The Riddle of the Sons

I enjoy a good riddle. Riddles are great because they give you a chance to think creatively around a problem and come to your own conclusion, and then ideally compare notes with other people. The point in riddles, as I see it, it not to get a definitively right answer insomuch as to get any answer which fits strictly within the parameters defined by the riddle. Any valid answer is fair game, and the one you come up with says something about who you are and what kinds of implicit assumptions you make.

So… have a go at this one. See what you make of it, and in just a moment we’re going to talk about how adding multiple choice answers fundamentally changes the question.

Assume there is a 50:50 chance of any child being male or female.

Now assume four generations, all other things being equal.

What are the odds of a son being a son of a son of a son?

David Allen Green

logic_boat

By Randall Munroe, licensed under Creative Commons 2.5

What did you get? 50 %? 20 %? 6.25 %? Very interesting, that’s not what I got! But before I talk about my solution and what I think the riddle reveals about people, let’s talk about the bit I cut off above. The riddle was a poll, and the poll gave four possible answers:

(i) 6.25 %

(ii) 12.5 %

(iii) 25 %

(iv) 100 %

What do you think, now? Did that change your answer? Funny, isn’t it? When you don’t know what constitutes “possible” answers, you get to think more creatively about the problem. You think for yourself. So, step through the problem and explain why the multiple choice changes the riddle itself.

The riddle asks you to identify whether if “a son” is selected, are they “the son of a son of a son”? Well, some people read this and think it’s asking whether “a person” is “the son of a son of a son” (let’s just shorten that to “a great grandson”, shall we?). A person is a son only 50% of the time (per the question), so you’d think it’s 50%. Is it? No! Because you’re not asked about any old person – you’re told that the person is a son. The 50 % gender balance is a red herring.

So, if a person you know is male then they are a son. If they are a son, their father had a father… and so on. They will inevitably be a great grandson. So, since that is a possible answer, is it the answer?

whoaresons

The males are in pink, obviously.

Well, 100% is definitely the intended answer. I have to concede that. The riddle was supposed to be making a point about how good people are at reading and understanding the question – spotting the fact that “a son” is the subject of the question and inferring from there.

But we’re not here to find the intended answer – we’re here to find an answer which follows the rules, so we need to check whether that is consistent with the rules.

The second rule is that there are exactly four generations. Exactly. Not “at least”. There is a generation 1, which did not have parents because there was no preceding generation. Those are the parameters of the riddle – remember that it’s made up, it  doesn’t have to actually correspond to real life. So generation 1’s males are not sons. Does this matter?

It matters a lot! If Gen 1’s males aren’t sons, then their sons (males in Gen 2) are not sons of sons (grandsons), they are only sons. Gen 3 are grandsons (sons of sons) but not great grandsons. Only Gen 4 are great grandsons.

not_great_grandsons

So, if you pick a son at random, what is the probability that he is a grandson? That is the same as asking: If I pick a male from generations 2, 3, & 4, what is the probability that male is from generation 4?

The answer is found by adding up all the males in generation 4 (n4), and dividing by the number of males in generations 2-4 (n2 + n3 + n4). Generation 1 doesn’t matter at all, because they aren’t sons. So now you have to know what the distribution of population is throughout the generations. But wait! Before we do that, let’s point something out – it doesn’t matter what the actual values are, as long as there is at least 1 male in generation 2 and generation 3 combined, the number can’t ever be 100 %.

eqn1

eqn2

So, as long as the only thing which produces a generation is the generation before it (an assumption you could lift if you wished), the intended answer is necessarily wrong. The rules of the riddle implicitly rule this out. What should the correct answer be? Well, since the riddle states that “all other things are equal” then there is the same number of people in each generation. On the average, you’d expect the number of males in each generation to be about the same – so the answer turns out to be 33.33 (repeating, of course) percent. This is not a “possible” answer, and I think that means that the person setting the problem didn’t solve the riddle for themselves before deciding on the answer.

eqn3

No, I think that the question was designed with the intended answer in mind. Of course, David Allen Green was trying to make a point about reading comprehension, and he came up with a cute device to force his audience to read carefully. The problem is, if you’re going to make a point like that then you need to be really sure that you’ve got a definitively correct answer. It would only need a very small change to the wording (“at least four generations”) and David’s intended answer is perfect!

But by setting the answer before he worked through the question, David showed us something else. Nobody could have voted for what I’ve established is a totally reasonable solution to the problem, which means that in my opinion everyone who answered the question was wrong. Including myself. It just goes to show that when you set multiple choice questions, what you gain in quantifiable data could lose you an accurate appreciation of the truth.

brexit_pollcards

That applies to more than just riddles – good survey design is difficult for precisely this reason, and skilled statisticians know how to avoid (or exploit) this implicit bias. You only need look as far as the likes of the Daily Mail or Fox News to see that in action. This kind of implicit bias affects how public opinion is expressed, which ultimately affects what laws get passed. You, as a political agent, have been affected by this kind of implicit bias, and it has often been deliberately used to mislead you.

If only there were a topical example where the British public had been forced into a constrained choice which ended up massively changing public policy, and society, as a result. Wouldn’t that be a convenient comparison to draw here!

blue and yellow round star print textile

Photo by freestocks.org on Pexels.com

About stoove

A physicist, researcher, and gamesman. Likes to think about the mathematics and mechanics behind all sorts of different things, and writing up the thoughts for you to read. A competent programmer, enjoys public speaking and mechanical keyboards. Has opinions which might even change from time to time.
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