A polygon puzzle that really isn’t
June 17, 2025 at 9:13 PM by Dr. Drang
This is another post about a puzzle in Scientific American. I confess that this and my previous post have just been placeholders, things that I’m putting here because the post I really want to write is giving me trouble. It started when I read this article in Ars Technica about dropping eggs. The more I thought about it—and the paper it’s based on—the more I felt I should say, and now I have a couple of weeks’s worth of notes and calculations that I’m struggling to organize. Posts like this are much easier to write, so here we are.
The puzzle is this one. There are eight regular polygons with increasing numbers of sides, triangle through decagon. The first seven have numbers in them, and you’re supposed to find the number that goes in the last one.
Because of the house of cards puzzle I discussed several months ago, I decided to set up a difference table, like this:
The numbers in the difference column are obviously a series of prime numbers, so I figured the next difference would be the next prime, 29, and therefore the missing number would be 129. But I had no clue as to what that had to do with polygons.
It turns out that the number for the triangle is the sum of the first three prime numbers, the number for the square is the sum of the first four prime numbers, and so on. The (slight) geometric aspect of the problem is the number of sides being the number of prime numbers you should add. This is what makes the difference table work out the way it does and how I got the right answer without really solving the problem.
If you’re wondering, yes, this sequence is in the Online Encyclopedia of Integer Sequences: A007504. Mathematicians really like their prime numbers.