1- I think an eternity of sadness is worse than a finite time of sadness, however long, therefore I would choose board 2 for question 1. For question 2 however, I’d take the chance that I’d be one of the humans very far away from the drop and choose board 1.

  • As the God: Create the sad board with the drop of happiness. Initially, yes, everyone will be sad, but the happiness will continue to expand infinitely, and happy after sad > Sad after happy

    As the person: This is tougher, actually. Due to the nature of the word infinite, I could be on the happy board and hever expect the sadness to reach me, but I could also be on the sad board with the same problem. Not sure.

  • Dógui da Marina

    I once read that the best happiness is the one that comes after the suffering, like when you study a lot first (suffering) and get an A on a test (happiness). The opposite is like when you drink a lot first (happiness) and then you’re drunk and have a hangover later (suffering). Therefore, I would choose the drop of happiness. And, of course, because once it hits you, you stay happy forever.

  • Life_XP

    How do people know they’re sad or happy before even experiencing the other feeling?

  • There are two opposite ways of thinking about this:

    1. If the board is infinite then the drop will always be infinitely smaller than the board. So the happiness board is the right choice.
    2. If there is an infinite amount of time then if you choose the sadness board each human will eventually become happy (and vice-versa) for the happiness board, so the sadness board is the right choice.

    The solution:
    I think it’s a mathematical issue for which I don’t have enough knowledge to solve it. There are different types of infinities in mathematics, and some are smaller than others by defition (altough they’re obviously infinite).
    So to solve this you need a mathematician who specialises in this sort of thing, and he will probably be able to prove his solution (or to prove that a solution doesn’t exist). 🙂

  • Nathalie Gil

    I’d go with option 2 for both questions. The movement of the drop of ‘sadness’ is determined, and theorically will always be trying to catch up, therefore there will always be a bigger quantity of happy people then sad people in this context. Which is better in my world if I live there, but also, as I am a God, as my freedom to chose one context over the other is also a Godly decision, therefore even if I need to drop sadness in it, I am reassured that there will be an infinite (minus drop) amount of people happier than sad.

  • Jonas Betzendahl

    My take would be the following.

    No matter where I am on an infinite checker board, my distance to the origin of the drop will always be finite. That means if I’m originally feeling the one feeling, I will do so only for a finite amount of time. The drop expands at finite speed and hence reaches my finitely-distant square in a finite amount of time. I then have an infinite amount of time with the other feeling. So I’d choose to be on the sad board with a drop of happiness.

    Same idea leads to the conclusion that the god also should pick the sad board since the above is true for all individuals on the board. And if an infinite amount of humans all would rather be on the sad board with the happy drop, who am I to argue. It’s their suffering. This also “maximises happiness”, even though I don’t like that phrase.

    But I may be proven wrong with a whole bunch of cardinals and ordinals and some-infinities-are-bigger-than-other-infinities-talk. Looking forward to it. =)

  • Pam O

    Sad first, happy later. People who are sad always don’t necessarily appreciate that they really are sad – it is the norm. But to be deliriously happy, and then have it ripped away? Immeasurably more cruel.

  • lilyc123

    It doesn’t matter what you choose, because you will never finish making the board. If you have to create an infinite board, you will spend the rest of infinity creating it (by the very definition of infinity you will never finish).

    • abstruseAbacist

      What if it took you 1 minute to make the first square, 1/2 a minute to make the next, 1/4 of a minute to make the third, etc. This may seem impossible, but we’re assuming you’re an immortal god. You’d finish

      making the entire chessboard in 2 minutes.

  • Fridolin Saal

    Well… That is a very tough question. At first I thought Board 1 would be the better option, but I think it`s not that easy. Strictly mathematically speaking, one would have to choose Board 2 in both cases. As the drop expands outward with a finite speed, at any given time there would be only a finite number of unhappy people but an infinite number of happy people. Therefore, statistically speaking, the ratio of unhappy people is zero at any (finite) time. I`m not entirely sure about what will happen in eternity, though. I presume my human (and mathematical) understanding can not grasp the concept of endless time and space. However, it is clear that there will never be zero happy people, at best the ratio of unhappy to happy people would approach infinity divided by infinity, which could be any number. There are probably people who will never feel sad in all eternity whereas on Board 1, everybody would be unhappy at some point. Also, it just happens to strike me that the statistically expected time until the “changeover” on both boards equals infinity. So, Board 2 is definitely the one to go with.
    If I made any mathematical mistake, feel free to correct me. 🙂

  • gastromax

    I’d go for sad with a drop of happiness in both cases. If everyone were created constantly sad from the start and that was the only feeling they ever felt, then they’re not actually sad – they just are. They wouldnt know there was another way to feel. I belive the only way to know you’re happy is to at some point have experienced sadness.

  • Todd

    Before I contemplate this checkboard, was there an answer to the penny landing touching a black & white square on a checkboard? I can’t seem to find it here. (Was it not a dinner table discussion from WBW?)

  • tweinstre

    If the board expands at infinite speed (whatever would that be) and the drop at some finite speed,then the second option is better. Because you’ll always have an infinite number of happy people and only a finite number of unhappy people. I would choose to be at the second board because the probability of the drop ever reaching me is extremely small,approximately 0. But morally it is better to make people happy than sad so,morally,it is a tricky one…ehehehe…..I wonder is this your answer as well?

  • Sascha Jürgens

    IQuestion 1: In my opinion the best moral choice would be option2 because at any point of the timeline sadness would be finite and happiness would be infinite. So always more people would be still happy then the other way round. Question 2: I would choose option1 following Alex’ explanation.Because thinking as an individual hope and expectations are more important to accept suffering for a terminated period.

  • String userName=new String();

    For both questions, I’d choose the first board. Here’s my logic: since every human is living forever, any given human will be reached by the drop after a finite amount of time. From then to infinity, they’ll be stuck in that state, for better or for worse. Every person on board 1 will be sad for a finite amount of time and happy for an infinite amount of time, while the opposite is true for people on board 2.

  • Bryson

    First things first: Omnipotent gods having bosses seems a bit off. Is the boss also omnipotent? It’s not like you can be more omnipotent than an omnipotent god. How did the whole boss arrangement come about? Does the omnipotent god have low self esteem?

    As for the question: It’s hard to represent the problem, because the concept of infinity makes no sense to the human brain. The area around the drop is infinitely large, so no matter how large the drop is it would be infinitely small. Theoretically, the omnipresent god would have to be a pretty massive dick to go with the second option, because he/she would be screwing over an infinitely larger number of people. (although, to be completely honest, you’re pretty much a massive dick either way. This is what you’re doing with omnipotence? Seriously?).

    HOWEVER, context also has to be considered.

    sad (adjective)
    feeling or showing sorrow; unhappy.
    The definition of sadness is the lack of happiness. How can you be sad if you’ve never been happy?
    Remember that conversation from the last dinner table about how the brain can only interpret change? Sadness to the people in the sad area means nothing. If they are in a constant state of suffering, suffering won’t mean anything to them – so things can only get better. However, if they go from overwhelming happiness to infinite suffering, that’s going to suck. a lot.

    So, my answer for both options is board 1.

    • Fridolin Saal

      I would agree with you on the moral and psychological aspects. I think there are two ways to think about all this, an ethical one and a mathematical one. In my take on the question, I chose to go with mathematics. First I had similar thoughts like you, but after thinking a while I came to the conclusion that a dickish omnipotent being would surely find a way to make people unhappy even if they have never felt happiness. 😉 In that case, the moral aspects you considered would not apply. It’s all about interpretation.

      • Bryson

        It’s because these things don’t exist in our world.
        You know how it took humans until the 17th century to realize that objects that are moving move forever unless stopped by another force? It’s difficult for us to comprehend Newton’s first law in a world with forces that we can’t physically see such as friction and gravity.

        How are we supposed to know what infinite happiness and infinite sadness are? They don’t exist in our world.

  • D_Aiello

    Board 1, definitely. I would choose both as a god and as human. Specially as human.

    Yes, the board 1 initially sucks, but it gives you something that the board 2 can’t give: hope.
    I’d rather choose to live a thousand years of suffering with hope than live my happy life knowing that one day everything will fall apart.
    Of course, what makes this decision so difficult is the own nature of the word “infinite”. Being a human in both boards would suck either way. That’s way I’d choose board 1, it’s the one were more and more people become happy as the time pass.

  • Wes

    If I were the god, I would pick the happy board with a drop of sadness. Since the number of sad people will always be finite and the number of happy people will always be infinite, this option does the least moral wrong since the least number of people are worse off.

    If I were to be placed on the board, I would pick the sad board with a drop of happiness. The amount of time that I am sad will be finite, whereas once the drop hits me, the amount of time I will be happy will be infinite. Therefore, I will experience the most happiness on this board.

  • RJ

    I would choose the one where everyone is happy in the beginning in both the cases.
    Since the board is infinite and the drop spreads at some finite rate, at any given time the number of squares the drop had spread to is finite but the number of squares it hasn’t spread to is infinite. So at any given time, there will be finite sad people and infinite happy people. Also the probability of a person on a square being unhappy at any given time is zero because finite/infinite, which is zero

  • tass0s

    As a God I would be aiming for less people to be sad, and the second choice seems like the one that does it since the sadness is expanding in an infinite set meaning that not everyone will be sad at the same time.

  • Elena Gaska

    I’d choose the first board, sad with a drop of happiness, for both questions.
    There are very interesting answers so far. People who choose the second board have a very
    good point; at any given moment there are more happy people than
    sad people (happy people is infinite, sad people is finite) and, when they
    choose it for themselves, there’s a greater chance of landing in a happy square
    than in a sad square. But the second board seems better when we think about the
    infinite time + infinite life + expanding drop, which means that happiness will
    reach everyone eventually (although it will always be one step behind), and, for
    every individual, sadness will be finite and happiness will be eternal.

    • Jonathan

      I couldn’t have worded it better myself. This is my thinking too. Gaska for president of the universe!

      • Elena Gaska

        I humbly accept your nomination and promise to work hard for you, for the people of the Comment Section and for the rest of the citizens of the Universe!

    • AndersN

      Happiness will never reach everyone.

      • Elena Gaska

        That’s true (because
        of the infinite sad people vs. finite happy people) for any moment. But is also
        true that everyone on the board is either happy (reached by the drop) or
        sad-now-but-will-reach-eternal-happiness-eventually (because the drop is
        expanding). Taking infinite time into consideration (and that whole
        human hope thing) made me choose the first board.

        • yep, elena, i’m with you. especially assuming the humans know what’s coming.

      • Ksenia Kolchina

        Neither it will on another board, as the whole world will start with a minimum of one unhappy person, which is already not everyone by default. We can only talk about the absolute majority in both scenarios

      • Martin Nick Smolík

        whom will it not reach? keep in mind that no such place as [infinity,infinity] exists

        • Tom Miller

          It will not reach an infinite amount of people.

    • JunoEven

      “I’d choose the first board, sad with a drop of happiness… for every individual, sadness will be finite and happiness will be eternal.”

      Pefectly phrased.

    • Tom Miller

      “happiness will reach everyone eventually”

      Actually I don’t think that’s mathematically correct. There will be an infinite amount of people it doesn’t reach, who will be sad for infinity. Because infinity – infinity = infinity.

      It’s a bit like a number sequence. If you take 1,2,3,4,5… for infinity as list A, and then take every 10th person from list A and call them list B. Logic suggests that list B is 10% of the size of list A, however, both are of equal length (infinity).

      • WW

        Actually I’m pretty sure it does because of infinite time. We can collapse the problem into 1 dimension, and set 0 at where the drop lands. You are assigned a number x for the number of people between you and that drop. X is forced to be an integer, because you can’t have partial people between you and the drop. So now you have an integer, and the happiness level is increasing to eternity from 0. Eventually it’ll reach you no matter what number you are, assuming people live for infinite time and the drop expands eternally.

        • Tom Miller

          Infinite time doesn’t matter. Even if you forget the drop altogether and just instantly change everyone from sad to happy, you STILL have an infinite number of sad people! Crazy, I know, but it’s mathematically accurate.

          • WW

            If you instantly change everyone from sad to happy, you definitely have no sad people left… by definition. Would you please walk me through the mathematical proof of what you’re saying? I get your argument for the 10% thing, but you’re assuming that instead of a dot, you’re converting people by bands. That’s a different problem, because although you’re talking about the same infinity, you are creating a different mapping problem. Your argument about 10% shows that it’s not sufficient to show that the two infinities are equal to solve the problem, but it does not prove that there are people that will never be reached by the dot.

            • Tom Miller

              Well, break it down into smaller chunks. Let’s say you instantly change 10 sad people to happy. How many sad people are left? Infinity. Essentially, you haven’t changed the “number” of sad people at all. Infinity – 10 equals infinity.

              If you change 10 to the power of 20 (bloody large number) sad people to happy people, how many sad people are left? Infinity. Infinity – 10 to the 20 = infinity.

              So if you change an infinite number of sad people to happy people, how many sad people are left? Infinity. There is still an infinite number of sad people.

            • WW

              You’re making a jump in your logic. Let’s take the real numbers from 0 to 1. there’s an infinite number of them. If you draw one thin line at 0.5 with a highlighter, there’s still an infinite number of real numbers not covered. If you draw a thicker line covering 0 to 0.5, there’s an infinite number of real numbers not covered. If you highlight the line from -0.1 to 1.1, then all of the numbers of covered.
              You’re missing the point that at the end of infinite time, there will be an infinite number of people, but this problem by construction covers everything. To use your last paragraph. If you have an infinite number of people. If you convert an infinite number of them.. let’s say you convert every person that has cells in its body (which is an infinite number right?), every single person would be converted.
              Another example. If you have a field of infinite dots. If you convert every dot that is a dot, you will convert every dot.

            • Tom Miller

              Ahh, you see I think you’re making a jump in *your* logic.

              “…at the end of infinite time…”

              Well, there is no “end” of infinite time is there? That’s the problem. So your 0 to 1 analogy breaks down. It’s 0 to infinity. And in that scenario, you can’t draw a line, because the line is also infinite!

              I’ll give you the fact that my last scenario IS on thin ice. “Converting” people probably isn’t the best wording. You may be right, but maybe not. I’m not 100% on that. I’m not convinced either way 🙂

      • Matias Frank Jensen

        Well, if the area of the drop is non-converging that is there is no limit to the size of the area, every single person will be hit at some point. There might be an infinitely amount of people, but every single individual is only a finite amount of space from the starting point. It is like if you counted up from 0 for eternity: 0,1,2,3,4….

        There might be an infinite amount of positive integers, but every single integer is only a finite distance from 0, infinity is not an integer. It is the same principle here. It is actually kind of the basis for cardinality in set theory. The cardinality (the size) of the set of positive integers is infinite, but infinity is not on the number line.

        So with infinite time you will eventually hit every single person because every single person is only a finite distance from the starting point.

        • Tom Miller

          Yes, but you could also argue that there is a finite number of people who receive the drop, even though time is infinite. Therefore, there will also an infinite number of people who *don’t* receive the drop.

          • Matias Frank Jensen

            It is absolutely true that at every single point in time there are an infinite amount of people who are sad, but that is the problem with infinity. It can be quite counterintuitive.

            Mathematically speaking it actually does matter that we have an infinite amount of time. There is an entire branch of mathematics called set theory which tackles stuff like this, and I would guess calculus can do the same thing. We are talking about limits here, and limits dont act “normally”, but can be quite weird.

            The problem with the intuition is that it is definitely true that at every single point in time, an infinite amount of people will be sad contra a finite amount of happy people. But every single point in time corresponds only to a finite amount of time that has passed. It can be analogous to numbers on the number line. For every single number on the number line there is only a finite amount of integers smaller than it, but an infinite amount of integers larger.

            But then again, then we are not actually using our infinity if we look at an actual number on the number line, since no numbers are infinite. But at the limit as we travel towards infinity, we will hit every single number. The limit is not a real number, it is something different, because real numbers dont describe infinity.

            So because of our infinite time, at the limit as our time goes towards infinity, we will hit every single person on the board. But then again, we are not talking about a specific point in time since these specific points are not infinite, just like our numbers. The limit as time goes towards infinity is something different.

            So yeah, infinity is quite counter intuitive and it took mathematicians all the way to Newton and Leibniz to figure out how to approach the concept correctly. My explanation here is by no means rigorous either, but more an outline of the intuition behind it. If you want to be rigorous about limits and infinity, calculus textbooks are probably where you will find proofs of it.

            • Tom Miller

              Yes exactly. I watched a really good documentary on infinity (I think if you google BBC horizon infinity) you should find it. In fact, there are numerous mathematicians who deny infinity is actually real and that it’s merely a human construct that cannot actually exist.

              The way I see it, is that there are two finite numbers; the number of people who have become sad, and the time elapsed.

              There are also two infinite numbers; the number of happy people, and the time *still to elapse*. So there will always be time that hasn’t happened yet, and there will also always be happy people that haven’t become sad yet. Both of those “numbers” (if you can call them that) will always be infinite, by their definition.

              I see it like Zeno’s Paradox, but in reverse. In Zeno’s Paradox, Achilles will never catch the tortoise, even though he’s twice as fast. In this thought experiment, time will never “catch” all the happy people, because they’re infinitely large.

              Maybe… I dunno 🙂

            • Matias Frank Jensen

              Right yeah, thats how you would be inclined to think about it, that there is always a time where people haven’t become sad yet, and it actually does feel like a reverse Zeno’s, but this intuition again contradicts modern mathematics.

              Zeno’s paradox was actually resolved with Newton and Leibniz (as far as I am aware), where they showed that it is in fact possible for Achilles to catch the turtle by getting infinitely close to it, so close in fact that he actually, truly catches it. It is not particularly intuitive and requires some mathematical rigor to prove, but that is really the fundamentals of calculus.

              So in our checkerboard experiment, at some weird point in time, in calculus it is called the limit, even though there really are an infinite amount of people, every single one will be hit by the drop. But it really is quite weird because this point in time is arguably not really an actual point in time, but a bit different concept. But if we want to use our infinity we have to introduce this weird, arguably non-existing point in time, otherwise we dont actually use infinity.

              But again, this mathematics took Newton to establish and prove, and if you dont have a background in mathematics or is otherwise familiar and comfortable with calculus, it really might not seem as the answer which makes the most logical sense.
              The concepts and implications surrounding infinities are really fun to think about and they have puzzled the greatest thinkers for thousands of years because of their really peculiar nature.
              If you really want to, you could try to study a bit on limits and the basics of calculus, you might find it interesting 🙂

        • Titus

          I was thinking that at T=infinity, the drop would just continue to expand, it would not have reached everyone though

      • Elena Gaska

        There are two ways of looking at the sets of happy / sad people.

