Let M be a set of six distinct positive integers whose sum is 60. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding 1 to the numbers on those faces. Determine the number of sets M for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number. geometry3D geometrycombinatorics unsolvedcombinatorics