How many ways are there to arrange the numbers $1$–$10$ in a circle such that each number is not greater than the sum of the two numbers adjacent to it?
My Progress: Not much really. When I can solve a problem, I can solve it, and if I can't, it's usually because I don't know how to start.
I thought a little bit about what would be an automatic no.
10 can't have two numbers less than 6 next to it. 1 and 2 can't have a number in between them unless that number is 3.
etc... Clearly this doesn't work. I need to find a way to just straight up count the ones that work and not subtract the ones that don't work.
So I looked at recursion.
So for 1 there is 1 way For 2 there is 1 way (rotation) For 3 there is 2 ways: 1,2,3 and 1,3,2 For 4 there are however many there are such that 4 isn't between 1 and 2.
But past that it already gets very messy. Does anybody have a slick solution?