        The first, considering the sets at a precise time frame, for example, 10 seconds after the
        drop. Then sad people is an infinite set and the happy people is a finite set.

        A: all people {∞}
        B: happy people {y} a number determined by the function of the drop reach
        C: sad people {∞ – B} , an infinite set

        The second, including that the drop is expanding through infinite time in the definition of the happy people set, then both sets are infinite:

        A: all people {∞}
        B: happy people {1, 5, 9, 13 … n}, an infinite set
        C: sad people {∞ – B} , an infinite set

        So both B and C are infinite sets contained in A. I think they’re Dedekind-infinite sets, but I’m not sure. Set theory includes the notion of some infinites being bigger than others, for example the set of natural numbers N= {0, 1, 2…} is bigger than the set of odd numbers I = {1, 3, 5…}, and both sets are infinite.

        I don’t know how to compare B and C mathematically to determine which is bigger (or if it
        possible to do it), but B is expanding and C is shrinking at the cost of the B expansion.

        Of course, in the end, it’s a moral question, not a math problem where there might be an
        unequivocal right answer; math is just a tool to understand the scenarios of
        the boards and make a choice. Math aside, I’m more comfortable with an expanding
        blob of happiness than an expanding blob of sadness.

        • Milokot

          Both natural and odd numbers are same “size” (Aleph 0, smallest infinity). There are bijection between them, and between all infinite sets of whole numbers. Real numbers “bigger”, but when counting living humans we only use natural. So “after infinite time” there are as many “changed” people (set B) as… Set A. And in all finite time |C|=|A|.
          It sounds crazy, but while not all people are happy on happy checkerboard, size of set of happy people is exactly the same until you “reach” infinity time, when it drops to 0.

          Read it up, this stuff is literally AMAZING.

          • Elena Gaska

            Yeap, my mistake; some infinites are bigger than others but the example is wrong. I forgot that the set of Naturals is considered the “smaller” infinite, probably because I’ve never understood exactly why that is and the fact that two different infinite sets can be added to form a third always confused me; it creates the intuitive and persistent notion that the three are not equal in size.
            Following your Aleph 0 lead I’ve been reading about Aleph 1, the continuum hypothesis and other really amazing stuff I hadn’t thought about since high school and didn’t really get back then (I think I’m starting to get it now), and I read and remembered the story my dad used to tell me about poor George Cantor and his infinity-related depressions.

      • Blrp

        It is indeed not true that there exists some T such that everyone will be happy after time T. But it is true that for for each person there exists some T such that that person will be happy after time T. See pointwise / uniform convergence. In other words, no-one will be sad forever, but there will always be someone who is sad (infinitely many, in fact).

    • PaintedSlate

      True. But after Graham’s number of years, the chances that any given individual will have switched from Sad to Happy is still effectively 0 (any finite number/infinity). Not that this solves the problem, either…but it’s a tough perspective to take.

  • Bindle

    Hmmm . . . I think I start with the board of happiness. At t=0, you have an infinite number of happy people and zero sad people. At any time short of infinity, there will be more happy people than unhappy. When t=infinity, it doesn’t matter which board you start with: there will be an infinite number of people happy and an infinite number of people unhappy. But that’s just a non-mathematician’s guess.

  • Commenter4376

    A paradigm where people are not in control of their own happiness is terrible (and communist?). I refuse to participate.

  • Link Elek

    1. number of people inside of the drop is ALWAYS finite
    2. which means, there are infite number of people always OUTSIDE the drop
    3. happy+drop of sad is the best option

  • mtjces

    Initial state: Happiness
    Latter state: Sadness

    Sadness and suffering are good motivators for societal and technological advances. They would all try to improve their lives and therefore explore what might exist beyond.. Kinda like what Earth is like, but more radical.

    However if I had to live there, I’d choose the opposite. Sadness -> Happiness. It’s my life and the 2D world seems irrelevant so I’d want happiness for everyone.

  • Ksenia Kolchina

    To me it’s effectively answering this question: would I rather suffer for a while knowing there’s an absolute happiness ahead of me, or would I rather taste what a happiness is only to eventually loose it forever? To me, the former is an award for all the pain endured, and the latter is a lingering torture. I even doubt I’d be able to enjoy happiness at all, constantly dreading that it can be over any minute.

    So I choose infinite sadness with a drop of happiness – a mere thought of it will keep me going for a long while.

  • James E Hudson

    To know infinite happiness and loose it seems far worse. I would definitely choose sadness with a drop of ever expanding happiness

  • Shani

    Either scenario I would choose the board made with infinite happiness- as the God I would take an infinite amount of time deciding where to drop the sadness on the infinite plane.- as the person on the square I would rather go into infinite sadness having known happiness so as to have hope.

  • TCGM

    You’re an omnipotent being and you have a boss?

  • Rob Armstrong

    As an omnipotent god id have to go with the happy board with a drop of sadness. that way there will always be an infinite number of happy people and only ever a finite number of sad people. on the whole, i would have created a world that the majority of people would be happy. i couldnt sleep if id created a world of infinitely sad people.

    As a person on the board im torn between the two. i may be at the far edge of the infinite board of happiness and the drop of sadness will never reach me due to the nature of the infinite. alternatively, there is the chance that the drop will land right on me, in which case i would rather start out sad and never knowing happiness. when happiness does arrive its going to be awesome. but to only know happiness and then suddenly feel like crap for eternity, im not sure if i could deal with it. it would be like having depression suddenly and anyone thats been through it or living with it will tell you how much it sucks to feel like shit and not really knowing why. i think id have to go with the sad board. id rather risk an eternity of sadness without ever knowing happiness than to have to deal with it suddenly being taken away from me. also everyone around me would be sad too, so maybe laughing at their misfortune would make me feel a little better.

    and now i can see why most gods are such selfish dicks. might be time for god to take one for the team and deal with the terrible feeling of creating a world with an infinite number of sad people for whom life can only get better.

    • Laura Bowers


      • Rob Armstrong

        I find it ironic that you would correct someone’s grammar like with a sentence fragment. I’m also rather disappointed that you only picked up on one of the grammatical errors that I made. I’ll attempt to complete the corrections in list form for for you.


        I also forgot to write “I dont give a shit about my grammar because this is the internet”.

        • Laura Bowers

          That was beautiful. I fully accept my horrific mistake, although I should note that if one doesn’t use grammar how can one expect for his/her notions to be seriously taken?

          • Rob Armstrong

            i guess that mixing with people for whom english is a second language for so long, i look past the inconsequential nature of grammatical errors and try to focus more on the subject matter and what the person is trying to say than the flowery way in which they might be trying to say it in. but maybe im wrong. maybe the opinions of people whom are bad at spelling really dont matter.

            • Laura Bowers

              I suppose that you are indeed right that if someone cannot pick up on what you intend to say from what you type than their opinions are inconsequential; however, I must ask you this: How do you know that most of the people on this website have English as their second language?

            • Rob Armstrong

              haha sorry, i should have been more specific. ive spent much of the past 4 years in SE Asia. it wasnt a reference to this website specifically. as a silly little side note though, most people who do speak english as a second language consider American english and British english to be dissimilar enough to make the distinction. so assuming the majority of wait but why readership traffic is from the US, it could be argued that english is a second language for all the american speakers here

            • Laura Bowers

              While it could be argued that English would be considered a second language for all the BRITISH people here, that would be an idiotic argument to make. Considering that the biggest difference between American and British English is pronunciation, there is no such problem with that as we are online. Therefore, I rebut that American English is not a second language to all the Brits reading that. Not to mention that this entire tangent of the argument is based upon the assumption that the majority of readers on this website live in the UK.

  • Nathan

    You create the happy board, and drop the sadness afterwards.

    With an infinitely large happy board, it doesn’t matter how large the circle of sadness gets, because there will always be infinity happy people beyond the reach of the circle. This is not the case with the sad board and happy droplet.

  • Tom

    Board 1.
    Because even if the board 2 continues to expand as well as the drop, at a point the amount of sad people will be greater than the amount of happy people at a point (or will it? I didnt bother checking).
    Also, you need to experience sadness to appreciate happiness.

  • Luckyus

    Let me see… I’m omnipotent, but answer to a boss? That tells me I’m not as “omnipotent” as I thought, eh?
    Next I decide ‘how much’ bliss or misery, too. Well, I would choose the least happiness which will then expand in all directions through the sadness, presumably covering an ever expanding creation. Best to have ever expanding happiness than ever expanding misery.
    As for the other ‘God’ I would like to think he has created the same, especially when I am placed into his creation. Yet, t that time I will be placed on the board by another.
    Why be any type of ‘God’ if your purpose is to spread misery? Sounds too much like, well here…

  • TNH

    I’d rather choose board A because I think the people who live in board A will be stronger people than the people living in board B. You know the story about the Buddha whom found eternal happiness within himself by giving up his kingdom to find his true self. I think I would like to see as much Buddhas on my board.

    If I am one of the people living forever in one of those boards, I would like my damn god to drop a little bit of both sad and happiness once in awhile for me to become as human as possible. I don’t like the idea of being sad or being happy forever, I like to be bit of both because it will help me grow as an individual.

  • Miko Kuta

    I will chose board 1 for both questions and here’s why:
    every single person has a position on that board, meaning it has a finite, determined coordinate, if we start with a drop of happiness then no matter where you are on the board, eventually the happiness will reach you. The only people that it won’t reach are at infinity, which by definition, mean that they don’t exist.

  • Colin Rafferty

    1. Board 2. At all times, there will be an infinite number of happy people, and only finite number of sad people.

    2. Board 1. I’ll only be unhappy for a finite amount of time, and happy for an infinite amount.

    Boy, this sucks.

    • Pranav Srinivasan

      Exactly what I thought.

    • Martin Nick Smolík

      Why the sad drop for the “moral obligation” question?

      Every person on the board would like more to be on the “happy drop” one. Why would you deny it to them?

      • Colin Rafferty

        With the sad drop, there will always be an infinite number of happy people and a finite number of sad people. With the happy drop, it would be reversed. So with the sad drop, there will always be infinitely more happy people than sad, no matter how far out in the future we go.

        • Vasco

          The people in the sad drop are not finite. The drops expands forever, or infinitely, in time. You can’t consider just moments in time, it seems to be you need to consider that the infinitely expanding drops will eventually cover all the infinity of the board. And since when it does touch a person, that person will have the effect of the drop forever, the board with the happy drop is best.

          • Colin Rafferty

            “Eventually” is the key. We never reach infinity. A million billion trillion years from now, there are still infinite happy people because the sad drop has only reached a finite number of people. There will always be an infinite number of people who are always happy.

            • Couldn’t you say that you do “reach infinity” if you had infinite time? The sad drop will reach each and every person in the board, won’t it? It has to, it has infinite time to do so. I might be wrong, I’m no mathematician, but this makes sense to me.

            • Colin Rafferty

              That’s a mighty big “if”. 🙂

              The thing of infinity is that you actually never reach it. Not infinite time, or infinite distance.

            • So when considering time, aren’t both the drop and the board “equally infinite” then? 🙂

  • Grant Peterson

    It doesn’t matter, because either choice yields an infinite amount of both happy and sad people for an infinite amount of time.

  • Tyler Leedom

    Board #2. The infinite board #2 is ever expanding and although the sadness drop may also be ever expanding toward infinity, it will always be smaller than the infinite board because it started as a finite point on limitless board.

  • Daniel Patel

    The important question here is number two. Number two is one based on statistics, so the answer applies to every human that will exist on the board. I’m a fan of democracy, so my kind of god would listen to the unanimous decision of an infinite number of people.

    Let’s talk about board one. No matter how large the area of the drop, it is a finite space within an infinite one. If we were to take a poll of everyone’s attitudes, the “happy” crew would never gain more than 0% of the vote. Yes, everyone on the board could take solace in the fact that a forever of happiness lies sometime ahead of them, but the thing is that with an infinite space, the time that it will take the drop to reach the average person is also infinite. True, once happiness comes, you will be happy forever. But the concept of infinity means that since you are endlessly far away from drop of happiness, it will never actually come. When one infinity comes after another, the second infinity never arrives.

    The people should want board two, and morally, the god should create it.

    Infinity is so cool.

  • Laura Bowers

    For both questions I would choose board A. “From great suffering comes the greatest people.” On board A I would be sad for an indefinite period of time but I would eventually get to feel bliss for the rest of infinity. On the other side if you pick board 2, you would feel happy but at some point you would enter a state of sadness, able to recollect the feelings of bliss but not able to experience it again. That would be pure torture.

  • OneUser

    From an economic standpoint, I’d say that it is better to have infinite happy people but an increasing number of finite sad people (2nd board). Just because infinity! As a God I might be concerned with what people think of me and even though an infinite number of people are happy, many would know what is coming to them: Sadness.
    Also from a viewpoint of an individual I might prefer the 1st board because I would be used to being sad and I knew that a time would come (maybe never, because infinity) that I become happy.
    So infinite happiness is the best!!

    Also don’t forget to be united in our uncertainty! 🙂

  • Wiremu Hohepa

    I’m wondering how the drop could behave while “rapidly” expanding.
    I’m not sure if time matters in a universe with infinite people.

  • Vasco

    Regardless of the board, the drop will always reach everybody eventually. When it does reaches someone, that person will stay in the drop’s state forever, which is always more time than the time they stayed in the previous state. Therefore, the first board, where the drop gives happiness, is better in both questions 1 and 2.

    • ericsp23

      Actually, that might not be true. If there are an infinite amount of people and the number of people who are under the “drop” is a finite number that increases over time, it is not mathematically valid to say that every single person will eventually be under the drop, for the same reason that you can never count to infinity. There will always be some finite number of people under the drop and there will always be an infinite number of people outside of the drop because of the fact that you cannot subtract from infinity.
      This problem is asking us to work our way out of a paradox created by dealing with an infinite quantity, which is not really possible in real life. I think I would choose the happy world with expanding drop of sadness because there would always be an infinite number of people happy and only a finite number of people that are sad.

      • Vasco

        You have to consider that time is infinite too, and the drop expands infinitely. So in that sense, the drop is not finite, it expands infinitely, and will therefore cover all the “infinity” of the board. The argument that says that “there will always be more people outside the drop” seems invalid to me, because you can’t just look at the board’s state in one moment in time, you have to consider that the drop is infinitely expanding and will eventually “cover all ground”.

        • I think time being infinite makes question 2 easy to choose the expanding happiness because you’ll be an actual person that just needs to wait it out. But I don’t know if an infinitely expanding circle can actually ever “cover all ground” in an infinite world. The way I see it, the amount of people outside the expanding circle being infinite does overpower the idea of it expanding infinitely from a moral perspective for the greater good.

          • I might be wrong, I’m no mathematician, but if the drop expands for infinite time, doesn’t it necessarily have to reach each and every person on the infinite board? It has infinite time to do so. It seems to me that when considering time, both the drop and the board are infinite, no one is bigger than the other.

            • Tom Miller

              actually no – There will always be in infinite amount of sad people and a finite number of happy people if you start the board with sad people, regardless of how fast and for long the drop expands. Essentially, you can never get to infinity even if you add numbers together FOR infinity.

              It’s a bit like halving a number, and then halving that number, for infinity. Even with infinite time and division, you’ll never get to zero.

              At least that’s my understanding.

            • Couldn’t you say the same for the size of the board? Where exactly is this “infinite distance”? The thing that I think that matters is that, whatever point of the board you can imagine, the drop will get there. It will get to ANY of the infinite points of the board.

            • It’s been over a decade since I’ve taken calculus so this is purely arm chair philosophizing. I do see what you’re saying and I guess the crux of my argument is that my infinity is bigger than yours, which makes me hope some certified mathematicians put us all straight.

        • ericsp23

          Again, I’m not sure that is true. Its not like you can say that at some point in time an infinite amount of time has elapsed and at that point all of the infinite number of people will be under the drop. Having an infinite amount of time just means that time never ends and so the drop never expands to all people. At any given point in time there will always be only a finite amount of people under the drop and an infinite amount of people not under it.

          • In the same way that you say “at any given point in time there will always be only a finite amount of people under the drop”, I can say that “for any person you choose, the drop will reach them”. They just both feel infinite to me (the drop and the board), but I’m not a mathematician; I’m not sure how one would view this.

  • Theodore Grizzwold

    If I’m omnipotent than I can make the rules. I pick number two and ignore the boss’ request to use the sadness dropper. What’s the point of being omnipotent if you are limited to only two options?

    • Zach

      You my dear sir, have pulled a kobayashi maru, and have won. Congratulations

  • Jiri Roznovjak

    1) Happy with a drop of sad. Since the board is infinite, at any point in time there will always be an infinite number of happy humans, no matter how much the drop expands.

    2) For an individual, sad with a drop of happy is a better choice. Even though you’re sad at first, the happy drop will reach you in a finite amount of time and then you’ll be happy forever.

    I can imagine arguments can be made contradicting my answer. This is just another problem from the class of problems involving infinity, since infinity is not a number so our normal reasoning breaks down. We can reason about it how much we want but we might not get a satisfying answer.

    • Gold Skulltula Hunter


  • Gold Skulltula Hunter

    If the board is infinite and the drop is finite (however large it becomes), there will always be more (infinitely more) people outside the drop than inside. Therefore, the second option is way more beneficial to the people on the board.

  • Martin Nick Smolík

    For both the obvious answer is the happy drop.

    Why is it better for the second question is obvious (finite time sad, infinite time happy).

    Concerning the first question the answer that does come up most of the time is the second board, because more people are happy at all times. But given that society is made out of individuals and each and every individual would like to be on the first board, what kind of god would I be to deny it to them?

    • Colin Rafferty

      Your second paragraph is exactly my moral conundrum. Should you do what will make the most people happy, or do what the most people want you to do?

  • sadpumpkin

    Board 1 results in all people experiencing a life of 100% happiness, but at any given moment, 100% of the people will be unhappy.

    Board 2 results in all people experiencing a life of 100% unhappiness, but at any given moment, 100% of the people will be happy.

    The question is unanswerable. Anyone claiming that one choice is better than the other is arbitrarily deciding that some facts matter, and others don’t. Is the infinity of time more important or less important than the infinity of people? We cannot possibly know.

