Stick Figure Math

I’ll never forget the first time I saw the pattern

1, 2, 4, 8, 16, __

and was dumbfounded to learn that the missing value was 31, not 32, because the pattern was not meant to represent the powers of 2, but rather, the number of pieces into which a circle is divided if n points on its circumference are joined by chords. Known as Moser’s circle problem, it represents the inherent danger in making assumptions from a limited set of data.

Last night, my sons told me about the following problem, which they encountered on a recent math competition:

What number should replace the question mark?

Well, what say you? What number do you think should appear in the middle stick figure’s head?

Hold on, let me give you a hint. This problem appeared on a multiple-choice test, and these were the answer choices:

1. 3
2. 6
3. 9
4. 12

Now that you know one of those four numbers is supposed to be correct, does that change your answer? If you thought about it in the same way that the test designers intended it, then seeing the choices probably didn’t change your answer. But if you didn’t think about it that way and you put a little more effort into it, and you came up with something a bit more complicated — like I did — well, then, the answer choices may have thrown you for a loop, too, and made you slap your head and say, “WTF?”

For me, it was Moser’s circle problem all over again.

So, here’s where I need your help: I’d like to identify various patterns that could make any of those answers seem reasonable.

In addition, I’d also love to find a few other patterns that could make some answers other than the four given choices seem reasonable.

For instance, if the numbers in the limbs are a, b, c, and d, like this…

then the formula 8a – 4d gives 8 for the first and third figures’ heads and yields 8 × 6 – 4 × 9 = 12 as the answer, which happens to be one of the four answer choices.

Oh, wait… you’d don’t like that I didn’t use all four variables? Okay, that’s fair. So how about this instead: ‑3a + b + c + 2d, which also gives ‑3 × 6 + 7 + 5 + 2 × 9 = 12.

Willing to help? Post your pattern(s) in the comments.

[UPDATE (3/9/18): I sent a note to the contest organizers about this problem, and I got the following response this afternoon: “Thanks for your overall evaluation comments on [our] problems, and specifically for your input on the Stick Figure Problem. After careful consideration, we decided to give credit to every student for this question. Therefore, scores will be adjusted automatically.”]