Knight's Tours for Cubes and Boxes

摘要

The study of Hamiltonian tours of knights oa chessboards dates back more than a thousand years - and Euler found knight's tours as early as 1759 - and knight's tours have now been been fairly completely studied not only for rectangular boards but also for boards on other surfaces such as the cylinder, the torus, the Mobius strip, and the Klein bottle. In this paper we study knight's tours on the surface of cubes and boxes. In particular, we show that for any values of a, b, and c the box of dimension a x b x c has a knight's tour on its surface. We also introduce a 3-dimensional knight's move that is a generalization of the standard 2-dimensional knight's move and exhibit a Hamiltonian tour using this 3-dimensional knight's move that visits every cell of an 8 X 8 x 8 cube.