    • Daniel Kranich

      I would argue that the question is answerable and here is why:

      Humans can make nearly anything become normal. A constant state of unhappiness is not something a human can understand without having a good day to reference it to. Lets take the analogy of heroin. Technically, we are living a miserable existence compared to someone who is high on heroin. However if you have never done heroin, you have no concept for what that bliss even feels like. However, if you are a heroin addict, it is extremely difficult to stop taking heroin, because life feels miserable without it.

      Lets take that analogy to our boards.

      On board one, everyone is on heroin but one by one they start getting the comedown and will never be able to go back.

      On board two, you have normal “Miserable” humans, and one by one they are given heroin and never have to come down.

      Overall board 2 is clearly the preferable board.

  • stephaniesid

    1) Is this a math question, or a moral question? And does my (more omnipotent) boss allow clarifying questions? I hope so. It’s kind of a big decision. if I assume the question is “Everyone’s sad but eventually they’ll all be happy forever” vs “Everyone’s happy but not for long”, Board 1 is the no-brainer. But then there are the mathy factors, like, “How fast does the droplet spread?”, which kind of matters, since both scenarios are pretty shitty. But my biggest question is this: “Do the humans really not have free will regarding their emotions?” i.e., does their square simply contain conditions that most people consider terrible or awesome, or does the lot that I assign to them actually hard-wire their emotional state? If I assume they’re like regular humans, and only being acted upon by (semipotent) me, then I can’t be sure of any human’s emotional outcome or duration based on the conditions inflicted upon them.
    2) Moot, if we all have free will re: emotion. If no free will, then there’s no point to life on the planet, and either board is equally (un)attractive.

    • Don Reba

      It doesn’t matter how fast the droplet spreads, because it is infinitely far (farther than any given distance with 100% probability) from just about everyone on the board, which, remember, is infinite.

      • stephaniesid

        so @Don_Reba:disqus — you’re making me think. you’re saying that the distance between 2 adjacent humans is not constant? i may be showing my limitation in understanding infinity… but my assumption is that every human gets a fixed spot, just that there are an infinite number of squares in 2 dimensions….

        • Don Reba

          The distances are constant, but for almost all people they are very, very, very large. Take Graham’s number:

          There would be a finite number of people smaller than Graham’s number squares apart, and an infinite number of people farther.

  • Maria Antonia Marturet

    For both questions I would chose Option 1: I think that suffering for a determinated period of time before entering perpetual happiness is far better than the other way around.

    This belief gives my life sense in a lot of ways: I’m Catholic, and daily I chose to do and to abstain from a lot of stuff that sometimes go against my wishes or what would make an easier option. This is what I call sacrifice and I enjoy it because: a) it makes me a better person b) I know I will enjoy eternal happiness and satisfaction on heaven.

    • Don Reba

      There is a 0% probability of you changing state in any set amount of time, though. If you start out sad, you will certainly remain sad, because the number of people not having changed is always infinitely larger than the number of people who have.

      • Maria Antonia Marturet

        I know, but as time is infinite too, each human has the certainty that at some point they will be happy, and that the unhappy amount of time in their lives will represent 0%

        • Don Reba

          But this is people we are talking about. Many question if we need to live past a hundred years. Do we REALLY care what happens after a billion? What about 1 with a billion zeros years? A Graham’s number of years? Almost all of them will have to wait for more than a Graham’s number of years.

          From Tim’s post:

          Writing this post made me much less likely to pick “infinity” as my answer to this week’s dinner table question. Imagine living a Graham’s number amount of years.8 Even if hypothetically, conditions stayed the same in the universe, in the solar system, and on Earth forever, there is no way the human brain is built to withstand spans of time like that. I’m horrified thinking about it. I think it would be the gravest of grave errors to punch infinity into the calculator—and this is from someone who’s openly terrified of death. Weirdly, thinking about Graham’s number has actually made me feel a little bit calmer about death, because it’s a reminder that I don’t actually want to live forever—I do want to die at some point, because remaining conscious for eternity is even scarier. Yes, death comes way, way too quickly, but the thought “I do want to die at some point” is a very novel concept to me and actually makes me more relaxed than usual about our mortality.

          • Maria Antonia Marturet

            “Almost all of them will have to wait for more than a Graham’s number of years.”
            And all of them will spend a bigger amount of time being happy after the wait. I think the other way around is horrible, the wait, the anxiety, the infinity of the time you know you will be unhappy. I think my position is summarized in: finite time of unhappy + infinite time of happy >> finite time of happy + infinite time of unhappy.

            Regarding the monstrosity of an infinite life, I completely agree with you (and Tim), but I think it gets out of topic as both options guarantee it.

            • Don Reba

              finite time of unhappy + infinite time of happy >> finite time of happy + infinite time of unhappy.

              I guess I strongly disagree that an eternity of happiness is worth a million (or pick your number!) years of misery. I agree with Tim that there is such a thing as living “long enough”. And everyone who agrees with that should pick the happy world.

  • Question 1) I think the only choice from a moral perspective is board 2. At first I was thinking that board 1 would depend on factors like how fast the happiness spread, but the concept of infinity overpowered this logic. Board 1 could literally spread out at the speed of light, but there would still be an INFINITE amount of sad people. With board 2, sadness could also spread out in insane proportions, but there would still be an INFINITE amount of happy people. It doesn’t matter how long or fast sadness would spread on board 2, it would still be, without question, the more moral choice.

    Question 2) I thought this would be easy, in that I would just want to be on board 2 because of the above reasons. But I think it’s more tricky than question 1. Would I be placed on a random spot? Would I get to choose a spot number? The way I look at it is, even if board 2 is more moral for the greater good, the reality is if each place on the board has a number, I would be placed on one, with there always being an infinite amount of people further out than me. This makes the reality of the sadness/happiness wave actually reaching me a certainty instead of an infinite hypothetical. Will all this in mind, I would have to choose board 1 because if I’m placed in an actual spot, I could be certain that the happiness wave would eventually hit me and I would live the rest of my infinite life in bliss. It wouldn’t matter how long it would take to get to me either because what’s a million years of pain compared to eternity of happiness?

    • Guest

      Folks below are really missing that the boards spread infinitely.

    • Martin Nick Smolík

      So why do you feel that board 2 is a better choice for the first question? If every person would vote, they would unanimously choose #1. Why would you defy them?

      • Because I am a utilitarian god. Each person would clearly want me to choose option 1, including me as the poor sap on the board, but each individual cannot fathom the stakes that involve an infinite amount of people being sad. I would need to overrule the logic they (or me in their situation) would use in order to cause the least suffering. The nature of the problem immediately changes when you change from thinking of an individual to thinking of infinity. Either way, now I feel bad, and don’t want this damn job.

  • EZnded

    There are a couple of assumptions I made when I came up with my decision:

    1) there is only a single drop administered at the start and it is dropped at the center of the board. Since the board is infinitely expanding (I assume it is expanding like our universe) and getting the exact center might not be feasible, it will be dropped around the central area, not the edge.

    2) the speed of the ripple will be constant and its effect is exponential. I assume that a single drop will affect only one person. After a second, it will fan out to its neighbors in a square and so on.

    3) Since the board is infinitely growing, you cannot make everyone happy/sad all at the same time. The only instance everyone is happy/sad is at the creation time, before the drop is administered.

    4) the definition of happiness or sadness directly correlates with the fulfillment of an individual’s basic needs. I will use Maslow’s hierarchy of needs for this. Therefore, a happy person is one where all needs are met, while a sad person is someone who hasn’t met any or at least one need. Thus, an individual is sad if they have everything in life but is missing only one need, like self-actualization.

    5) each individual doesn’t know that their initial happiness/sadness will only be felt at the start–that the opposite feeling will be felt by them after much time has passed.

    Given all the assumptions, it will be logical to choose the happy board with a drop of sad, since it would mean that more people will be happy and fulfilled in the long run. But the problem is, the happiness is just at the start. The people don’t know that the happiness they are experiencing may be taken away after the next second, which makes it even more painful.

    On the other hand, the sad board with a drop of happy may seem kind of fulfilling, especially for the individuals in that board–they have suffered from the start, but a feeling of happiness will be multiplied since they know the feeling of sadness. I think true happiness can only be achieved by knowing sadness.

    In this scenario, I’d pick the sad board with a drop of happy. As the god in that realm, it would make me happier to see my people enjoying and knowing what it means to be happy since they have already experienced sadness. Besides, I’d rather have my people be happy indefinitely than see them sad for all eternity. It’s cruel to see them wallowing in despair forever without death. That’s just torture for me.

  • Robin

    It’s a fun question: I’d choose the second board if I’m a God, the first one if I’m a human.

    If I’m a God – I’d want to maximize the average happiness of the board at all time (granted, a very debatable choice). On board 2 at any given time there’s infinitely more happy people than sad ones (since finite number of sad people), so the mean happiness is maximum.

    If I’m a human – I’d want to maximize my happiness over my lifetime. If I’m on board 1 I would be at a finite distance from the happy-drop’s origin, so however far from it I am it would only take it a finite time to reach me – leaving me with an infinite time of personal happiness.

    The arguments can be reversed when picking the other board: board 1 leads to average misery, board 2 an infinite time of individual sadness. The difference is in the diverging goals the God and the human have.

  • anonymous user

    Answer 1: (Following Rules)
    Question 1: I would choose board 2, because 100% of people would be happy.
    Question 2: I would choose board 1, because I would eventually be infinitely happy.
    Answer 2: (Not Following Rules)
    I’m an omnipotent god. I fire my boss, and take over. I now make a happy board of humans with no sadness ever.

    • Daniel

      1) Board 1. If people never knew anything but sadness they wouldn’t know this state of emotion at all. Humans feel the delta.

      2) Assuming that as a Demi God I’d retain the information after my decision I would, as many others suggested, trade immediate bliss and fear of what’s to come with hopeful anticipation regardless of my proximity to the center unless I was convinced that the amount of likely suffering was breaking me forever.

      Extra thought: if I had the ability to commit suicide in this 2d reality I’d reverse my decision on 2). Not just because I could effectively live happily (I may not even know that I’m happy without knowing sadness) but because the value of life and time is governed by its scarcity. I would want my decision to merry the girl in square 3.1415*10^4.736 to matter and it only does if this decision matters. If I have all the time in the square world nothing would matter.

  • Tom Miller

    The fact that one starts after the other is the key variable here, even with infinite time.

    If you start with an infinite amount of sad people, you’ll always have an infinite amount of sad people, regardless of how fast the drop expands. You can never “reach” infinity.

    So I think you have to start with happy people, as there will always be more happy people than sad people.

    So it’s either that, or another possible answer is that they are both exactly the same. But I don’t see how it’s possible that the moral answer can ever be board 1.

    Infinity by it’s nature is extremely counter-intuitive and difficult to understand.

    One clever idea I’ve heard about is the infinite hotel. A hotel has an infinite amount of rooms, but also an infinite amount of guests. If you arrive at the hotel, the receptionist will say it’s full. However, if you ask everyone to switch rooms (the person in room 1 goes into room 2, the guests from room 2 goes into room 3 etc. etc.) you can actually squeeze into a room (in room 1), even though the hotel is full!

  • Don Reba

    There are so many people on the board that, no matter how fast the drop expands, the people inside it are always going to be an inconsequential minority unimaginably far away.

    No matter how fast the drop expands, any person on the board would have an impossibly large amount of time before switching states. Pick a number, any number, and that amount of time is greater than that with 100% probability.

  • Alexander Bazes

    My gut human response is to want to go with 1, but I know this is logically incorrect. Regardless of how long the drop spreads, there is always a finite number of people within it, which can be calculated based on rate of expansion and time elapsed. It doesn’t matter that it will expand forever. Outside the expanding drop, however, there is always an infinite number of people.

    So, option 1= infinite number of sad people (outside) and finite number of happy people (inside).
    Option 2= infinite number of happy people (outside) and finite number of sad people (inside).

    Based on the morality of “the needs of the many outweigh the needs of the few,” and since an infinite number will always be greater than a finite number (regardless of the fact that that number is growing), option 2 is the moral choice.

    And I would still feel bad about choosing it. Glad I’m not a god.

    This does illustrate a really interesting point, however: faced between choosing hope (option 1) and logic (option 2), our human nature really wants to go with hope.
    Hope is a hell of a drug…

  • Arnold Oswald

    I answer in reversed order for reasoning purpose :
    Q2 ) I would chose the 1 since wherever I am, I will eventually be infinitely happy.
    Q1 ) Considering all individuals as other me, I now chose also the 1 so that even if it maximize sadness at any given time, every single person knows they’ll eventually be infintely happy (I’d have told them)

  • IsAnyoneThere

    1) The moral choice is to make the people on the board happy, and the drop sad. We’re talking infinity here. The total percentage of people who are unhappy will be infinitesimal for eternity.

    2) I would like the people on the board sad, and the drop happy. It is better to be sad, but to know that infinite happiness is in the future, than it is to be happy, knowing eternal sadness is in the future.

  • BWR

    Q1 – The way I see it, the expanding circle is always a finite space, approaching infinity. But it will never actually reach infinity – it will just just keep expanding forever. The checkerboard, on the other hand, is an already infinite space. Therefore, no matter how long you wait for the dropper to expand it will always be covering a finite number of beings while an infinite number of beings will be uncovered (half of infinity is still infinity, after all). Therefore I would have to create infinitely happy people and use the sadness dropper. That way more and more people would be sad all the time, but the number of happy people would always be infinite.

    Q2 – I like Arnold’s answer below: I don’t know where I am on the board, but I would want the happy dropper because eventually I’m going to be happy.

  • Ekin K.

    Depends on the velocity of drop vs board expansion. If the drop expands faster, the obvious choice is board 1 since eventually drop would catch up to the board and everyone would become happy instantly. Otherwise, board 2 is better since drop would never catch up to the board and most people would never suffer. If velocity info was given, I would choose accordingly, same choice in both cases (being the god or the person on the board). If velocity info is not given, board 1 is more risk averse since if the drop is faster, board 2 would end up torturing everyone forever, so I would choose board 1 in both cases.

  • Michael J. Sieler Jr.

    We’re all making assumptions based on the little information we’re given. The pictures of the checkerboards and drops of happiness/sadness that we are given are a bit deceiving to me. The only reason I think Tim made the checkerboards so large in comparison to the drops, is so that he could add the text and arrows to illustrate the motion of the board and drops in the options/questions. But WHY does the board have to be so much bigger than the drop? I’m going to make two assumptions: 1) The drop was dropped right after the board was created. 2) The board was one square. Then it turned into 3×3, then 5×5, 7×7, 9×9, 11×11, and so on…

    Each time the checkerboard gets a one square bigger in each direction, the circle of the drop’s radius also increases by 1 square. Initially, there will be more sad than happy people. Eventually, around the 7×7 mark, happiness and sadness flip. There becomes more happy people than sad people. Happiness exponentially increases.

    Therefore, I would pick Board 1 for both questions.

    There would always be an increasing amount of sad people, yes. However, there will eventually be a moment were the total amount of happiness surpasses the amount of sadness and happiness would exponentially increase at a faster rate than the sadness. At some point, given enough time, the sad people would become happy forever. I believe that temporary sadness is an okay trade off for infinite happiness in the future.

  • tommo_montana

    Board #1 sounds better for a god, who might seek to inflict a painful “moral” lesson. Suffer, then be rewarded for suffering with happiness. I think this is both ignorant and stupid.

    Mathematically, the correct answer is Board #2. There are different kinds of infinity. For every point inside the drop, there are infinite points OUTSIDE the drop. Simplify from a circular drop on a 2D plane to a radial line going outward from Ground Zero. On this line, there are some points that have been glooped on by the drop. But there are infinity infinity points on the line that haven’t been glooped. For example, the drop is at radius r. There is a length along the radius r who’ve been glooped. Yet, just outside the drop there are 2r-r (r) people who haven’t yet been glooped. Then, there are 3r-2r (r) people who haven’t been glooped. I can go on counting infinity – r people who HAVEN’T been glooped, for every r people glooped. Even though r is growing infinitely, there is always infinity^2 more people un-glooped than glooped during this infinite time series.

    So to summarize, the only reason to choose Board #1 is if you think it teaches a moral lesson. I don’t. If we assume that a God is good, a God is all-powerful, and a God is all-knowing, and such a God seeks to maximize happiness and minimize suffering, Board #2 is infinitely better than Board #1.

    To choose Board #1, you have to first assume that either this God is evil, this God isn’t all-knowing (ignorant of math), or this God isn’t seeking to maximize happiness and minimize suffering.

  • ImSleepy

    Question 1: The moral choice is Board #2, since there is an infinite amount of happiness and a finite amount of sadness.

    Question 2: The better choice for yourself is Board #1, since you would experience a finite period of sadness, and, once the drop flows over you, an infinite period of happiness.

    • Joeswam

      this ^

    • Victoria

      I might not just be thinking this through, but how is there a finite amount of sadness in board two if the drop is continuously expanding outward?

      • ImSleepy

        I think it’s finite because you will always be able to count the number of people within the drop’s circle. You could go through them one by one, making a tally for each, and eventually be done with the project. By the nature of infinity, you could never do that for the people outside of the drop.

        Mainly, the drop-to-board ratio is relative. No matter how big the drop is, the board will always be infinitely larger. It will always contain infinitely more people. And, consequently, in the case of Board #2, it will always contain infinitely more happiness.

        Infinity is super weird. That’s why I think this problem is really cool to think about.

  • I wish I had time to hear all 117 people at this dinner table.

  • Leonardo Zuniga

    1. Simple: the second board, at least on t=0 everyone is happy, so everyone knew that state. Also the infinite board is always larger than the ever growing ripple of sadness. Let’s imagine the ripple’s diameter = n at any given time, we can always ‘zoom out’ the board to, let’s say, twice that diameter, making effectively its area larger than that from the ripple and happy/sad ratio always larger than 1.

    2. Far more complicated, but I’d say the 1st board, because it’s guaranteed that I’ll eventually be caught on the ripple of happiness, making worth suffering the initial pain. Kind of what religions promised :^)

  • i’m trying to learn here: if for question #2 i chose Board 1 because the happiness drop would get to me eventually, doesn’t it stand that any human on Board 1 will get the happiness juice eventually?

  • Victoria

    Wow, after looking through a few of the other posts this question has become much more confusing. For anyone else in the same boat, here’s a video my calculus professor mentioned a few months ago that may help (or at least is interesting and somewhat related).

  • Artyom Karapetov

    For both questions I would choose Board 1. This way everyone knows they’ll eventually be infinitely happy, even if it takes the “happiness drop” an x amount of years to reach them.

