Question 20
An integer is initially written at each vertex of a cube. A move is to add $1$ to the numbers at two vertices connected by an edge. Is it possible to equalise the numbers by a series of moves in the following cases?
(1) The initial numbers are $0$, except for one vertex which is $1$.
(2) The initial numbers are $0$, except for two vertices which are $1$ and diagonally opposite on a face of the cube.
(3) Initially, the numbers going round the base are $1$, $2$, $3$, $4$. The corresponding vertices on the top are $6$, $7$, $4$, $5$ (with $6$ above the $1$, $7$ above the $2$ and so on).
(1) The initial numbers are $0$, except for one vertex which is $1$.
(2) The initial numbers are $0$, except for two vertices which are $1$ and diagonally opposite on a face of the cube.
(3) Initially, the numbers going round the base are $1$, $2$, $3$, $4$. The corresponding vertices on the top are $6$, $7$, $4$, $5$ (with $6$ above the $1$, $7$ above the $2$ and so on).