IMO 1967 LL USS54

Origin: USS

Problem

Is it possible to put 100 (or 200) points on a wooden cube such that by all rotations of the cube the points map into themselves? Justify your answer.

Solution

Let S be the given set of points on the cube. Let x, y, z denote the numbers of points from S lying at a vertex, at the midpoint of an edge, at the midpoint of a face of the cube, respectively, and let u be the number of all other points from S. Either there are no points from S at the vertices of the cube, or there is a point from S at each vertex. Hence x is either 0 or 8. Similarly, y is either 0 or 12, and z is either 0 or 6. Any other point of S has 24 possible images under rotations of the cube. Hence u is divisible by 24. Since n = x + y + z + u and 6 | y, z, u, it follows that either 6 | n or 6 | n −8, i.e., n \equiv0 or n \equiv2 (mod 6). Thus n = 200 is possible, while n = 100 is not, because n \equiv4 (mod 6).