    Sure, more people will be unhappy at any given point, as the board is infinite. But everyone knows that at some point they’ll be infinitely happy, and that makes the suffering much more bearable.

    Don’t know about you, but if I was choosing which board to step on, I would choose to wait for the eventual infinity of happiness, even if it takes millions of years to come – rather than living in “happiness”, (or, uhm, dread), awaiting the impending infinite sadness and endless suffering (can your life be even considered “happy” when you know you’ll someday going to be sad and miserable forever?)

    I would do to others what I would want others to do to me. I would choose board 1. Statistically more unhappy people, but infinite number of people happy or waiting to be infinitely happy. Better than an infinite amount of happy folks shaking from the impending doom and upcoming infinity of suffering. Just seems like common sense to go with the happiness drop.

  • Alejandro Rojas

    Board one is better, even if sadness is infinitely bigger than happyness, because all that infinite sadness is sadness with hope and every one eventually will be happy.

  • Brouroboris

    I think it’s worth considering that the sad people on board 1 have never known happiness before the drop hits them, so their suffering would be relatively less than the board 2 sad people who have experienced bliss only to have it taken from them.

    Likewise board 1’s happiness would be greater than board 2’s as they would each experience it relative to their previous experience, which in 2’s instance would be nothing.

    The choice is then between an infinite amount of lesser sadness with a growing amount of greater happiness – board 1- or an infinite amount of lesser happiness with a growing amount of greater sadness – board 2.

    Based on the relative nature of the two binary states the answer to both questions would be be board 1.

    • Brouroboris

      Put mathematically board 1’s population scores a +2 for happiness and a -1 for sadness. There are an infinite number of somewhat sad people and a growing number of very happy people, but since we have an infinite amount of time the end (ha) result is also an infinite amount of very happy people. This gives board 1 an overall score of +1.

      The same method gives board 2 (unhappy -2, happy +1) a happiness score of -1.

      Thus the general happiness on board 1 is greater.

  • Ferival

    From a psychological standpoint, this discussion is very interesting. Reading through the comments, there is no clear consensus on what the god should do. Ultimately it comes down to what you believe a god should base morality on. Its actually quite interesting to dissect the thought process of the two groups.

    Drop of sadness people argue that because the board is infinitely large and the drop isn’t, there will always be infinitely more happy people than sad people. This maximizes the total happiness in the universe at any given time and is the moral choice. For every sad person there are infinitely more happy people; sacrificing the few for the good of the many. Interestingly, these comments more often than the drop of happiness ones contain references to math that support their claims.

    Drop of happiness people counter that while there will always be infinitely more sad people than happy people, you have to look at each individuals life. Since every person lives for an infinite amount of time, they will spend a finite amount of time sad before an infinite amount of time happy. This maximizes the amount of happiness in every person’s life.

    Ultimately it comes down to how you approach the problem, and how you believe the god should base their morals. If you take a step back and consider the well-being of the universe as a whole and believe that the god should maximize the total amount of happiness, you would choose the drop of sadness.

    Those that believe a god should focus on the details of the universe and consider the lives of each individual believe in maximizing the amount of time happy in every individual life. They also often consider other moral issues besides happy and sad, such as the fear of your happiness being taken away and the hope you will someday be happy.

    Drop of happiness users, when asked to consider what they would want for themselves answer the same as if they were the god making the decision. They stayed zoomed in on the individual life rather than looking at the big picture when asked to play god, which is what the drop of sadness people do. This is the major difference between the two answers. Drop of sadness users follow logical and mathematical deductions to decide what is best for the universe as a whole. Drop of happiness users use moral reasoning to decide what is best for each individual. An interesting paradox: more empathetic reasoning leads to a more selfish decision, while more impersonal reasoning leads to a more selfless decision.

    • JaapVerlinde

      I almost entirely agree except for the qualifications in the end. I think drop of sadness people do use moral reasoning, and that moral reasoning typically IS mathematical and logical. Morality =/= empathy. In psychological tests they have the question: “what do you think is more important, justice or mercy?” to distinguish the more logic-based morality perspective from the more emotionally based empathic perspective.

      • Ferival

        I’m not a psychologist, but I find psychology fascinating. Thank you for clarifying this, I was confused in how I wanted to portray this statement. What I was trying to say is that the thought process of many drop of happiness users is what they would choose happen to themselves. They base their reasoning for the fate of the universe on the life of a single individual, which is inherently selfish. And yet in doing so they are maximizing the happiness of everybody. Conversely, in trying to maximize happiness, drop of sadness users make the lives of every individual miserable.

    • Ferival

      As for what I believe to be the correct answer: the drop of happiness. In the interest of shaking things up, I’ll use math to support my reasoning. Interesting that not many other drop of happiness users rely on math to explain their decision.

      The main argument for the drop of sadness is that there is infinitely more sad people than happy people at any given time. The droplet will never cover even a fraction of the universe because it will always be finite in size. This is a logical fallacy. Yes at any finite time the droplet has finite size, but the lifespan of the universe is not finite. After infinite time, the droplet will expand to infinite size and there will be an infinite number of happy people. An infinite number of happy people out of an infinite number of people is indeterminate, so it is impossible to know if there are more happy or sad people after an infinite amount of time.

      Maximizing the amount of happiness in the universe is impossible, so I look toward the individual lifetime. I assume the droplet is happiness. Since the droplet will reach infinite size, every person will eventually become happy. Everyone will be happy for an infinite amount of time but some will also be sad for an infinite amount of time. Since it takes infinite amount of time for the circle to reach infinite size, the people at the edge of the circle are sad for an infinite amount of time. It is impossible to tell if their life was predominantly sad or happy. Everyone else is sad for only a finite time and their lives are definitely happy. The indeterminate people are located at the edge of the droplet, represented by the circumference of the circle. The happy people are represented by everything else: the area of the circle. Since area is a function of radius squared and circumference is a function of radius, area will always be larger. Infinity squared is considered infinitely larger than infinity. There are infinitely more people who have led infinitely happy lives than those have led both infinitely sad and infinitely happy lives. The drop of happiness is the better choice

  • Waahto

    My choice: Board 2

    If you choose Board 1, you can be sure that everyone will eventually be happy (see Artyoms response), and they’ll be happy for the end of eternity. But, the estimated time they have to wait to get there is also infinite (the distance between two randomly selected points on an infinite plane). There’s two infinities here.

    By choosing Board 2, everyone can be happy as long as the expanding drop hits them and, as said above, this takes infinite amount of time. And surely, the expanding boardverse has gathered enough momentum by then so that the speed of unhappiness is negated by the speed of expansion.

  • Yuchong Li

    I up-voted Victoria below. But I think we are trying to compare two countable infinities: the board size and the drop size. Then it is mathematically meaningless to compare them. People on each type of board have the same probability to be happy or to be sad… However:
    Q1) I would like to create a board of sadness with a drop of happiness. Because it feels better to turn people from sad to happy.
    Q2) I would still like to live in a board of sadness with a drop of happiness. Because it feels better to live in hope than in threat.

  • Scott

    I hate leaving such a safe dodgy answer, but given the information I think the two boards are morally . The moral choice is to choose whichever board will render most net happiness, but any logic to determine that breaks down because we are dealing with infinite boards. We can’t equate finite happiness using infinities. Here are two proofs proving why both board 1 and 2 are the best choice (disclaimer: I’m not a mathematician or a logician, so don’t dock me for proof syntax):

    Given: No finite number is greater than infinity

    Proof Board 1 is more moral:
    A. For every single person on board 1, there will come a time when they turn from sad to happy.
    B. Once a person turns from sad to happy, they will remain happy for eternity (i.e. infinity).
    C. If A and B, then every person on board 1 will have a finite amount of time being sad and an infinite amount of time being happy, which means every person on board 1 will spend more time being happy than sad.
    D. Using same logic above for board 2, every person on board 2 will spend more time being sad than happy.
    E. If every person will spend more time being happier on board 1 and sadder on board 2, then board 1 is the more moral board.

    Proof Board 2 is more moral:
    A. At any given point in time on board 1, there will be a finite number of happy peeps and an infinite amount of sad peeps.
    B. At any given point in time on board 2, there will be a finite number of sad people and an infinite number of happy people.
    C. If A and B, then at any given point in time there will more happy people and less sad people on board 2 compared to board 1, meaning board 2 is the more moral board.

    These two proofs contradict each other, but I don’t think either one of them them is invalid, we just can’t use infinities in determining finite solutions.

    All that said, my answers are:
    1. They are morally equivalent. As a god I would be indifferent to which board I choose.

    2. If I was a person and had to choose a board, I would choose board 1. Not for logical reasons, but probably due to my own human bias. The proof for board 1 resonates with me more since it’s done from the viewpoint of each person. I understand the idea of starting sad and eventually becoming happy for eternity better than wrapping my head around the collective happiness of infinite people.

    • JaapVerlinde

      Yeah I agree with the analysis and also with the choice in the end.

    • Ferival

      The problem with proof 2 is that this universe does not have a finite timescale. After infinite time, the drop will grow to infinite size, making the number of sad people compared to the number of happy people indeterminate.

      • Scott

        But nothing finite can turn infinite by adding a finite amount to it, and all the expanding circle is doing is adding larger and larger finiite amount to it. You have a point in saying that adding finite amount for infinite amount of time is infinite, which shows the contradiction, but doesn’t disprove the proof, since at any point in time the circle is in fact finite and the outer area is always infinite.

    • Waahto

      1C is not true. The average time it takes to get from sad to happy is infinite. (The edge of the drop is – on average – always infinitely far. Just because the drop is expanding doesn’t actually have any significance on this.)

      • Scott

        If 1A and 1B are true, then 1C has to be true. I think the issue you bring up is with A, that every person will have a finite time of sadness.

        Are you saying that Board 2 is the more moral choice?

        • waahto

          From 1A and 1B it does not follow that 1C. Infinities are strange. Let me try to rephrase your argument.

          A) After infinite amount of time, everyone will become happy.
          B) Then they will stay happy for infinite amount of time.
          C) Thus, the time before happiness is finite.

          You can think this by completely forgetting the time and asking, how far from each other are two random points chosen from an infinite plane. Or you can simplify and think it in one dimension: how far from each other are two points selected from an infinite line.

          You can think this line as a series of numbers. Then the shortest distance is 0 and the longest distance is ∞. Then the average distance is ∞/2, which is, you guessed it, ∞.

          And so, my answer is Board 2 (see below (my answer was few posts earlier)).

          • Scott

            You misread 1A. 1A is saying that after a FINITE amount of time, everyone will become happy. Now you may say that is incorrect, but again the issue, as i’ve said, is dealing with infinity.

            Your logic for choosing board 2 is equally invalid as board 1. Board 2 will have an infinite number of happy people AND sad people, just like board 1. An increasing number of sad people for eternity is equivalent to infinite sad people.
            ∑ (1 + 1 + 1 …) = ∞

            • waahto

              Didn’t misread, this is what your reasoning implies (if you take a point that is infinitely far from the drop, the time it will take the drop to expand there is infinite, not finite). Because the time it takes for a single person to turn sad is, in average, (0+∞)/2=∞, I see no point choosing Board 1.

              What comes to Board 2, there is no mistake. If we take eternity, the number of sad people becomes infinite. But this infinity is only a subset of a greater infinity that houses all the sad and happy people. And never does the ratio of sad/happy reach 1. Nor does sum(sad) reach any meaningful value beyond zero, being a smaller infinity than sum(happy).

              Now, mathematically speaking, I _might_ be completely wrong with my reasoning 🙂 But as far as I can see, I find no fault…

            • Scott

              I think we are letting the concept of infinity get the best of both of us. I think mathematically you are completely right, but this concept of infinity can’t actually apply in real life. If the happy drop is a real think on the board and you are a real square on the board, then you can’t be an actual infinite distance a part. There are no two real numbers whose difference is infinity. I just made another post because I think it explains my point better than this one. We can continue our discussion on that post 😉

      • Scott

        I edited my answer saying that both proofs, while appear true, are invalid, but do show that both boards are morally equivalent, or maybe more appropriate to say they are both morally indeterminate.

  • SGrint

    I’d go for Board 1: the sad people here will always have hope to become happy, the happy people on Board 2 will always be afraid to become sad.

  • JaapVerlinde

    I agree with a lot of people here that there is more than net happiness to consider. To me, this is not necessarily an appropriate measure for quality of life. Also, I think it anyway becomes weird territory when we start thinking about the meaning of happiness in an infinite life. I think you can argue that happiness exists by the grace of the finiteness of life. Without wanting to sound too mushy, I think aspirations, and fulfilling those, becomes meaningless when there is infinite time.

    In short, I have trouble with the premise, which makes me fail to adequately consider this. I don’t believe these infinite people could really be happy, and if they are, I don’t know what that happiness means. Alternatively, the agents on the checkerboard are not actually people, in which case I also don’t know what their happiness means.

    So I’m gonna be an erratic old testament god and flip a coin.

  • Las10

    1. I’d go for board 1
    2. I’d stay on board 1

    Now we cannot add the quantities to calculate net happiness or net sadness for the entire collective of humans since we are dealing with infinities here.

    But there is also a time factor involved where everything is going to exist for eternity. Since everything is going to be there for eternity – the board, the humans and the happy-drop expanding & touching the humans, according to Board 1’s conditions, eventually everything will turn to happy, and stay that way for eternity.
    The exact same logic applied to Board 2, which also has the exact same conditions except reversed, will result in eternal sadness.

    • Ramil Magerramov

      That’s not precise.Infinite board is infinite(yeah) and drop of happines will expand for ever. It would never make everyone happy. It doesn’t matter how much area it covered because it would always be much more to go.

      IMO variant 1 is better because waiting to get happy is better than the the opposite.

      But we should consider this situation other way. Imagine picture that Tim draw with checkboard and quiet big circle of happiness. Now imagine that this circle is actually a dot, very small dot, ifinitely small dot. And such image would represent current situation.

      Despite the fact that happiness expand exponentially there would always be infinitely more area to cover.

      • Las10

        There would. But we are talking about eternity here.
        Out of the given choices, Board 1 is eventually going to end up happy, and be that way for eternity. So, that was the point I was trying to make.

        • Scott

          The issue is infinity. Another way to explain the boards is, “No matter what, board 1 will always have an infinite number of sad people, and board 2 will always have an infinite number of happy people.”

  • Thomas Carrier

    Board 1
    Board 1

    It s all about improvement (make things better)… and hope… (if the information/rumour gets to first!)


  • Tough one….

    I’d go for the second board, concerning the first question. As the board is infinite, there will ALWAYS be an infinite number of happy people, and the tiny drop of sadness will, proportionally, always remain infinitely small, no matter how far it will reach. It will always be a smaller infinity, if such a thing exists (another good question for the dinner table?).

    As for the second question, well, it depends where that god would drop me. If it’s near the centre I’d happily go for the first board, if not, well I’d hope he’d drop me as far out as possible on the second board.

    And anyways, if EVERYONE in your infinite neighborhood is sad….. well, it will just be the norm and you won’t even realize you’re sad, no? Just get used to it, man, and don’t start crying.

    • user_kp

      yes exactly. I often post things in social media with the hasthag #science. In this case I should post my response with the hashtag #math or #logic.

  • user_kp

    Easy answer. Board #2 without question. This is not an ethical or emotional question. It is one of mathematics.

    If you foolishly choose board #1 (initially sad board) and increase happiness in some finite way (no matter how fast), you will NEVER EVER make a majority (or even a significant minority) of people happy. You can increase your happiness drop at a million miles per hour for a million years and your board will still be more than 99.99999% sad. Such is the nature of infinity.

    On other hand, if you wisely choose board #2 (initially happy board) and increase sadness in some finite way (no matter how fast), you will NEVER EVER make a majority (or even a significant minority) of people sad. Again, no matter how fast or for how long your sad drop increases, your board will be more than 99.999999% happy effectively forever.

    For anyone who chose board #1, please review your high school math.

    • Ferival

      After an infinite amount of time, the drop will grow to infinite size. It is impossible to tell whether the board or the drop will be larger after an infinite amount of time.

      • user_kp

        Right but you’re missing the point. Infinite time effectively means never. You’ll never there. You’ll never get half way there. You’ll never get 25% there. You’ll never get 1/10 of 1% there.

        • user_kp

          So by your own estimation, choosing board #1 means you are subjecting the infinite # of humans to abject sadness effectively forever.

          • Ferival

            The drop expands forever towards infinity. Every person in the universe will be affected by the drop at some point in time. After that time, they will spend eternity in the effects of the drop. Option one subjects everyone to a finite amount of sadness and infinite happiness.

            • user_kp

              I think I’ve addressed this. See my comments.
              This is a fun discussion. I hope everyone enjoys it.

          • Valiri

            There will be an infinite number of sad people in board #1, but you cannot state that they will be sad forever. That is the real paradox (or should I say f*ed up thing about this problem). No one will be in their original state forever, yet the original state will be shared by an infinite amount of people at all times.

        • Ferival

          Since it is stated that the universe lasts infinite amounts of time, time=infinity is a time that will be reached.

          • user_kp

            By definition, you can never reach infinity. It is a concept.

            • user_kp

              Look at it this way, if infinite time means you’ve gone so far into the future that you’ve run out of time (i.e. there is no time left because if there were you haven’t reached infinity) that means that for effectively all of existence, your people have been sad.

            • Ferival

              There is no end to time. By definition infinity is never ending.

            • user_kp

              Exactly by the same token you’ll never reach infinity. So your drop will remain insignificant in relation to the board forever.

            • Ferival

              Yes you will never reach infinity, which is why you need to incorporate limits. As time approaches infinity, the drop’s size will also approach infinity. Therefore after infinite time, the drop will be infinitely large and no longer insignificant compared to the board.

      • Depends on the propagation speed of happiness versus sadness…. I think the drop could never outgrow the preexisting board as the board is the foundation for the drop of happiness or sadness

    • Jacco

      You seem to be concerned with the happiness of the majority of people, but you still make each of them sad forever essentially. Sure there will always be more happy people than sad people, but all those people will eventually become sad forever. And there will still be more happy people than sad people, but they too will become sad. Ad infinitum

  • karl

    Can time be infinite? At any point in the time line there will be an exact amount of time that has passed and there will be an exact number of people that now have become either sad or happy, depending on what board they live in. But if there is an infinite number of squares there will always be an infinite number of people. So board 2 will at all times have an infinite number of happy people and a finite number of sad people and in board 1 there will be an infinite number of sad people and a finite number of happy people. I would choose board 2 for both questions because there will always be more happy people than sad people.

