# It’s the Friday Puzzle….

73

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.

## 73 comments on “It’s the Friday Puzzle….”

1. Immediately, but I’d seen it before. No, I didn’t get it that time.

2. Liam says:

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!

3. Steve Jones says:

This problem strikes me as non-trivial.

• Steve Jones says:

Having said that, I have a quick answer. I’ll have to check my intuition later.

• Yat says:

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.

• Steve Jones says:

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.

• Yat says:

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.

• Steve Jones says:

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.

• Yat says:

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%.

• Steve Jones says:

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.

• Yat says:

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.

• M says:

This is getting close to a spoiler…

• Yat says:

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.

• Steve Jones says:

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.

4. John Loony says:

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.

5. Lesliet says:

Seems too easy. I must be missing something. I had a picture in my mind in about 15 seconds.

6. Seen it before, but I do like this one

7. ritesh says:

This is not a puzzle but a question on probability distribution. PS I do not do homeworks and this puzzle sucks.

• Yat says:

You are gonna love the answer on monday. 😉

8. Nigel Alexander says:

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!

9. JimC says:

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”.

10. Moray says:

The answer is “What is China”

11. Edgar says:

I think this is the first time I’ve been bored with a puzzle.

12. Charles Sullivan says:

When people don’t die that means they live eternally, right?

13. zatytom says:

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.)

• Yat says:

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.

• ivan says:

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.

• Ken Haley says:

[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.

14. Tony says:

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

15. Yat says:

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 ?

16. There is one word in the puzzle I’m not sure how to interpret…

17. Luis says:

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.

• Yat says:

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.

18. Dharmaruci says:

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.

• Yat says:

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.

19. Gib says:

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.

20. Nick says:

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 — https://richardwiseman.wordpress.com/2010/12/03/its-the-friday-puzzle-88/ — 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.

21. Costas Vlachos says:

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!

22. Berhard says:

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…

23. Lazy T says:

I think I got it, whilst reaching for a pen.
Reading the comments and I think Yat’s telling me I’m right, ta Yat.

24. AMWhy says:

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.

25. Paul Durrant says:

I have the answer. I got the answer immediately, but then had to spend a few minutes making sure.

26. Costas Vlachos says:

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.

27. Susqueda says:

If, for whatever reason, the children’s age in the final question need not mentioned then the answer is immediate.

28. 10 seconds wondering how the series would converge, 5 seconds wondering if there’s an easier way and finding it, 10 more seconds to convince myself the two answers were the same. Nice.

29. One Eyed Jack says:

30. You Tube Wannabe says:

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

31. Two minutes- couldn’t express it mathematically but drew a logic tree and went down each path applying the 50% rule in the question. I get the same answer no matter how many levels I cascade down through.

32. Duncan says:

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

33. Martha says:

It depends how much they want that one son. After 4 or 5 daughters, most of them’d give up, right?

• Nick says:

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.

34. physicalist says:

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.

• physicalist says:

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.

• physicalist says:

Or maybe we don’t. Curses!

35. fluffy says:

Feels like a trick question. I think I got the answer right away.

36. Dharmaruci says:

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.

• Nick says:

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

• You Tube Wannabe says:

The question is about children not percentages

• You Tube Wannabe says:

I meant to say the question is about children not families

37. Dharmaruci says:

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.

• Nick says:

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.

38. Emir Mulabeg says:

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 …

39. Jeez! “People” in the sense used is plural, i.e., “the people consist.” “Amount” of children? How about “number?” And some sexism thrown in, to boot? Oh, Richard!

• Maarten says:

Sorry about my English… I’m not a native English speaker

• Maarten says:

But admit: There are hardly any interpetation problems this time.

40. z says:

Guess there’s an implicit assumption that everyone’s heterosexual here in the puzzle, right?

• rolyh says:

actually, does not matter. Just like some other sentences in the problem dont matter. Wait n see

41. John Loony says:

The puzzle has six sentences. Only the third sentence is relevant. The rest is distraction.

42. Anonymous says:

Consider only one-second-old children at a given moment; there will be n births, of which….

43. Thanks for game. It will give good fun and learned lot of tricks to play this game and addicted to this

You can also play more puzlles games clickazoid puzzle games

44. Jerry says:

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.

45. I believe this is an absolutely converging series. I don’t know how to solve that, so I’ll sit back with a beverage and wait for others to do so.

46. Michelle V. says:

Is that THE Maarten ‘t Hart, my favourite Dutch writer?