Maarten ‘t Hart has sent me this lovely puzzle…..

Imagine there is a country with a lot of people. These people do not die, the people consists of monogamous families only, and there is no limit to the maximum amount of children each family can have. With every birth there is a 50% chance its a boy and a 50% chance it is a girl.  Every family wants to have one son: they get children until they give birth to a son, then they stop having children. This means that every family eventually has one father, one mother, one son and a variable number of daughters.  What percent of the children in that country are male?

As ever, please do NOT post your answers, but do say if you think you have solved the puzzle and how long it took.  Solution on Monday.

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.



  1. I think I’ve solved it. I’m also pretty sure there’s a trap I avoided falling into. It took around 10-20 seconds to solve in my head which probably means I’m way off target!

    1. Well, it strikes me as very trivial 😉
      I bet your quick answer is right. Not much to check, though. A random sequence does not change its properties because you take another coin each time you get tails.

    2. There’s an interesting real world issue in that in some conservative societies, there is a considerable surplus of boys over girls. Selective infanticide and abortions play a part (do a search on “gendercide”). However, it probably doesn’t require such extreme action to skew a population mix. So maybe this isn’t as theoretical a problem as it might appear.

    3. I disagree. Properties of a 1 dimension random walk are well known, if every birth is just a 50/50 draw, there is no way to keep the boys ratio on one side of the 50% limit.

    4. It’s only trivial of your are starting from a position of some known properties. From first principles it’s not.

      In any event, variations on this do not lead to the same answer. For instance, if the families give up trying after the fourth child it does affect the distribution. It would be fairly catastrophic for a society with deaths, but among a population of immortals it would not and would have the useful side effect of limiting the population to a finite number of individuals.

    5. I don’t really understand everything in this last comment, but what I know for sure is that when you say “For instance, if the families give up trying after the fourth child it does affect the distribution”, you are mistaken. Just do the maths or try it with a spreadsheet.

      zatytom’s variant would affect the distribution, but only if you consider the first generation (or any generation separately), because it allows you to catch a moment where the ratio is over 50%. On the long term there is nothing you can do to prevent the ratio from converging to 50%.

    6. I have done exactly that modeling and if everybody gives up trying to have a boy after the first four are all girls, then the distribution is different.

      Indeed, try the example where they give up after just two if a boy hasn’t been produced. It’s a very different result to if they go onto infinity.

    7. Ok.

      1/2 of the families will have a boy immediately.
      1/4 of the families will have a girl, then a boy.
      1/4 of the families will have two girls.

      Number of boys per family : 1/2 + 1/4 + 0/4 = 3/4
      Number of girls per family : 0/2 + 1/4 + 2/4 = 3/4

      With 4 tries, I get
      1/2 + 1/4 + 1/8 + 1/16 + 0/16 = 15/16 boys
      0/2 + 1/4 + 2/8 + 3/16 + 4/16 = 15/16 girls

      If they go onto infinity, quick excel cheat shows it converges to 1 boy and 1 girl per family.

      Please tell me where I am wrong.

    8. This is right, I am sorry. I was giving my solutions to a variant of the problem and did not realize the infinity case was actually the original problem.

      It would be fine if we could edit our post.

    9. Ok – have to admit to an error in my modelling sheet. It was a misalignment, that didn’t matter on convergence to infinity, but does for the individual case. If I’d just run through some actual numbers, I’d have seen my mistake in laying out the sheet.

  2. Immediately, instantaneously, except for the time it took for my eyes to read it, my brain to process the image, and for my natural cautiousness to wonder why you are asking such a simple question.

  3. I followed the population of potential offspring, giving that each had a 50:50 chance of having a boy. The result is uneasing, but it took me one minute to get the answer. I think!

  4. My answer seems to require that some children are half boy and half girl. To quote Fagin via Bart “I think I’d better think of that again”.

  5. I have seen this puzzle before, and like it. A possibly more-challenging version is what would happen if each family stopped having children when they had more boys than girls (or vice-versa.)

    1. Your version seems interesting. I guess the one dimension random walk thing implies that each family will eventually stop having children, and the total population will actually have more boys than girls. This may just be an extension of this random walk to the whole population, if the boys and girls never grow up to procreate.

    2. We have to assume that the children themselves do not start breeding, otherwise that provides an ever-growing population of new familes embarking on procreation, who outnumber and outweigh those who have stopped breeding, and who therefore allow the 50/50 generational distribution to persist.

      But suppose only the original population breed, the children themselves never start breeding, not an unreasonable assumption for an immortal population. Eventually only a small number of people will be still having children. With only a small numbers breeding, substantial deviations from 50/50 distribution will at random occur, and eventually everyone will stop breeding with more boys than girls. If you have random fluctuations, and selectively do something different in those fluctuations, then you can seemingly defeat the random result – consider card-counting Blackjack playing strategies, where you bet more when the odds are temporarily skewed because the pack has become skewed.

    3. [POTENTIAL SPOILER ALERT] No decision about when to stop having children can have any influence on the boy-girl ratio of children already born. The answer is the same as the original puzzle.

