Consider a cube and let$ M, N$ be two of its vertices. Assign the number $1$ to these vertices and $0$ to the other six vertices. We are allowed to select a vertex and to increase with a unit the numbers assigned to the $3$ adjiacent vertices - call this a movement. Prove that there is a sequence of movements after which all the numbers assigned to the vertices of the cube became equal if and only if $MN$ is not a diagonal of a face of the cube. Marius Ghergu, Dinu Serbanescu