    • user_kp

      exactly. see my answer below.

  • Florian

    Do the people on the board know about their situation?
    If so, i’d go for 1!

    Better have a lot of sad but hopeful people than a lot of happy people feeling something dark is lingering in the shadows… + All the sad people with know it will never end, while all the sad people in board 1 will have the hope that it could end someday!

    • Awareness is the key.

      Are the sad people aware there is such a thing as happiness?

      Are the happy people aware there is some sad doom speeding their way?

      If not, it ultimately does not matter which board they are on, no?

      • After thinking a bit more about this I guess the people can not be aware of their situation. There would be no really happy people if they knew their state would end some day.

    • Florian

      Even if unaware of the overall situation board 1 is indeed the better choice, as you only have 2 states to be in and you will not be able to know the charachteristics of the other state before you have transitioned to it, but you will remember the original one after the transition

      So in board 1 you will be sad, but you will not know happy until it arrives, while in board 2 you will have known happy once you go sad and crave for it…

      + everybody knows the concept of mathematical infinity is flawed, certainly if you try combining multiple infinities in one physical/philosophical though-experiment, you should not just look at it with current mathematics. @Tim The current state of infinity in science, math, physics, philosophy,… could be interesting for a post 🙂

  • Coco

    My imagination tells me both are quantitatively the same. So the question for me becomes, ‘would you rather be sad first or happy first?’

    I’d choose sad with a drop of happy. People who are sad first appreciate happy. People who are happy first will be sad and live in a depressing state of the ‘best is behind us.’

    I’ll recant my answer if my imagination is wrong after reading some comments.

    • Coco

      Yeah, sad with a drop of happy. This is not a math problem at all.

  • Board #1. If you choose board #2, the end state for eternity will be sad, with knowledge of being happy.
    Even if you have to wait for eternity to be happy in board #1, it still is the end state, while appreciating it, because you know sad and suffering.

    • I think you’re wrong. There’s no end state to infinity!

      • Ah, but the drop of happy is expanding eternally, hence it should reach all at some point.

        • Naaaa, can’t reach infinity, want to bet?

      • Jacco

        The time untill the drop hits you will be finite, it may seem like an infinite time. but it will hit you. and because of this the time after the drop will be infinitly bigger than the time before the drop.

        • Artyom Karapetov


  • It depends on the PoV.

    If you’re looking at the problem entirely in numbers, Board #2 is the better option because you will always have more happy than depressed people.

    If you’re looking at it from the PoV of a person who lives on the board, #1 is far better. Why? Because the circle of happiness will reach you after X time. As it is infinite, X will take the value of 0 to infinity, but it will be smaller than infinity. So you will have Infinity – X depressed period and Infinity happy period.

    • Kestrel

      But time never gets to infinity, by definition.

  • Zolo

    Board #2
    The amount of happy people is always infinite, while the amount of sad people (no matter how huge) is always countable, and so, smaller.

    @elisealpha:disqus said it best, the answer for the second question is Board #1

  • Frankimahri

    Board 2 for me.
    I think it’s really hard to comprehend ”infinity” in a moral question. But basicaly how I see it:
    With infinity squares/people after 1000 years less then 1% is sad on board 2. Even after 1000000000 years , there are still more happy people then sad people if you choose board 2.
    If you choose board 2 you will have a world where most people are happy all the time. Board 1 will only have a very small group of happy persons if you compare it the infinite amount of unhappy people.

  • Jonathan Wells

    I tend to agree with the Buddhist conception that suffering = ignorance, that there is already a state of overwhelming joy called enlightenment, and that only our ego, which is our ignorance of the true state of reality, keeps us from being there all the time, which we already are, but don’t know it yet. So my world view is pretty much in keeping with Board 1. Part of the drop of joy is knowing that you are not stuck forever in a single checkerboard square but are part of the infinite oneness of the checkerboard. I wouldn’t wan’t to be the god to do it, but given no choice I would make Board 1. Doesn’t the notion of omnipotent god preclude having a boss, by the way?

  • shwetha jagannathan

    First table- although everyone is sad at first, it ultimately ends with everyone being happy forever

  • Kestrel

    Board two for case 1 and 2.

    In board 2 there are always infinite happy people, and only infinite unhappy people at time = infinity. Which I guess it never gets to.

  • Richard Kenneth Niescior

    What is the time difference between the creation of the board and the application of the dropper?

  • Valiri

    Board #2 defenders state that it is the only board able to provide maximum hapiness, which is true. The way the question is asked actually pushes us towards answering Board #2, because it states that each human on the board is identified by its state of hapiness (1) or sadness (0), which automatically makes the number of drop affected people quantifiable, thus infinitesimal compared to the vast infinity of the ckerckerboard. Board #2 essentially stands on an utilitarian moral ground. It is the way to go if you consider your checkboard of human only in terms of those numbers. If you stop time and observe, you will always have a maximum of hapiness.

    I may be wrong, but the maximum hapiness fact is only true if we consider a given time (that is : stopping time and taking a picture). It doesn’t take in account of time itself reaching infinity. Like previous posters stated, the drop will be infinitely smaller than the board on any given time, but only when stopping time and observing the drop. If we consider time growing towards infinity, then the drop’s size is also growing towards infinity. In a infinite time, the drop will be, as well, infinite. Now this needs verification, but from what I gathered, you cannot compare the checkboard’s infinity and the drop’s infinity. Thus in no way can you state, when considrering an infinite amount of time, whether there is more happy or sad people.

    So it seems we stumble upon an indetermination, in the very long run, no method comes as absolutely good. Whatever board we choose, over time we cannot say whether there are more happy or sad people. There are two ways to lift the indetermination. The first one we have already been through : stopping time. Stopping time makes the drop finite. This method comes in favor of board #2.
    The second way is actually answering to question #2, that is choosing a point on the checkerboard. When you choose a human on the board, the distance between the human and whatever state of the drop becomes finite, however big. On this human’s point of view thus, the drop will eventually be here before a finite amount of time. This goes in favor in Board #1.

    Which board do I defend ? I wanted to defend board #1 prior to writing this down, but now I can’t make up my mind. Board #2 is all about the present moment, it might seem managerial (counting hapiness as one would money or staff), but some very valid non-occidental ethics would gladly oblige as well (I think mainly of Chinese or Japanese morals where the notion of self is down-rated). Board #1 is all about individuals, it might seem egotist, but can also derive from empathy, humanism, and is certainly emotion-driven.

    • Valiri

      I find this problem odder and odder, and I’d like to share a thought on how absurd all of this is : there will always be an infinite amount of sad people on board 1 and happy people on board 2. To say that humans will be hit by the drop is saying there is a finite distance between him and the drop, and this can only be achieved by focusing on a finite number of humans. Say you’re in board 1, the drop converts a finite amount of people to happy, retrieving a finite amount of sad humans to a infinite amount of sad humans still makes an infinite amount of sad people. This vastly goes towards choosing #2

    • Artyom Karapetov

      I like your answer. So basically pick board one if you actually care about the humans, because that’s what each and every human would choose if they were to decide their own fate, or pick board two if you care about getting good statistics (until the time hits infinity?)…

  • DBoddu Santhosh

    As mentioned earlier Infinite happy people beats finite happy people. So I am going to select 2nd board for both of the cases.

  • Julien Boyer

    From the God’s point of view, making board 2 is better. At any given time there’ll be more happy people than unhappy.

    From the point of view of someone standing on the board, board 1 is better. Because you’ll spend a finite amount of time waiting for the drop to expand to your square followed by an infinite amount of time of happiness.

    • João Gonçalves

      What a fucking paradox

  • Shruti

    Questions – do the humans on the board know about the coming of the drop, in either case? Is there a difference in the rate of expansion of the board and the drop?
    Assuming answers to be yes and no in that order –
    Board #1 in both cases. It is easier to bear suffering knowing that happiness will come, when you are going to live on infinitely. The argument in support of board 2 (of finite number of sad people at any given time, but infinite happy) makes no difference to the individual, only the god player, because time = infinite, so you’d rather be sad for a finite time and happy infinitely after that.Also, imagine living on a checkerboard feeling happy but waiting for the wave of sadness to hit but not knowing when or where it is going to come from! Nightmare.
    As a god player I would prefer people to experience sadness but only because it allows them to appreciate happiness more. Both the board and the drop of happiness are expanding. It’s only a matter of time before happiness gets to each new human on the board.

  • Thomas Dirscherl (Primer2004)

    Seems like most people get this wrong by arguing about hope.

    Some of you argue that being happy whilst knowing that at some point they might turn sad would diminish their happieness and vice versa being sad whilst knowing at some point they might become happy would give them hope. Now what would you prefer: Being sad and knowing you will turn happy never or after a finite or infinite amount of time? Or the other way round? Wouldn’t that mean it would be better to be sad, because only then you can hope for an improvement?

    This line of thought totally runs contrary to the intended szenario. Sad is really sad, happy is really happy. So if you need to think in terms of hope and whatsoever: Just imagine that the sad people know that they will become sader and sader forever (making them truly sad) and the happy people know they will become happier and happier forever (making them truly happy). Then think again which szenario you would chose.

  • Len Errera

    #1. A person’s sadness in this case is a finite amount of time. At some point the drop will reach him/her and they will have infinite happiness. Even if it takes a really, really long time for the happiness drop to reach me, it will eventually and then I will live the rest of eternity in happiness.

  • Dorel

    Basically you have something (the drop) expanding in an infinite space. On one hand, the space remaining outside the drop should always be still infinite along with the number of people outside the drop. However, the drop expands for an infinite amount of time, so this would mean that is should eventually reach the boundries, which is impossible. I believe in maths this was a “undetermination” case or something like that when calculation stopped. So any further thinking on this is just philosophical.

  • Katarina Bešir

    Maybe I am missing out on something but, using only the info from the question and not adding any other factor whatsoever, I would go for #1 in both cases. As stated, the drop spreads rapidly and infinitely so it will reach everyone eventually and finite sadness followed by infinite happiness sounds better to me regardless of the time it takes to reach everyone.

    • Katarina,

      As there’s an infinity of people, the happiness can’t reach most of them EVER. And for most it would reach them only after an infinity of time…. tough

  • Alex Chiriac

    1. #1. If there’s a boss, there’s a system; systems can be rigged. I would make a type 1 board and, being almighty as a god, drop more happy drops on an infinite number of points.
    2. #1. If human on board AND know what’s going on, well, I still would want to be happy. And there are infinite chances my board god turns out to be a rebel and rig the system.

  • I guess I won’t sleep tonight, and when I get home this evening, I’ll drag my whole family into this thing!

    In fact, Board 1 is the foundation of nearly every religion: Rejoice, for you will be happy one day…. perhaps.

    Therefore I strongly think The people on the board don’t know of the magical drops. They are happy, they are sad… period.

    Happiness and sadness resides only in the eyes of the gods. The people on any board, in any of the situations are unaware of anything that might change their lives (apart perhaps for those immediately at the event horizon, when the drops rush at them and they see smiles or tears on their neighbor’s faces – but that will be quick). The sad people will not know they are sad as there’s nothing else…

    So the question is just, those people who are reached by the drops WILL remember their former situation and will be happier or sadder. So it’s only their situation that’s important. In that case Board #1 will be the correct one.

  • Blyxx

    I’d put the drop of sadness onto the infinitely happy board. Since the board is infinite, think of the sadness as a program that makes a black dot in the middle of your screen that slowly gets bigger and bigger, but has an infinite zoom function…when it starts to look big, just zoom out until it’s tiny again, which you can do infinitely. The amount of happiness will always dwarf the amount of suffering. I would also choose to be a human on that board because I would rather have real concrete happiness immediately than endure suffering for an unknown amount of time before being happy. And the odds of the sadness dripping onto my square on any given day would be infinitely small.

    • Frankimahri

      You described my thoughts really good! Totaly agree with this!

  • Richard

    If I was the god I wouldn’t prefer either, because why should I care about the happiness or sadness of the human beings? I’m a god, i’m not the one whose duty it is to (from some kind of utilitarian view) make the most people happy. So I’d be indifference.
    Besides from that. What omnipotent god has a boss? Why in the world would you want to create such a world?
    From an ethical point of view this is a fun discussion. From all other points of view it’s an empty one that needs no answer.

  • Evelyn Sztojanov

    Your question is confusing, can you clarify? You say the board “extends forever in every direction in a 2-dimensional plane” – did you mean every direction or both directions? Is this happening in 2 dimensions and if yes is time one of them? If not, taking hindsight prediction / retrodiction into account the two options are the same as the future state of happiness / unhappiness will have effectively created the past state, meaning our very human selves are creating and defining our state, which is essentially omnipotent divine presence who created / is creating the board to begin with. Then the only question remains who the hell is this boss person?! cheers

  • Pyriax

    To me this has a mathematical answer (i.e. maximizing average happiness), and it depends on how the drop expands with time.

    Case A: If the drop expands with its radius increasing at a constant rate, then Board 2 is the better choice.
    Case B: If the drop expands with its area increasing at a constant rate, then Boards 1 and 2 are equally good.

    (Note that if you assume that for question 2, you have no control over where you are placed on the Board, then your preference will match exactly with the answer to question 1. So I haven’t made a distinction between the two.)

    The reasoning behind the above is as follows:

    We start by giving everything finite dimensions. Treat the plane as a circular disc, radius L. The drop, which we also treat as a circular disc, at time t has radius r(t). The drop expands from t = 0 until t = T such that r(0) = 0 and r(T) = L.

    Now, we model the average happiness at time t, h(t). This is the percentage area of the disc which is happy:

    Board 1: h(t) = r(t)^2 / L^2
    Board 2: h(t) = 1 – r(t)^2 / L^2

    Now we average happiness over all time. Average happiness is H = 1 / T * integral of h(t) dt from t = 0 to t = T.

    Board 1: H = 1/TL^2 * integral from 0 to T dt of (r^2)
    Board 2: H = 1 – 1/TL^-2 * integral from 0 to T dt of (r^2).

    This is as far as we can get without considering what function r(t) actually is. We now look at our two cases.

    ** Case A (constant increasing radius) **

    r(t) = kt for constant k. But we know that r(T) = L so kT = L so k = L/T.
    Hence r(t) = Lt/T.

    Then by simple integration:

    Board 1: H = 1/3
    Board 2: H = 2/3

    Hence Board 2 is better. This answer is independent of L and T so we can let those tend to infinity and our answer won’t change.

    ** Case B (constant increasing area) **

    r(t) = k t^(1/2). But we know that r(T) = L so k T^(1/2) = L so k = L / T^(1/2).
    Hence r(t) = L (t/T)^(1/2).

    Then again by simple integration:

    Board 1: H = 1/2
    Board 2: H = 1/2.

    So the boards are both equally happy. Again this is independent of L and T so the infinities don’t matter.

    ** Further notes **

    If r increases at a rate slower than t^(1/2) then Board 1 is the better choice. I won’t prove this but I don’t think it would be too hard to show using similar math to the above.

    Comments welcome!

    • JCosta

      This seems counterintuitive. If the radius grows at a constant rate, the area of the circle would grow exponentially, which should give more value to board 1 (expanding circle of happiness) over board 2 (expanding circle of sadness), I’d think. The math clearly doesn’t show that, however. Any idea as to why that might be?

    • bszert

      You’re assuming that those are the only two possibilities. Maybe the radius grows quadratically, or exponentially, or tending toward a constant (that would be such a dick move from God).

  • Sylvain ROBINE

    Definitely the board 1 for both question, since the amount of humans in either will be the same anyway (we are here talking about infinity and eternality, so the numbers don’t actually matter, they are both infinite no matter what). And it is more enjoyable to feel happy after having suffered than being happy and then suffer ! Moreover, if you know that the happiness drop will come eventually, you can accept your suffering a little better, whereas if you know the sadness will come, you will enjoy less your happiness. So no hesitating.

  • Titus

    Board #2 in any case. To me it’s fairly simple: The check board (square) is infinite. The drop surface (circle), despite expanding infinitely, will always be finite. Therefore, at any given time, there will be infinitely more people outside the circle. As a result, the fact that people live infinite lives is irrelevant. This is clear if you grab the notion that both the following statements are false: “if you are outside the circle you may never get reached by it”, and “Eventually, everyone will be reached by the circle” as “never” in an infinitely expanding dynamic and “everyone” in an infinite population, are both irrelevant terms. So it must be board #2.

  • DanCoz

    I would choose option B both times.

    1°) The number of happy people will be infinite, while the sad people would be finite. Always, no matter what.
    2°) For the same reason, if I had to be a square in the board, I have infinite more possibility to be happy than sad.

    • DanCoz

      after reading the answers I changed idea 😛 I would say 1° B and 2° A…

      After thinking a while it came logical.

      • tread

        Can you explain why you changed your mind?

        • DanCoz

          That’s because the question makes you look from the other side of the window. In the second situation, choosing B, the sad time will be finite and once the drop gets me, the happines time will be infinite.

  • Titus

    Tim, if you withdraw your pictures, the percent of board #1 new supporters will significantly drop 😛

  • ST Bruce

    Board #1 inspires hope. Board #2 inspires dread. With the dropper of happiness we know that at some point everyone will find happiness. With the dropper of sadness the first place the dropper hits will leave those people with almost never having experienced happiness. There is always an infinite number of sad people but we know they will all one day find happiness. Thats better odds than this world.

  • David Zussman

    Board #2. In Board #1 many people will only know sadness for a long time, and then they will only know happiness. I fear they will forget sadness (a la ‘Brave New World’). The alternative world of Board #2 is a world where people know sadness but maybe can recall happiness (a la ‘1984’). I have always thought the Brave New World scenario is worse because unrelenting happiness has no meaning or value. Suffering is terrible, but at least induces hope.

  • Nicolas Voyez

    If the goal as a God is only to maximize the number of
    happy people at a given time, then I have to choose board 2, cause the number
    of unhappy people will be:

    UnHappyPeopleCount = (Pi * NumberOfExpansionTimePeriodsSinceDropLaunched²)

    while the number of happy people will be:

    HappyPeopleCount = Infinity – UnHappyPeopleCount = Infinity

    So HappyPeopleCount (infinite) will always be superior to the number of UnHappyPeopleCount in the drop of unhappiness
    which is a finite number.

    If my goal as a God is to maximize the happiness of every single human, I’ll
    choose the 1st board, because in that case, a human will be unhappy a finite
    amount of time :

    UnHappyTime = (DistanceFromTheDropInitialPoint * ExpansionTimePeriod),

    while it’ll be happy for :

    HappyTime = Infinity – UnHappyTime = Infinity.

