Counting the n-Chromos of I. J. Schoenberg

In a series of papers, I. J. Schoenberg [1 3] studied motions inside an n-dimensional cube with side length one, centered at the origin, of a point particle which moves like a billiard ball, that is, the particle always moves along a straight line until it encounters the boundary of the cube, when it is reflected in the usual way. Only non-trivial motions, that is, ones not contained in any hyperplane parallel to a face of the cube, are considered. In the second paper in the series, Schoenberg [2] considered the following: Problem. Find the largest n-dimensional open cube