# 63-gon and the sum

by Numeriprimi
Tags: 63gon, geometry, product, side, smallest value, vertex
 P: 229 You have $d$ edges with different numbers on the vertices (product =-1) and $s$ edges with the same number on each vertex (product =1), and you're constrained to have - $s+d=63$; - $d$ an even number (because of the cyclical arrangement); - $d\leq s$. You want to make $s-d$ as small as possible, subject to the above constraints, i.e. $$\min_{d\in\{0,...,63\}} [(63-d) - d] \text{ subject to } d \text{ even}, \enspace d\leq 63 -d.$$ It's straightforward to check that the solution to the above problem is 3. Indeed, the constraints require that $d$ be an even number which is $\leq 31.5$, the largest such number being $30$. Since minimizing $63-2d$ just amounts to maximizing $d$, the above problem is solved by $d=30$, which yields $s-d=33-30=3$. If you want an example of a labeling that attains this, try labeling the vertices $((1,1,-1,-1)^{15},1,1,1)$.