    Personally, I think it’s morally more
    profitable to let all humans experience more happiness then sadness in their
    life, than having a world containing more happy people than sad ones, while promising
    them an eternity of suffering in the end.

    So, as a God or as man, I’ll choose board 1.

    (My only hesitation would be considering that the suffering prior to the happiness
    may break you and make you unable to enjoy the happiness, thus letting you in a
    state of suffering. But I feel that this possibility leave the specter of this
    dinner table so I won’t take this in consideration.)

    • dimik

      Wouldn’t DistanceFromTheDropInitialPoint be infinite for virtually all people?

      • Nicolas Voyez

        Nop, it’ could be very very long, billions of years maybe, but still way shorter than infinity.
        Every person has a given position, so he also have a fixed time to wait.
        That’s the problem thinking about something infinite, we still imagine boundaries in our mind (I’m the 1st one to do that), and so we imagine that there will be guys “at the end of infinite chessboard”. But there will always be a guy after us which will wait 1 more period to be happy, and then stay that way an infinite amount of time.

        • tread

          It can never be actually infinite, but the expected value will still be infinite.

  • Arty Lee

    1. If was an omnipotent God, I’d wipe out all the humans and have a nice game of infinite chess. I don’t give a fuck about useless humans.
    2. I don’t believe in omnipotent Gods.

  • Cate

    The better moral choice is to create board two. Since the board is infinite, there will always be more happy people, untouched by the dot, than the finite number of people caught by the expanding sadness. However, if I were an eternal human, I would want the god to create board one and I would just pray that I was towards the center of the grid.

    • tread

      The thing is, even as an individual, there is literally a 0% chance that the sadness will have reached you at a given time, so you shouldn’t worry at all about choosing board 2.

  • Matias Frank Jensen

    Definitley 1 for both questions. For every person on the board with 1 that person will be sad a finite amount of time but happy and infinite amount of time. It is reversed for 2.
    So if I want to be happy for a longer period of time than I want to be sad, choose 1. Being sad for a trillion years and then happy for eternity is infinitely better than being happy for a trillion years and then sad for eternity.
    And if we talk about the moral answer, since every single person would agree with the fact that being happy for an infinite amount of time is infinitely better than being sad a finite amount of time, you should choose 1 to maximise happiness for each individual.

    You could make the argument that 2 is better for the moral question since at every point in time there are infinitely many more people happy than sad, and thereby you maximise happiness. But then you only maximise happiness if you look at the entirety of the board as an organism where the individual is irrelevant. For the individual, as I have explained above, 1 is the better choice, and since the board exist of individuals, 1 should be better.

  • WW

    Definitely the drop of happiness in the infinite board of sadness for both.

    Let’s simplify the question. You’re a person on the board. The drop lands at point 0, and you are assigned a number x defined as the number of people between you and 0. X is an integer because you have no partial people. In effect, all people at the same X has the same happiness/sadness history, so for the purposes of this experiment, they’re the same person.

    Now you have the whole numbers 0, 1, 2, … infinity where the drop starts at 0, and let’s say it expands 1 every day. So now we’re working with a 1 dimensional board, which is easier to think about.

    If you’re person X, no matter what X is, you’ll be unhappy for X days, and happy for an infinite number of days. You know that the dot will reach you. I’d like this scenario even more if you know where the dot lands, because you can anticipate it.

    There is no such thing as X being infinity, because the person’s space on the board doesn’t move, but the dot does. The person is always a finite distance from the center of the dot, so any single person will always be reached.

    As for all the discussions about multiple infinities, because this is a board dealing with distinct people, this is all the same infinity, aleph-naught, so you don’t have to worry about one infinity not covering another.
    So in summary, if you choose the infinite sadness board with the happiness dot expanding to infinity, any given person will know that they will be reached in time X, which is finite, and they will live all the days after X (which is infinite) in happiness.

    • Maria Antonia Marturet

      I completely agree with you.

    • Titus

      Do you realize that when you write “the person’s space on the board doesn’t move” and “The person is always a finite distance from the center of the dot”, you finitize X? It is self evident that at the moment you pick a single person, it will someday (or in X days) be happy/sad.

      Despite that you can place an X wherever you want and then be able to count the distance, the concept of infinite includes that you can also “place” an X at an infinite distance. Except that it’s not like dropping a pin. I think that trying to image this ruins everything.

      • Nicolas Voyez

        You’re on the board. you look at yourself. You’re existing, so you obviously GOT a position. you’re not at a “infinite place”, you’re at X,Y on a chessboard. It can be 1,1 1M,10M it doesn’t matters, you got a fixed spot.
        So depending of the rule you choose, you either got a finite time unhappy and an infinity of timehappy, or the opposite.
        Up to you to choose which one you prefer.

        • Titus

          Yes, me myself I am at a X,Y. But if you talk about me you’ve dropped the pin. Can you fix an infinite number? You can’t.

          Imagine a circle. There’s a dot on the circle and it can travel along
          the circumference. For simplicity lets assume that when the dot travels
          one time the circumference, it counts for one person changing from sad
          to happy. Obviously the dot will never reach at an end. The thing is
          that we can answer when the dot will have cycled X times but we can’t
          answer when the dot will have traveled half the distance because there’s
          no such thing as “half the distance”.

      • WW

        You can’t place X at an infinite distance. This is a checkerboard. The moment the drop appears, we call it (0,0). At that second, every single person on the board no matter how far away, has a coordinate, and any point has a finite distance to the dot even if the board is infinite.
        So back to my example. If you have the number line, even if you say it’s – infinity to infinity, but you have to assume that it’s integers, because people are distinct, then the moment you drop the dot at a point, call it D, any given point has a finite distance to D even though the line itself is infinite.

  • vacations

    Ultimately, I’d choose sadness with a drop of happiness for both questions. I understand that on the aggregate, the second board is better because at any given time, you have a finite number of sad people and a infinite number of happy people. But, from the perspective of someone on the board, you’d have a finite time in sadness, followed by an infinite time in happiness. So I think it would be the more compassionate choice to give people infinite happiness…even though when viewed with detachment, a board having infinite numbers of happy people seems to make more logical sense.

    Taking it a step further: I think this is why governing bodies fail to serve their people in the most compassionate way. When viewed with cold logic detached from the human aspect (ahem, Republicans), things can seem like great ideas in the big picture, but when experienced at the individual level… they kind of suck.

  • Akash Gupta

    My answer – Happiness with a drop of sadness

    Assuming the problem is based out of 1 dimension with each box representing one person and it expand 1 day everyday.

    Let’s look at both the scenarios.
    Sad with a drop of Happy:-
    As happiness begins to expand,there will be a linear graph between happiness and “number of days it take to get happiness”.Initially there would be few happy people which would be same as number of days passed, but infinite sad people.As we approach infinity, there would be infinite happy people starting from 0, but there will also be infinite sad people left. This is because, infinity-infinity=infinity and not 0. No matter how many people are turned to happiness, there will always be infinite sad people left.
    Before we reach infinity – few and countable happy people but infinite sad people.
    As we approach infinity – we will have infinite happy people and infinite sad people.

    Happy with a drop of Sad:-
    Again,using the same concept, there will be a linear graph between sadness and “number of days it take to get sadness”.As we approach infinity, there would be infinite sad people starting from 0, but there are also infinite happy people left. No matter how many people are turned to sadness, there will always be infinite happy people left.
    Before we reach infinity – few and countable sad people but infinite happy people.
    As we approach infinity – we will have infinite sad people and infinite happy people.

    At infinity , the situation is same in both the scenarios i.e. both have infinite sad and happy people. But before we reach infinity – “happy with a drop of Sad” sounds much better, because there would be limited number of sad people in this case.

  • DrFil

    I would pick Board 1 for many of the reasons already given (primarily finite sadness and the very powerful notion of hope for eventually becoming happy forever; could you imagine living under the constant fear of becoming sad…. forever?)..

    However, this made me think of another issue. If you are popped into existence feeling sad, do you really know any better? You have no frame of reference for any other type of existence. Are you really suffering if that is just how you were for your entire life, and all you have ever known?

    If accepted, this means that board 1 and 2 are identical prior to that drop. Now imagine the experience of going from a completely miserable existence to one of bliss. Consider the reverse with board 2. I think this is a strong argument for picking board 1.

  • Aaron Barbee

    I’m going with Board 1 on this: Sad with a drop of happy. I’m a super-optimistic and cheerful person, and I’d hate to be sad for eternity (the ultimate result of Board 2). I’m creating my own theory of how the checkerboard works, since not implied or directly stated, and will assume that the drop can be placed at any time (= immediately) and that the expansion of the “Happy” is nearly immediate as well.

    Smiles and laughter are infectious. So, why can’t a drop of “Happy” be that contagious, as well?

    So, for my Board 1, once the board is created and people placed in their squares, I will immediately place my drop of happy (in the center, because of my OCD tendencies) and then its happiness will radiate out for eternity as soon as the first person is impacted.

    Love it. 🙂

  • O’B

    If you’re the benevolent sort, board 2 all the way. At all times, there will be a finite amount of sadness in an infinite expanse of happiness. Therefore, at any point in time, the fraction of sad people will always be ~0%.

    To say that “eventually” the sadness will get you represents a misunderstanding of “infinity”.

  • The_Postindustrialist

    The funny thing is that I think both will yield the same results.

    I originally thought #1, because of the aforementioned “each person will be said for a finite period of time but then happy” reason.


    The board is infinitely expansive, and the drop may be infinitely expansive, but for every square the drop hits, there’s one more square that hasn’t been given happiness yet, so the positive is always racing to catch up with negative. There’s always that +1 person that’s going to be sad.

    On the other hand, the other board works similarly, so there’s always 1 more happy person beyond wherever the sadness goes. So the negative is always trying to outrun the happiness…. And there’s always +1 person happy rather than sad.

    So I guess it’s more of a question that isn’t so much to reveal who is smarter and who can figure out how to get more happy people on the board, but whether you are the type of person to focus on the happiness of the now, even if it means sadness later, or if you are willing to delay a great amount of happiness (possibly even for what seems like forever) and that the end result is worth the means necessary to get there.

    (also, time for this experiment is kind of irrelevant. at any given time, for one person who is happy, there is going to be one that is not in either experiment. For every person that endures sadness for X amount of time, there is one that has had proportionally the same amount of happiness and vice versa…)

    Kinda reminds me now also of two similar math problems.

    The one where you have to add up say, the numbers in a sequence, say 1-10. You can do this the long and hard way, or you can do it the way Gauss did by lining up the numbers like so
    1 2 3 4 5
    10 9 8 7 6

    Which reveals…
    1 2 3 4 5
    10 9 8 7 6
    11+11+11+11+11 <—— The results are all the same

    And then the other one is how 1+1-1+1-1…… = .5

    Kinda zen really….


    I think I'd still rather pick situation A though because I can get through sadness with a fair amount of hope and assurance of a good outcome, rather than be happy and then having it all taken away without ever seeing it again.

    (now you can also compare that to the religious concepts of "heaven" I'm sure… And the whole experiment itself I'm sure has heaps upon heaps of use in apologetics in western christian doctrine back from when the churches tried to reconcile faith with the greek philosophers.

    Though, mind you, not all religions have a heaven and a hell, and the concept in itself, is NOT the main reason people have faith. In fact, very few religions are based purely on what happens after you die.)

    • The_Postindustrialist

      Additionally, I don’t think the second question is much needed, as most answers are showing people are mainly empathizing with the people on the board as they choose which one to create. I think that they all will choose the board that they view as most moral as the board they would most like to be on.

      Isn’t empathy amazing? 🙂

    • Kingfisher12

      Mathematically they would be the same, but for a person on the board experiencing time linearly the sad board would be the better option, since the time ahead of you will always exceed the time behind you.

  • Kyle

    Q1: I go with option 2 as infinite happiness will always trump finite (even infinitely expanding) sadness. Same answer for question 2.

  • Adrian Canto

    Assuming that you can create a 2×2 checkboard from start (It will start as 2×2 and then begins expanding) and then use the drop on the 1,1 square (seeing it as a coordinate), here are my answers:

    1. I would choose eventual eternal happiness for everybody, that´s option 2. The first 4 humans will be sad, but just for a short period of time (then become happy thanks to the drop). Every new human will be sad, but not for a long period of time (since the bord will be now 3×3, 4×4, etc).

    2. I would like he choose option 2, since I will be sad for a period of time, but eventually become happy for the rest of eternity.

  • Erik Anderson

    Relatively speaking, a person’s enjoyment of existence has to do with deviation from the norm. So, constant sadness and horrible suffering being transformed into blissful happiness leads to a large increase in ‘good’. On the flipside, having everyone being constantly happy and then being thrown into eternal suffering with no hope of escape leads to an increase in ‘bad’.

    In both cases, you have an infinite number of sad and an infinite number of happy people, but only in the second case do you have people who were once happy and then made eternally sad. That suffering will be greater than any suffering any person on Board 1 will experience, because they will have known happiness and will never achieve it again. Only on Board 1 will you have people who have known suffering and then achieved eternal happiness.

    As for the questions, if I’m an omnipotent god, I will choose to create the first board but never create it. In essence, I would take an infinite amount of time to make the board.

    I would want to be on board 1. I would suffer for a finite amount of time and then enjoy eternal happiness. If I were on board 2, I would be happy for a finite amount of time and then suffer eternally. In answering these questions and choosing board 2, one assumes that they will be far outside the sadness dropper. What if it lands on you a microsecond after the board is created? A minute? A year? A millennium? You could be happy for a trillion years before the dropper hits you, or just for a moment. The only certainty is that it will hit you.

  • Vincent Van de Langenberg

    I don’t think looking at the checkerboard as being 2 dimensional can solve this paradox. In board #1 it is true that the finite happiness circle will always be smaller than the infinite sadness surrounding it (as Blyxx perfectly described think of it as an expanding dot with an infinite ‘zoom out’ function), so making a snapshot at any given time will show # of sad people > # of happy people. But how does a snapshot hold up in infinity?

    Let’s add the value of future happiness to the equation by elevating the happy squares and lowering the sad ones. Each individual happy square now has an infinitely high tower (=time) of happiness, while the sad squares each have their own individual finite depth of misery ahead of them, before becoming an eternally happy tower.

    Now because the sadness squares are infinite, its own infinity causes the depth of misery to decrease by infinity with each passing moment, thus squaring an infinity out. On the other hand the total happiness
    will increase for infinite time, with infinitely high towers.

    This also holds up for the second question. Even though the board is infinite in size, the moment you place the happy dot every individual square on the infinite board will have its own measurable distance to the happy dot (infinity-1). So your future infinite happy tower will always be larger than the finite depth of your misery.

  • Without getting all technical and mathematical like I’ve noticed in some of the comments, I’d just say I’d rather start as a miserable person and become happy forever than the reverse. Once you’ve known infinite happiness, that crushing sadness would be unbearable. But if all you’ve ever known is sadness and depression, there’s nothing to compare it to. No what ifs. That’s just your constant state of being. You don’t know of anything better. And then when the drop of happiness reaches you, suddenly your life is sunshine and rainbows and you’ll know that happiness forever.

    I’d hope the god would think the same way as I do if I was on the board. I’d rather be sad and know nothing of happiness and someday be happy forever, than to start out happy and get unbearably sad and know that a better state of being existed.

  • Nil Indef

    I guess if the premise is undefined (“infinite”), there is no way to give definitive answer. We can do things like this: My answer is “infinity mod 2 + 1”.
    I don’t know…

  • Adam

    I really can’t decide what to answer, but I do have some thoughts:

    (1) People saying ‘the people who start off sad won’t feel as bad because they’ve known no other state of existence’ are missing the point. These imaginary people are permanently unhappy. They don’t get used to it. They’re in “a constant state of horrible suffering”. One could also say the exact same thing about happy people just getting used to it, but people seem to be using this as an argument against choosing board 2.

    (2) I don’t understand the distinction between question 1 and question 2. Presumably, we’re assuming benevolence in question 1 [and therefore trying to maximise happiness], so why would that answer differ from one where we happen to be someone on the board? If you said ‘if you knew you were going to be placed on the square where the medicine will be immediately dropped…’, that might make some selfish people change their minds from board 2 to board 1, but we’re not told where we’re going to be placed.

    (3) I just don’t understand infinity [other than in a few purely theoretical limited mathematical contexts]. I don’t understand how the universe could be infinite and I can’t understand this infinite checkerboard. It just doesn’t make sense. There isn’t a right answer.

    • Luisa

      (1) really? how do you know? Maybe god did create this board, with the eternal sadness, and it’s called earth. And once we die we achieve eternal happiness. We don’t know how miserable our existence here is because we can’t compare it to heaven.

      ha, just kidding, I’m an atheist.

  • Kostas

    I think the problem tries to mess around two issues that human nature has core inabilities to perceive :

    From A) infinity point of view a more mathematic approach would be suitable. If we consider as a given that drop will expand with a measurable speed (even if very big), the solution to the problem is rather simple. In any given countable time, the happy people in board type 2, would be a lot more that the expanding sad people.
    Issue gets more complicated if we consider that this universe will live infinite amount of time… Now we need to consider if the infinity order of time is higher than the infinity order of Space. In case infinity order of time is higher then we should consider type 1 Board as better solution.

    And now we go to a morality B) point of view : In the above analysis it is clear that time and change is the key factor. Imagine a world that is very happy at first and becomes more sad as time passes (board 1). This world gets worse and worse as time passes. People even if happy would have no hope but to expect an endless missery-sadness that is coming.
    Instead on board 2) no matter how sad you are you can hope that in the near (or far far far far away future) you will live the endless happiness!!! Interesting!

  • Simon

    Q1 – board 2: you will always have infine happy
    people against a finite number (even if expanding) of unhappy people. So the
    moral choice would be to maximize the number of happy people with board 2.

    Q2 – board 1:
    as an individual on the board, this would mean a finite time of unhappiness and
    an infinite time of happiness. That’s clearly better than the opposite.

    So we have
    this nice paradox: each and every one of the individuals on the board, trying
    to maximize his own happiness, would prefer board 1.

    And still, the
    sum of all the rational individual choices would not be the optimal solution….