      The blackjack analogy fails because, in blackjack, cards played change the odds of what will happen with the unplayed cards (until the next shuffle). No such skewing occurs as children are born–the odds of each birth being a boy always remains 50-50.

  6. The logic here means that there are a small number of families who are gonna need a pretty sizeable house with LOTS of bathrooms..!

  7. Wow, that was an obvious one.
    I guess we are supposed to fall in some kind of trap, maybe try to work out the distribution of each family size, things like that ?

  8. I read this puzzle as having the three following assumptions:

    1) The population of the country at generation 0 is at least in the hundreds, preferably in the thousands.
    2) The men:women ratio at generation 0 is 1:1, or something very close to that. If there was any significant skew in either direction, its effect would propagate through several generations.
    3) Once people start having children, we don’t measure the men:women ratio at generation 1, but rather at generation 50, 100, 500, or some other suitably large number.

    If we agree that this much is implicit in the puzzle, then ritesh is correct: this is not a puzzle, it is a question of probability distribution. Please check your statistics textbooks for an answer.

    1. Considering most people here seem to think we need math to calculate the answer whereas it is totally trivial, it obviously IS a puzzle. And a good one. Too bad it did not work on me, I love to be told an obvious answer when I have spent some time with complex reasoning.

  9. Obviously all families will, once they’ve stopped breeding, have one more boy than girls.

    But that means not all those boys will have a girl to breed with in the next generation.

    But still, the total pop in the next generation will have one more boy per family, plus all the non-breeding uncles.

    Hard to see how to make that a percentage though.

    1. This is about zatytom’s variant, right ?

      To reach that conclusion you need to separate generations (no girl or boy from generation n is allowed to procreate before every family of generation n-1 has stopped), and count the population between generations only. In that case you will just observe the state of the random walk only at times when it is guaranteed that the ratio of boys is over 50%.

      In a continuous situation where generations are not separated or you count the population at any time, then there will be non finished families with more girls than boys. The ratio will converge to 50% no matter what.

  10. I cheated with Excel, and was a bit surprised by the answer. I’ll be keenly looking forward to how I was meant to be thinking about the answer.

  11. It took less than a second to remember that I’ve seen the puzzle before.

    The puzzle is a bit like the classic ‘fly flying between two cars’ puzzle — — in that you can either sum the infinite series, John von Neumann-style, or spot a simpler solution. 🙂

    You get the same answer if you say that each family continues having children until either (a) they have a son, or (b) they are no longer able to have children.

  12. Elegantly smart, surprisingly simple!

    Extremely easy to solve, if you approach it in the right way. No knowledge of probability theory required! You will know it’s the right answer once you find it…

    Thanks, Mr. Wiseman!

  13. I think i had the solution at the third approach… took me 10 minutes before i saw the solution… In an ex-post consideration the question for the average amount of childen would have been much more challenging…

  14. I got an answer instinctively as I read the question and after a minute thinking about it, I reckon it’s right. I don’t have paper or pen to hand to verify it though.

  15. No need for pen and paper! No need for complex mathematics! The answer is extremely simple and can be found almost instantly… A careful read and common sense is all that’s needed.

  16. Got an answer fairly quickly.
    Was then going to post a clever comment about twins to display my undoubted genius when I realised that my answer was wrong.
    Went back to answer the question in a simple straightforward way – modelling a result – none of this fancy probability lark.
    Now confident I have a sensible answer

  17. I got it straight away, but I keep coming back to it worrying that I have missed something… this could ruin my weekend!

    1. No, they never give up, even after a million daughters: they’re immortals, remember! 🙂 But actually, the answer would be the same even if they did give up.

  18. I’ve seen discussions of this on the internet recently, so I’m corrupted. It took me a minute to see how the specifications of this variant play out.

    1. Oh, and I’m assuming that each family is complete (i.e., they have their son and are done having children), since otherwise we’d need to know what percentage of the families are still working to produce a son.

  19. I think some people are overthinking this.

    Every family is guarantted to have two boys (the father and the son they’ve striven for).

    Yet each family is only guaranteed to have one girl (the mother), though they may have more.

    So it is twice as likely to be boys than girls. The answer follows from that.

    1. But the question is about the percentage of *children*, so since each family is guaranteed one son but no daughters, the answer is…?

  20. That is most hard to answer Nick. I can see that some families will have many girls children.

    But none will have more than 24 girl children, and perhaps less.

    After 24 the youngest girl is an adult. The wiseman question is to count the childen, so children who have become adults no longer count.

    So, on birth of guaranteed boy, families will have between 0 and 23 girl children, no more.

    1. I think here “children” means “sons or daughters”, so there is no age cutoff. As Richard says: “This means that every family eventually has one father, one mother, one son and a variable number of daughters.”

      But if there were a cutoff, it wouldn’t change the answer.

  21. This is a great. It is non-obvious but little excel table solves the problem (not sure if this is cheating or not but … ).

    This actually gives a hint to solution on bigger question “Why more male babies are born after great wars”. There is some statistical evidence of this but I haven’t found obvious answer in 5 min google search.

    This is worth investigating a bit more …

  22. An interesting problem. My intuition gave me a pretty quick answer. Proving it took about five to ten minutes of remembering how to deal with infinite series and, in particular, geometric progressions.

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