Puzzle Party
Delicious N-Gony
This puzzle is an excellent example as to why mathematicians should have editors.
I first came across this problem in The Mathematics of Oz, in which Dorothy is asked to find the number of sections in a fully-connected tridecagon (the answer being 21,480 sections). Since I saw about three or four ways to solve this problem, I went ahead and presented it here. But, while difficult, at least some people enjoyed this puzzle thoroughly, so I don't consider it a loss. And, since I know some people must be curious, there is in fact an equation to find this number for any polygon. It's surprisingly short.
(n4 − 6n3 + 23n2 − 42n + 24)/24
+ (−5n3 + 42n2 − 40n − 48)/48⋅σ2(n)
− (3n/4)⋅σ4(n)
+ (−53n2 + 310n)/12⋅σ6(n)
+ (49n/2)⋅σ12(n)
+ 32n⋅σ18(n)
+ 19n⋅σ24(n)
− 36n⋅σ30(n)
− 50n⋅σ42(n)
− 190n⋅σ60(n)
− 78n⋅σ84(n)
− 48n⋅σ90(n)
− 78n⋅σ120(n)
− 48n⋅σ210(n)
Where σm(n) = 1 iff nmod m ≡ 0
And σm(n) = 0 otherwise
This gem of an equation was only proven in 1998 by Bjorn Poonen and Michael Rubinstein. Our Puzzle Party contains the cutting edge of mathematics!