    Guess it’s
    Tim Urban’s version of the Tragedy of the Commons

  • Panda

    Two solutions based on two different assumptions.
    First: As a god, infinity is trivial. You can make it happen instantly.
    Q1: In this scenario, picking the board is dependent on how fast the drop of happy/sad can spread. Naturally if the rate of spread is infinite, board 1 is the way to go. Everything would happened instantly. Else, board 2.
    Q2: Same as Q1.
    Second: Even as a god, infinity is something you cannot overcome. You still have to mold the dam clay every time you want to make a new human.
    Q1: Board 2. The god would take infinite amount of time to create the human so god won’t be dropping the sadness therefore everyone is happy for eternity.
    Q2. Same as Q1.

  • redpill2010

    This one is easy. Happiness with a drop of sadness. Why? Because if the checkerboard is infinite, there will always be an infinite number of happy people and a finite number of sad people. If you do it the other way, there will always be an infinite number of sad people and a finite number of happy people.

    • Antoine Aublin

      I think this thought experiment is quite simple indeed… It’s actually just maths ! And here, I guess one should just be rational.

      Actually the only case in which I could choose the other board is that since sad people only know sadness, they maybe could not compare it to anything better… And sadness could actually seem normal to them.
      But then again, I’d rather say Happiness with drops of sadness for both answers.

    • String userName=new String();

      But if you’re on the happiness with a drop of sadness board, you will be happy for a finite amount of time and sad for an infinite amount of time.

      • redpill2010

        Not if you’re infinitely far from the sad drop 🙂

        Reminds me of the argument that time must have had a beginning, because otherwise there would have been an infinite number of seconds before now and we’d never reach the present!

        • String userName=new String();

          You can’t be infinitely far from the sad drop. Real numbers don’t work like that.

          Think of the board as a Cartesian coordinate plane, with the sad drop hitting the origin. A point on the board is located at (x,y). Since the board is infinitely large, x and y can be ANY real numbers. The distance from the drop at the origin to a square at (x,y) is equal to sqrt(x^2+y^2), which is a finite number for ANY real x and y. Therefore, any person on the board is a finite distance from the drop. Since the drop expands at a constant rate, any square on the chessboard will therefore be reached in a finite amount of time.

          • redpill2010

            Infinity is not a real number, it’s a concept.

            • String userName=new String();

              I know it’s not a real number*. That’s what I’m saying. You can’t have a point located at (infinity, infinity) in the coordinate plane because infinity isn’t a real number. Even if the board goes on forever, no matter what real numbers you choose for the coordinates, the distance to the origin will be finite.

              *Real numbers:

  • Andy O.

    Is this an exercise in delayed gratification and its evil twin, delayed disappointment? Hope vs. despair, inevitable satisfaction vs. impending doom for each new human on the board? If so, the answer has to be #1.

  • Matt Lichter

    Q2) Sad with a drop of happy. I will eventually be infinitely happy. Guaranteed.
    Q1) Sad with a drop of happy. Each individual will eventually be infinitely happy. Guaranteed.

    However, the benevolent (moral) god who lives vicariously through his subjects will always be sad for eternity, because the ratio of happy subjects to sad subjects will always be zero. The moral god will sacrifice his/her own happiness for the sake of his/her infinite subjects.

    People who are answering happy with a drop of sad are conflating individual happiness with aggregate happiness. There’s no such thing as aggregate happiness.

  • Brent Goss

    Neither of the questions matter…If I’m an “omnipotent god”, then I would not have a “boss”.

    • Annie

      and here we have a sardonic internet comment in its natural habitat

  • Canberk aygün

    Well, after pondering on this thought experiment for a while I guess it is time for me to watch ‘Inside Out’ especially for characters ‘Joy’ and ‘Sadness’.

  • John Kepley

    Everyone on the board will become happy at one point with the “happy drop”. Everyone will become sad at one point with the “sad drop”. The correct choice to both questions is obvious: the “happy drop” is way better because any singled out individual will eventually become happy.
    And besides, happiness is meaningless without sadness. Therefore, the “happy drop” scenario just makes more sense. To add reassurance to all the sufferers, the sadness experienced first is not meaningful because there is nothing to compare it to! Sadness is all they had ever known until “Shazaam!” —- “happy drop”!
    By the same logic, the “sad drop” scenario would be just TERRIBLE. Not only is the happiness meaningless but then here comes the “sad” drop that gives meaning to what you no longer have. Awful.

    • “Everyone on the board will become happy at one point with the “happy drop”. Everyone will become sad at one point with the “sad drop”. ”

      wrong, the board is infinite and thus there’ll always be infinitely more sad people on the sad board and happy people on the happy board. The amount of fields covered by the drop will always be finite.

      • John Kepley

        Correct. But I am right in saying that an infinite number of people WILL BECOME HAPPY. The question is really: Would you rather an infinite number of people BECOME SAD or an infinite number of people BECOME HAPPY? The answer is therefore obvious.

  • Yaroslav

    I would chose the 2nd one as a god because at any given moment there would be more happy people, and 1st one as a person because I couldn’t be truly happy with knowledge that one day something extremely sad happens to me.

    by the way, it is really interesting that most of the comments are with really elaborate answers, and i wonder how time spent on thinking about the problem would correlate with probability of posting a comment.

  • mtjces

    is there any place I could read more about the infinite checkerboard quandary? Or is it original content?

  • O’B

    After some more thought, it seems the sad board with happy drop is better in both situations. The fraction of happy people is a function of time, like this:

    HappyFraction(t) = HappyArea(t) / TotalArea(t)

    We need to take the limit as time approaches infinity to see if it converges (endless sadness) or diverges (endless happiness). This limit is undefined, but we can apply L’Hopital’s rule:

    lim t->inf HappyFraction(t) = HappyArea'(t) / TotalArea'(t)

    Since HappyArea is actually a function of t and TotalArea is just a constant (infinity; but no matter how you slice it, this is of lower order than the expanding happy area), the limit diverges.

    Therefore, since HappyFraction tends toward infinity as time approaches infinity, sad board with happy drop is the benevolent choice for both questions.


    • String userName=new String();

      The problem is TotalArea(t) isn’t a real number for any t, so I’m not sure if what you did is valid.

    • Artyom Karapetov

      I agree with you that the answer for both questions is a sad board with a drop of happiness, but mostly because everyone will anticipate happiness at some point rather than live in fear of the impending infinite sadness (which is a horrible way to have fun).

      So at the end of the day, everyone are at least waiting to be happy, vs. waiting to be infinitely sad. Hope for a better future is better than counting down your remaining happy hours, so if morality plays a role in this decision, I would pick board one.

    • thu nguyen

      Hm doesn’t the limit converges to 0. As limit t->infinity, if you keep on taking the differential of happyarea(t), it’ll equal a constant (say..constant A) since it’s finite. And totalarea(t) will be infinity so limit t->inf HappyFraction(t) = A/infinity which will converge to 0.

  • Pyriax

    A lot of people (including Tim) have said for question 2 that Board 1 is the better choice because as an individual, I am always a finite distance from the drop and so it will reach me in finite time. Hence finite period of sadness + infinite period of happiness is the better option.

    But think about this a bit more. Let’s say I get placed on Board 1 somewhere. I’m waiting for the omnipotent god to plant the ever-expanding drop, but I don’t know where he will choose. (I really hope he will choose somewhere close to me.) So we can imagine that I’m at the origin (0,0) and the drop is going to be planted a distance r away from me.

    We can ask the question – what’s the probability that the god is going to plant the drop within a circle of finite radius R away from me – so that the happy zone reaches me in finite time? That is, what is the probability that r < R? Well, if the god is picking completely at random, then the chance of picking within the circle is proportional to the area of that circle, and the chance of picking outside the circle is proportional to the area outside the circle. Unfortunately the area outside the circle is infinite, whereas the area inside the circle is finite. So the chances of the god picking inside my circle are 0. No matter how big the "target circle" is, the god is always going to have an infinitely greater chance of missing it than hitting it.

    So in fact the statement "I am always a finite distance from the drop" is incorrect. It is in fact 100% probable that I will be an infinite distance from the drop, and so will have to wait infinite time for the zone of happiness to reach me.

    The only way to get around these paradoxes is to treat everything as finite (as I have done in a previous comment) and figure out which is the better board given finite time and a finite board. Once you have done that you discover that the choice of which board is better doesn't actually depend on the size of the board or the length of time, so we can let both of those dimensions tend to infinity and the analysis still holds.

    (This, by the way, is how Newton, Leibniz et al. derived calculus. You start by treating everything as finite, come up with an answer given finite parameters and then let everything tend to infinity. If you're lucky then the answer doesn't change with the parameters, so you are done.)

    Another side note: the distinction between time and space is a bit of a red herring. In this analysis the time dimension behaves exactly like a space dimension.

    • Kestrel

      Yup, fully agree. Also, space/happy people in board 2 is infinite from the start. The drop grows with time, but never reaches infinity (by definition). So there are always infinitely more happy people than sad.

    • String userName=new String();

      With all due respect, you’re wrong, assuming we’re using standard analysis. Think of it as a Cartesian coordinate plane, with the drop falling on the origin. if you’re at the square (x,y), where x and y are two real numbers, the distance between you and the origin is sqrt(x^2+y^2). This number is finite for ALL real x and y.

    • Kingfisher12

      You are making the mistake in considering zero and infinitesimal to be the same. For all finite problems this is true, but for an infinite problem infinitesimal is greater than 0.

      Your other mistake is in considering ‘takes an infinite amount of time’ to be the same as ‘never’. The total time is unbounded so ‘never’ doesn’t apply here unless it is impossible. Since it is possible for the drop to reach any square on the board, it is necessary for every square on the board to be reached at some time.

  • thanks for this, tim – and it has me considering collective action, too. there’s a sort of “virtue of selfishness” theme bubbling for me. great discussion.

  • Reuben Hopper

    Pretty much every comment has been either a restatement of the conclusion, those who didn’t finish the post, and those who didn’t understand the post.

    • Yuri Matheus

      He posted this first yesterday without his answer, as to encourage us to think for ourselves before his opinion, so there’s a lot of older posts with dicussion.

  • Stijn

    I would just say, it’s better to have the sad stuff first and the happy stuff later. This way, you have progression which is good by itself.

    Also, what does an infinitely large board look like after an infinite amount of time?

  • xvorsx

    I think one of the tricks is in mathematical incorrectness of intention to place one human on the infinite board with equal probability to every cell.
    If such probability exist, it must be less then any Epsilon, so it should be 0 for every cell, which doesn’t make sense. Therefore, there is no correct probability distribution.

    I think Tim’s arguments still hold therefor and my choice is sad board in both cases.

  • Margling53

    I, for one, have had it up to here with these questions about omnipotent gods. Where does this obsession come from? Isn’t the very concept finite in itself? So therefore, who cares? We learn nothing about ourselves or anyone else from this kind of discussion.

    • Margling53

      Okay, Margling, since almost no one else will do so, I will stand with you. This discussion does not advance our collective understanding of life or love or culture or world politics or peace or war or international understanding or intergenerational conflict or human diversity or economic inequality or the fragility of the earth or exploration of the exosphere or anything else with an actual impact on our lives. It is, ipso facto, a waste of time. A puzzle now and then is good mental calisphenics, but discussing one into infinity is like a dog chasing its tail. Isn’t the symbol for infinity a drawing of that very doglike phenomenon?

  • I am still not convinced. The drop has infinite time to “reach” the infinite size of the board. The size of the drop is always limited at a specific time, but what about in “infinity time”? In the same way that there are always more and more people outside the drop, there is also always more and more time. Considering that the drop would, in a way, “cover” the infinite size of the board (since it will reach any point of it), wouldn’t a god see this and think of the drop as having the same infinite size as the board?

    But maybe I’m just talking nonsense… interesting thought experiment, at any rate.

    • String userName=new String();

      Problem is, since the drop is expanding at a constant rate, it will never reach the edge of the board. Time infinity can never be reached. Saying something will take an infinite amount of time is equivalent to saying it will never happen.

      • Scott

        Something increasing at constant rate does reach infinity (Sum of (1 + 1 + 1 … ) = infinity), which is why I think on either board you have an infinite number of both sad and happy people.

        • String userName=new String();

          Infinity isn’t a real number though, so it’s mathematically incorrect to say (1+1+1+…)=infinity. The infinite sum (1+1+1+…) diverges, which means it has no real value.

  • Ah.. I am late. I bookmarked this yesterday and now you have answered it. I have to say that I 100% agree with you.

    Serious answers aside, since I am an omnipotent god, can I just tell my boss the get the f off? :p

  • Raj Kulkarni

    I like you Tim, but this is the worst post here. Utter shit.

    • String userName=new String();

      What are you talking about? The apparent ethical paradox is going to keep me up at night for days.

  • Milokot

    There is another aspect of this problem – knowledge and it effect on present feeling. First checkerboard will suffer, but without point of reference may live miserable “normal” lives; If they know big picture hope can give them strength. Then, they will remember that relatively short time of suffering and enjoy that happiness even more. Second group, however, can think that happiness is something natural and only after drop catastrophe fully understood what they lost – and this memory will only make things worse. Or, if they know what will eventually come, they can fall into hedonism or depression…

    Why this sounds so much like “Heaven for poor, Hell for rich”?

  • Daniel Kranich

    Ok so here is what I see. In both boards, there will be an infinite number of both happy and sad people. The difference is what happens to the people who receive the drop.

    Humanity has the strange capacity to make nearly anything normal. If I lived for 10,000 years being blissfully happy, there would come a point in that 10,000 years when being blissfully happy is just normal. Its like how people describe heroin. Being high and happy simply becomes normal for them, and then when the high goes away… well if suddenly my “Normal” of blissfully happy was then changed to profoundly unhappy, All I would think about was how much more unhappy I am now compared to my previously happy normal. The only quantifiable change in my entire existence went from positive to negative.

    In the opposite scenario, if I was profoundly unhappy, after tens of thousands of years the feeling of being unhappy would simply become normal. Life would simply suck and that would be how life was. However if the drop then hit me and made my life full of blissful happiness, I would always carry that as a positive experience. I would always compare my life as being better than before, and I would live for eternity knowing that my life could be worse but isn’t anymore.

    Therefore I see the only logical choice to be the sad world with the drop of happy. In both worlds the overwhelming majority of people will simply be living the only way they have ever known. But in the world with the drop of happy, one day, your life will get better instead of worse, and THAT is what matters.

    • Scott

      You are describing the “hedonic treadmill”, which of course I learned from WBW.

  • Iambic Pentakill

    Regardless of position, pick the happy with a drop of sad. Infinity is mind bogglingly large.

    Think about it this way, regardless of the rate of change, after a Graham’s Number of millennia, you know what percentage of people will be sad? ESSENTIALLY ZERO.

    Even if you took a Graham’s Number to the Graham’s Number of millenia, less than one of every Graham’s Number to the Graham’s Number of people would be sad.


  • Andrew Byde

    I consider the problem to be badly formed, and the resulting head-spinning to be a common consequence of trying to get one’s head round infinity. In particular we don’t and couldn’t live forever, and have no way of imagining what that means, or reasoning about morality in that context. Likewise, trying to weigh up the suffering of infinite sets of infinite duration lives (“what proportion of people are happy / sad at any moment in time”) seems like folly. For instance, suppose I told you that in version a of this problem it takes Graham’s number of years for the drop to reach squares at distance 64 and similarly g_n years to reach squares at radius n; while in version b the drop spreads at the speed of light. Perhaps you, like me, intuitively feel that this makes the sad drop less of a bummer in version a, and the happy drop more compelling in version b. But of course these two versions are in fact identical as far as this question goes (and equally undefined, I would say).

  • Scott

    I made a post earlier, but I can’t help taking another stab because these infinities are mindfucking us all, including me.

    The issue is this hypo takes the concept of infinity and has us attempt to implement it in real life dimensions. Infinity is a mathematical concept that works well in math land, where it plays nicely in math equations, but the concept often (guarding against mathematicians out there) evaporates when you try to bring it into the real world. There are no two actual points in the universe that are an infinite distance apart.

    So before I continue my mind into the abyss of conceiving of infinity in real life, this is the question I have: In board 1, are any two squares an actual finite distance a part?
    If the answer is yes, that means that every person on board 1 will be a finite distance from the happy drop and will wait a finite amount of time until the happy drop gets to them. Since sadness will be a finite time followed by an infinite time of happiness, Board 1 would be my answers.

    If the answer is no, that means that there are a people, an infinite amount of people, that will start sad and NEVER become happy, regardless of the expanding happy drop, since they are an infinite distance (whatever that means in real life) away from the happy drop.

    I am confident that even an omnipotent god would be mindfucked by this hypothetical. Thanks for posting Tim.

    • Daniel Kranich

      Let’s say your second assumption is true and that there are an infinite number of people who will never see happiness in the happy drop board.

      Is an infinite number of sad people who will never be happy, worse or better than a finite (but nearly infinite) number of people who were once blissfully happy but are then damned to unhappiness for the rest of eternity?

      On the flip side is an infinite number of happy people better or worse than a nearly infinite number of sad people who will now be happy for the rest of eternity?

      It’s enough to make my head hurt.

      • Scott

        You bring up a good point about valuing the level of happiness vs. sadness, as well as the change from happy to sad. In my post I assume a utilitarian morality of one happy person + one sad person is a net of zero, regardless if they were changed or not.

        I think you stumbled on another interesting moral question:
        If you had, as an OG (omnipotent god), the choice of having nothing exist, or x-number of extremely miserable people and x-number of equal but opposite happy people exist, what would you choose?

  • Gokul

    Tim, If all humans now are in the ‘sweet spot’ warm period between two ice ages, then aren’t we in the happy board waiting for the drop of sadness to hit? (No infinities at play here, but timescales big enough)

  • Gokul

    Tim, If humans are now at the ‘sweet spot’ warm period between two ice ages, then aren’t we in the happy board waiting for the drop of sadness to hit?

  • Titus


    I’ve found a way to overcome this statement: “Because the second you’re assigned an actual square on the sad board, the amount of time you’ll remain sad is officially finite”.

    So it is true that at the moment you’re assigned a specific square your life is over.

    And so I ask you to assign me at the square of which the number is the sequence of the numbers in “π” (=3,14…..). ASSIGN THAT!

    …and now let me relax in my infinite hapiness board with my infinite mojito not waiting for the sadness drop to arrive…

    • Panda

      1. It was never stated that you as the human has the rights to pick your desired location on the board nor that the god has to listen to your request.
      2. Omnipotent –> being able to do anything. Not being able to deterministically define what human knows as π contradicts Omnipotent.
      So no you didn’t find the loophole, but nice try.

