Knight's moves, graphs, and lattices (Q1326526)

After generalizing the knight's move in chess to \(n\) dimensions and to arbitrary integers \(k_ 1, \dots, k_ n\) as ``step sizes'' (instead of 2 and 1), the author determines which pairs of points in \(\mathbb{Z}^ n\) are mutually accessible to such a knight and gives an algorithm for finding a path from one point to an accessible one. To do this \(\mathbb{Z}^ n\) is regarded as a graph and as a module for the Weyl group of type \(B_ n\), consisting of the signed permutation matrices of König's theorem on sums of permutation matrices is proven.