Snake-in-the-box codes for dimension 7

Abstract

In the n-dimensional hypercube, an n-snake is a simple path with no chords, while an n-coil is a simple cycle without chords. There has been much interest in determining the length of a maximum n-snake and a maximum n-coil. Only upper and lower bounds for these maximum lengths are known for arbitrary n. Computationally, the problem of nding maximum n-snakes and n-coils su ers from combinatorial explosion, in that the size of the solution space which mustbesearched grows very rapidly as n increases. Previously, the maximum lengths of n-snakes and n-coils have been established only for n 7 and n 6, respectively. In this paper, we report on a coil searching computer program which established that 48 is the maximum length of a coil in the hypercube of dimension 7.