Problem:
a. A number is written at each edge of a cube. The cube is called magic if:
(i) For every face, the four edges around it have the same sum.
(ii) For every vertex, the three edges meeting at it have the same sum. (The face and vertex sums may be different.)
Determine if there exists a magic cube using
(a) the numbers 1 through 12, each only once;
(b) the numbers from 1 through 13, each no more than once.
Solution:
(a) The answer is no. If the cube is magic, then every triple of edges abutting a vertex has of course the same average. Since every edge belongs to the same number of vertices (two), these vertex averages are the same as the average of all the numbers on the cube, which is . But it is impossible for three numbers to have an average of .
(b) The answer is yes.
On the right we have subtracted from each number to clarify the structure of the solution. Every face and vertex adds to , as does each pair of opposite edges.