      • Titus

        ok on the omnipotence but anyway someone has to be assign to such a number.
        And since the board is infinite and it will take an infinity for the drop to expand everywhere (meaning never), infinite people will enjoy infinite mojitos to infinity. one of them is me because the probability of landing outside the circle is 100%.

        • Panda

          Again this assumes that an omnipotent god is also as clueless about transcendental numbers as we are, which you agreed that it is contradicting. Therefore, you can be assigned to any number we can/can’t comprehend. On the other hand, each and everyone can be assigned using the natural numbers which already hold under set theory to be countable anyway.
          Also, I wasn’t arguing against the case of infinity. I’m talking about the fact that you CAN be assigned to anything you can think of given the premises.

  • Bendariaku

    Personally, both answers would be the first board, the one with a board full of sadness, with a drop of happiness.

    I don’t want to be the guy that makes people sad while the rest are extremely happy. I want to at least make people feel happy after so much sadness, it would bring hope. And I would be okay if another god picked that. Feeling happy after sadness is a great thing. I’ve experienced it, and it feels so good. And I wish it lasted so long, but I can’t control that.

  • JW

    Time and space are not so different. They are just dimension, both of them.

    Why would a God be more concerned with infinity of space, then with infinity of time? The reason to pick the happy board in question one has a reasoning as follows: for every point in time, there are infinitely many people happy and finitely many suffering, so I do want this board.
    It might as well be: for every point in space (every square), there is infinite time suffering and finite time happiness, so I do want this board.

    There is no intrinsic reason to view time before space, or the other way around. Calling one of them the “logical” answer is not rational, and based on a mistake in reasoning.

    In the end, it is just comparing infinity to infinity, which, as often happens, results in a paradox. The question is, do you want to give all people a world they want to live in, or have a positive happy/balance at all times. I would prefer the first one, hence avoiding the question your friend asked: “How can it be morally right for god to create a world that no one would want to live in?”

    • tread

      This is the key to why I disagree with Tim on Question 2. You are forced to accept an infinity of space, since that’s in the question. However, even if the time never ends, you shouldn’t consider a time T=infinity, since that’ll never be reached.
      That being said, I can’t think of a way to explain this that is entirely convincing.

      • Kingfisher12

        But why should infinite space be accepted, but not infinite time? It is just another axis, and is implied in the word ‘forever’.

        The paradox is in comparing two infinities. The difference is that only the drop is unbounded everywhere for T = infinity, while the board is bounded by the drop at all T>0.

        • tread

          As I said, I don’t know quite enough maths to put this really rigorously, but forever can be handled fine by just considering an arbitrarily large finite time, and in general it is not considered valid to just substitute in infinity rather than taking limits with finite numbers. The exception here is that you cannot avoid the infinite space, because that is definitively in the question.

          • Kingfisher12

            I guess our disagreement is semantic, so there isn’t any remedy there. If ‘forever’ and ‘eternity’ are both defined as an arbitrarily large finite time then you are correct, if they are defined as the infinite passage of time then I would be.

  • realfastbill

    This is simple, it’s hope vs. despair. Happy drop, at least the people know happiness is coming instead of knowing that their happiness has a countdown.

    • girly freak

      You make this actually look like “Oh hey, I am sad now but I will be happy some time, when the drop reaches me. This makes me happy.” – But this is not the case. You will still be sad. There is no hope in those sad people as long as they are on a sad point of a board. And the happy people are really happy as long as they are on a happy point of a board. They would not think those negative “Oh, at any time I will be sad and this makes me sad”.

  • Kingfisher12

    I think Tim’s reasoning is sound, but I think there is an illogical premise in the setup.

    We cannot comprehend infinity in any form, almost by definition of infinite. But presumably a God capable of creating an infinite checkerboard would be capable of comprehending said infinity in ways we cannot even imagine.

    So there isn’t a divide between the God’s reasoning and the subject’s, because their separate concepts of infinity are different. To the subject it would be much better to be on the sad board for the reasons Tim states, but it would also be better from the God’s perspective where the infinite becomes finite.

    Like Zeno’s paradox of Achilles and the Tortoise, the seeming impossibility of infinite sums just sort of goes away. So from the god’s perspective, only the ultimate state of the board is of concern, even if that ultimate state is infinitely in the future. In the two scenarios, even though for any finite amount of time the state will be an infinite amount of the starting condition, at an infinite amount of time the state can only be the second one, even though like Achilles it is impossible to cross the line dividing the two states.

  • girly freak

    If I was an omnipotent god, I would create a third choice with a happy board without sad-drop.

    • girly freak

      Ok, I need to really answer the question, I think. But I start with Q2 because this is the easy one.

      Q2: As Tim has already pointed out, the logical answer is the sad board and I would absolutely agree to that. No need to go further in detail, because Tim already did.

      Q1: This is the REAL hard question. Of course there is that happy board, where you will always have more happy people than on the sad board. But if I would not choose it as one individual on the board, I would also not choose it as the god.

      But the problem with the eternity of space AND time here is that after Grahams Numbers years there will still be far more people be happy on the happy board than on the sad board. This actuall makes me think about Q2 again. Mind-Fucking.

      But actually my first answer (of creating a third choice without sadness) is the correct one. We are just thinking in a way an omnipotent god would not do.

  • JMBakker

    A moral decision should result in the same outcome in both question 1 and 2. How would you wish something upon others, if not on yourself?
    My answer would be 2, hands down. Considering the infinity of the board and the people, the Surface not covered with the sadness drop will always be bigger. Therefore, there will always be more people happy, and I would have a better oportunity to be happy untill eternity.
    I would say though, that I do not think one could truly feel an emotion without having experienced the oposite. I therefore asume that all the people on the board ‘know’ saddness and happyness, and they will not be in blissfull ignorance untill the happy drop rolls along.

  • HumanFarmer

    The phenomenon doesn’t occur in societies, since they aren’t infinite, neither do they exist forever.

    • mtjces

      Yeah I thought so too, but perhaps there’s something that is seemingly infinite or unfathomable. Couldn’t think of a real-world example of that yet..

      • JW

        Because there is no example. This paradox purely rests on the fact that you just cannot compare infinity to infinity. In real life, everything is ending and in principle comparible.

  • Brown

    If board is “can experience happiness” (life) and drop is “can’t experience happiness” (death). Then once born you have finite time you can experience happiness until death in which you have infinite time to not experience happiness. Assumes you don’t believe in after life.

  • Kingfisher12

    I think I can simplify this to answer the second question; the time ahead of a person on the board is infinite, while the time behind them is always finite. No matter how long a person has to wait for the drop, the amount of time beyond that point will be infinitely greater.

    This is a bit of a paradox if the wait time is infinite, but still in this case infinity >> infinity

  • Jake S

    It would really suck to be where the drop started on board 2.

    Then again, it would be infinitely improbable…

  • gatorallin

    You lost me at I am a God, and then have a boss… and then spend my time making infinite checkerboards. I make the rules, not follow them…… everyone is made happy… the End.

    Nothing really about moral decisions, rather how you understand infinity and the order of doing things.

  • Micha Köppl

    Happiness and sadness are subjective perceptions. And as you mentioned before happiness is equal to reality minus expectations, so lower your expectations to an infinite small amount and the problem is solved for yourself as an individual. 🙂

  • Florian

    Normally i love these kind of though experiments, but after this one I believe infinity is just a concept of human imagination (& math trick) and not physical reality => thought experiments with a lot of ifs that are physically impossible just seem a waste of time no? => doesn’t mean we can’t have thought experiments, should just be possible in this universe 🙂

    • Florian

      thought* experiments

  • Peter

    Board 2 because as the board is infinitely full of happy people with a drop of depression coming from the centre the numbers will always be higher in the area outside the drop no matter how large it gets

  • Neelesh Naik

    This isn’t really a mathematical or logical problem as much as it is an emotional one.

    The board with the happy drop has HOPE. Every person on that board who is sad knows that he/she won’t be sad forever and even though things maybe bad now, they will get better.

    On the other board it’s exactly the opposite, the board with the sad drop has DESPAIR. Every person knows that things will eventually get worse. So either they have turned sad, or will turn sad at some later time.

    So this isn’t really a choice between Happy or Sad but HOPE and DESPAIR.

    • This completely hits the nail on the head for me. It would never be an option for me, as a god or an individual, to choose the board that’s happy with a sad drop — as Neelesh said, it would be devoid of hope. People are pretty good at surviving horrible things when they know it’s temporary and there’s hope for it to get better.

      Also, consider the concept of FOREVER. It doesn’t really matter how long people on the sad board suffer waiting for the happy drop, because one way or another they’re waiting a finite amount of time. No length of suffering can compare with being happy FOREVER.

      • jonathan

        I just logged on to say this exact point. I just thought it would be important to have something to look forward to.

    • Adam

      This is a very nice point but it assumes that everyone on the board is aware of their situation. Are they told what’s happening to them? Can they see the drop coming? Can they communicate with each other? I was just viewing them as entities whose sole state of being is binary happy or sad, so the state of hope isn’t possible.

      • Neelesh Naik

        The second part of the question is clear about the fact that people know what the boards are about and they have an opinion about the choice made by the omnipotent god. So we must assume that they are capable of the full range of human feelings.
        If this was only a mathematical question, then the choice given to the board people doesn’t make sense. Where would “you” want to be?

  • lung’afa patrick

    I am of the view that let the god put me wherever he wishes. Then I will think.
    Have my answer from here,

  • Wiremu Hohepa

    -Oh, so on a happy board, there would be infinitely many people outside of the drop but zero people (or an undefined number) infinitely far away from the drop. It just depends on whether or not time is factored in. I still think that having infinitely many people defeats the phenomenon of time. Maybe you and your boss are standing at an edge of the endless board. Maybe it would be interesting if time was replaced with a third spatial dimension.

    • Wiremu Hohepa

      But then again, they could be all infinitely far (along in time) from the dot (If the dot moves practically infinitely slowly.)

  • Julian

    Brilliant thought experiment, but you fail to adress the infinity of time.
    You see, if the omnipotent God is trying to maximize happiness, and if we say that that means having most happiness not just at one (or any) instant, but over the entire arch of infinite time, both boards are equally viable picks. It is true that at any point in time, there will be more happy people on the happy board than the sad board. But it is also true that on any square, the amount (or duration) of happiness experienced on that square over the entire arch of (infinite) timewill be far smaller than the amount/duration of sadness. Which you also have to take into account as a God.
    (You could probably construct some geometrical argument based on the fact that you have two spatial dimensions and just one temporal one, and thus the spatial considerations must be weighed double, but I don’t know enough about mathematical infinities to know if that’s possible)

    To me personally it seems logical to choose the happy board as a God (but only if the abovementioned geometrical argument is possible) and the sad board as an individual (provided you will spend an infinite amount of years there, otherwise you should always go with the happy board)

  • wobster109

    Without reading any other answers — I’m really stuck as God because if you look at it from the perspective of a point vs an instant, it gives the opposite answer. Any point is always better off in option 1 because it’s sad for some finite time, and then happy for infinite time. But any instant is always better off in option 2 because it has finite sad people and infinite happy people. So which is the “right” way to look at it? By point, or by instant?

    Question 2 is very biased. If I’m a person on the board, then my perspective is as a point (as opposed to an instant), so I’d pick option 1. Any person would rather be sad some finite time and then happy forever. But asking a question like that is directly saying to use the point perspective.

    I remember in a math class, I was taught that when summing infinite terms, you could only rearrange/regroup if the sum converges. So, for example, look at the series 1-2+3-4+5-6. . . . This won’t ever converge. Instead it will swing between positive and negative, getting further from 0 as it goes. We could add it as is, taking every pair at a time, and so that’s (1-2)+(3-4)+(5-6)+ . . . = (-1)+(-1)+(-1)+ . . . = negative infinity. Or we could group differently, doing 1+(-2+3)+(-4+5)+(-6+7)+ . . . = 1+1+1+1+ . . . = positive infinity, and now it’s a different answer. I think that’s what’s happening here. Thinking about the chessboard as points vs. instants is grouping the total happiness in two different ways, but grouping isn’t allowed (because the sum doesn’t converge). Both sums are wrong, and total happiness is impossible to calculate.

    Based on that I have to say that as God I’m indifferent to the two options.

    • JW

      Very good explanation and good example 🙂
      Being indifferent does not get you very far. The fun part is: you get to choose which grouping you take. Time-perspective vs person-perspective. The “total happiness”-approach obviously does not work here, because comparing it boils down to comparing infinity to infinity, which does not make sense (as you explained very clearly). So, why not just create a world everyone wants to live in, instead of one no one wants to live in? I would go with the happy drop for sure.

  • Bradley

    I would choose the sadness with a drop of happiness because the people on the sadness board were born in sadness. They do not know what happiness is so to them life is just normal. Once they experience happiness for the first time it will last forever.

    If you were to choose the happiness board then you would be giving people happiness just to take it away. Those people would know what happiness was and life would suck forever once they lost it. That’s just cruel. That’s my opinion.

  • Sid

    If we assume the drop increases in radius at a fixed rate, the rate at which the plane is covered is the same as for a line in terms of fractions. At first I thought you might be able to get some difference in cardinality between the spatial side and the temporal side, but the grid part avoids that neatly. So the intuitive idea that every space will eventually be covered holds, unless there is some point where the drop’s expansion converges, which would imply that it would stop reaching new people after some time. (If you don’t put the people on a grid and instead put them on a point in real two-dimensional space with no distance constraints, you can have the drop converge somewhere while still continuing to reach new people, and the happy plane becomes vastly better).

    With that bit of mathematical silliness out of the way, I think that this thought experiment is problematic because of how it deals with people. First of all, if you say that someone who is rooted at one point indefinitely is happy, I don’t agree that you can call that a person. People get bored after some finite amount of time. So I propose that you consider the plane to be more like some kind of voodoo model that affects people that live in a world somewhere.

    Infinite lengths of time with respect to people are still problematic, I think. Certain other people have commented that people get used to time passing when enough of it passes, so waiting a trillion years of torment in order to be happy indefinitely afterwards would eventually feel like it wasn’t all that long. There’s a potential corollary: what if an infinite amount of time also doesn’t feel that long once you’ve spent a long enough part of it? I agree that for smaller numbers, 10 years of suffering and 100 years of happiness is very different from 10 years of happiness and 100 years of suffering, but 1,000,000,000 years of suffering and 10,000,000,000 years of happiness might be pretty much the same as 10,000,000,000 of suffering and 1,000,000,000 years of happiness. Once you run out of things you’re interested in doing, the actual length of time might be meaningless. You suffer for a while, and then you are happy for a while. When what you do is not linked to your happiness/suffering, you might actually get apathetic relatively quickly.

    I would go for the happy plane in both cases, because you spend the time before you get interminably bored enjoying yourself, and then you get to have those memories for the rest of the time. Generally, the happy time for any person would be incredibly large, so the likelihood of still caring at all by the time the sad drop reached you might be quite low. However, it could be critical that nobody was aware of the sad drop until it reached them, since dread of the drop could keep people sane.

  • Nicholas Scull

    I love these kinds of thought experiments. Math and philosophy converge nicely here. I’d have to say a truly benevolent god would have to pick the board where everyone starts off sad for two reasons, 1) doing it the other way is extremely cruel when you consider any point on the board, and 2) all people on the board will eventually be happy and stay that way for all eternity. I’ll explain, but first I’ll list some assumptions.

    I assume you are using the term “human” loosely, and that all these “people” perform only the two possible functions listed: to be perfectly happy or perfectly sad, with the exception that they do remember what it was like to be in the state they started off at.

    Ok, so keeping that in mind, starting everyone off as sad is better if the god is considering the perspective of any/all humans on the board and his goal is not as simple as just achieving the highest number of happy humans at any point in time, which the initially happy board would do. To me, that is not the most enlightened, thought out position for a god who understands his humans.

    For the initially sad board, all people on the board will eventually be happy, and they will remember what it was like to be sad. For the rest of eternity they can say “it sure is fantastic to not be sad anymore!!!” It might be Grahams number plus one centuries until they are happy, but infinity will eventually dwarf even that in comparison. If you do it the other way around, every single person on that board will eventually be sad for the rest of eternity, an infinitely larger amount of time than what they were happy for, and they will remember what it was like to be happy, which means they are most certainly worse off than a person who never even knew happiness to begin with, making their sadness even worse in comparison.

    I would have to say it is better to not condemn any person to infinite sadness and still have an infinite number off sad people in existence than to have a finite number of infinitely sad people who once knew happiness, and an infinite number of finitely happy people who will eventually join their numbers.

    This does get a little messy when you consider the people at r= infinity because their time in the initial state is also technically infinite, and then you get into competing infinities and whatnot, but dammit Jim, I’m an engineer, not a mathematician.

  • Priscilla Burns

    The problem with the god metaphor is it makes you look at individual existence statistically. But if you’re making decisions for individuals in favor of their mental state,
    you can’t look at it statistically. You have to assume the role of the
    individual. Mental state is an individual experience, not a communal one. It
    follows that we can only base our decision off empathizing with a random individual
    on either board. The choice then becomes a decision between two sets of roles.
    Both exist infinitely, but in changing frequencies.

    The happy with a drop of sad board leaves us with:
    Role 1: Happy with a drop of doom
    Role 2: Eternal misery

    The sad with a drop of happy board, however:
    Role 1: Eternal Happiness (Ideal role)
    Role 2: Misery with a drop of hope

    Both roles exist
    infinitely, so the volumes that they exist in are irrelevant. Unless the
    individual’s misery is actually worsened by the increasing numbers of other
    miserable people, everyone exists in one of two roles. Based on the assumption
    that most humans would value being happy over being sad, I would say the answer
    is reflective of the person’s mental time zone. A present-oriented might prefer
    the immediate gratification of landing on a happy square that will eventually
    fail them, while a future-oriented person might risk the pain in the knowledge
    that they can still have hope, which is a much more comfortable thought than
    impending doom. The decision someone makes is then completely reflective of
    their sense of time, as well as how strong the effects of hope are to their
    moral. I would even tentatively counter with the question: would you rather
    live in a world of slow growth or impending decay?

    I was also
    fascinated by the question, “How can it be morally right for got to create a
    world that no one would want to live in?” I would want to chew on it some more,
    but my immediate thought is that if we hypothesize the existence of a god, any
    sense of morality that would exist in the world he creates would be inherently
    dependent on him. So it MUST be morally right if the god did it. I would love
    more thoughts to chew on about that.

  • Alejandro Rojas

    You have to forget this is a circle. Think in 4-dimentional space, and the happyness/unhappiness shapes are cones. Then do your moral decision.

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