Each of the $8$ vertices of a given cube is given a value $1$ or $-1$. Each of the $6$ faces is given the value of product of its four vertices. Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$?
Problem
Source: Norwegian Mathematical Olympiad 1993 - Abel Competition p4
Tags: cube, combinatorics, combinatorial geometry, Sum, Product