I recently came across a riddle:

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Guess and test

Delightful! That's enough information to answer the question? I'd have expected more rules.

Well, it's delightful if you don't think too hard about the effects this might have on an interviewee that has experienced e.g. the one-child policy.

Maybe the wording "everybody wants to have a son" biased me, but I am pretty sure that such a country would have more sons than daughters. I am also skeptical of my intuition, since Ellenberg's bread and butter is making you confident you understand something, and then pulling the rug out from under you. He's basically a bully, but we all have Stockholm syndrome.

So let's do some mental math: if there are 100 families in this country. 50 of them will have a boy and no girls. Another 25 will have one of each. And there we are: 75 boys to 25 girls, and only 25 families left to consider, and all of them will have boys too. It seems difficult for the girls to make a comeback. That's the feeling.

Next, we can approximate the answer with a calculator. Since all families have a boy (it seems like people generally ignore the possibility of families that have daughters, but haven't yet conceived a son), we can say that there is 1 boy per family. Look at us. We're doing math. Girls are a tiny bit more complicated. One quarter of families have one girl, an eighth have two girls, and so on.

Whoa! They are making a comeback, and we're only 9 terms into this infinite sum. I renounce my intuition and would now guess that somehow, the amount of girls precisely equals the amount of boys in the country. Well, on average.

But why? How?

Since we're here, this seems like a reasonable time to stop guessing and "do math".

The amount of boys per family is 1 by definition, but since we're doing math, and we wouldn't want someone to accidentally understand us, let us obfuscate:

And here comes the magic. We can multiply every term in the sequence by 2:

We can see that the nth element of equals the (n+1)th element of .

And being a little clever, we eliminate all those pesky infinite terms:

So now we're back where we started, with the added benefit of alienating some people. Let's keep going; maybe it'll be useful.

We can use the exact same reasoning as above to get a more general formula for the sum of an infinite sum:

so long as the series converges, it'll converge to this.

Now let's formalize the amount of girls in this country:

This sum has a slightly different form, so we can't use the general formula we just came up with. But, you might notice that , which does match the form of .

So let's follow that clue:

And shuffling some things around:

So here we are. We've been good. We didn't just take formulas from an oracle and plug things in; we derived them. I feel confident that the answer is correct, and that I would get a ✓ if this were math homework. Still, doing all this symbol manipulation brings me no intuitive satisfaction as to why or how those two infinite sequences converge to the same value.

Let's try something less fancy.

But why? How? (Round 2)

Let's try looking at the values of . Remember that

If we think about it, we can show that each term

So that's how I'll calculate each line below: take the previous line, divide by 2, and add the next power of 1/2:

And now we stare.

Somehow when we sum all the lines together, (in the reduced form) there are exactly two 1/4s, adding up to 1/2, two 1/8s, adding up to 1/4, two 1/16s, just one 1/32, but three 1/64s to make up for it.

I see how this is mimicking , and I get another little kick of dopamine from my brain, having somehow connected two seemingly unconnected things. But for all that, I just don't see the pattern. I see that it works, but I don't understand why it must.

But why? How? (Round 3)

Let's just start drawing things?

I began drawing by hand, but quickly ran into the limits of my ability to draw straight lines. So, not unlike Donald Knuth, I made playbox to generate the box images below.

Starting basic

As before, let's calibrate this tool on something relatively simple (population of boys), before we move on to the more advanced.

Recall that the number of boys per family can be expressed as So if we were to draw a square, we could show it filling up by shading in 1/2 of it, then another 1/4, then 1/8, and so on:

Each subsequent box has half the height of the previous, hence half the area. And still, we see the box filling up to the point that drawing additional boxes is difficult even with computer assistance.

Here's another equally valid way to visualize :

Here, we alternating halving the height and then width of the subsequent boxes, but the effect is the same: the whole box is filled with color, and we can wave our hands and claim this means the sum of the series approaches 1.

Something a teeny bit different

For comparison, let's try visualizing an infinite sum that doesn't converge to 1. That is to say, something that won't fill the box up all the way:

Now we'll want each subsequent box to have 1/4 the area of the preceding one.

I'm not totally sure what to make of that, and I think we could make a prettier visualization. Let's try:

At this point, I notice that besides being prettier, this second visualization of our series is a perfect complement to our first one. Well, almost perfect. We'd need a third to completely fill up the box.

Each of the three colors in the above image represents our sum .

And 3 of them fill up the box: .

Therefore .

My intuition is appeased. This makes sense.

Another way

Before we move on, there's another intuitive way to show that .

Because the images we are using to visualize are self-similar, we can do a little bit of magic on them. Notice that the bottom-right quarter of the above image (and, of each of our other visualizations of ) has exactly the same shape as the whole square, just shrunk by a factor of 4. In fact, we could refer to the bottom-right quarter as .

Once we've done that, we can get rid of all those infinite boxes. If we subtract the areas shaded in the bottom-right quarter from the whole square, the only thing left shaded is the top-right quarter. Its area is 1/4.

This is a visual representation of the "magic" we performed in the first But why? How? section.

The main event

But enough diversions. We are here to figure out why / how

So let's draw it:

It's pretty. And it seems to be filling up the box? Well, except for that top-right corner. What's going on there?

The shape doesn't have any obvious complements, so we can't rely on the method we found in Something a teeny bit different. But it is self-similar!

Just as in Another way, we can see that the top-right quarter of our box is exactly the same as the entire box, just shrunk by a factor of 4. And the top-left quarter looks just like , (which we've already shown to equal 1) shrunk by a factor of 4. Miracles!

The bottom-right quarter is exactly the same as the top-left, just rotated 90 degrees clockwise. The last bottom-left quarter is shaded in completely, so its area is 1/4.

This should be enough. Let's obfuscate!

The entire box is made up of 4 quarters:

And they sum up to :

🎉🎉🎉🎉 We did it! We drew a picture. And it turned out to be made of parts that were either infinite sums whose values we'd already calculated, or itself shrunken down.

Now I can stare at this picture and say "Well, that quarter is fully shaded, so I know its value. These two quarters are clearly converging to filling up each of their quarters, and this last one is self-similar to the whole square".

I'm still delighted by the puzzle, though (because?) it remains counterintuitive. But I now feel like I kind of understand how the girls' population catches up. And I added a couple more arcane spells to my collection. It's been a good week.

Epilogue

Something exciting we could have done is take the intuition we gained from visualizing how the girls' population converges to 1, and then formalize it.

I say this is exciting not because formalizing things is fun, but because it would demonstrate how you can use visualization to improve your understanding, which is then translatable even to formal contexts. Visualization is not merely a toy. It can be as valid a medium of thought as is conventional symbolic math.

There's a lot more we didn't cover here. Why does the top-right corner have so much more unshaded space than previous visualizations? Does the answer to this question depend on the population of the country? What about the case where families have conceived girls, but hadn't gotten to a boy? (I'd say that the main reason mathematicians find this riddle so contentious is this detail).

Sometimes we have to be content with what is, and not dwell on what isn't. By which I mean creating just this writeup took enough time and effort that I'm feeling pretty done with this puzzle.

Thanks for reading!

Thank You

... to leegao/readme2tex, because github doesn't natively support latex in readmes.