Pebbling in Hypercubes

This paper considers the following game on a hypercube, first suggested by Lagarias and Saks. Suppose 2 pebbles are distributed onto vertices of an n-cube (with 2 vertices). A pebbling step is to remove two pebbles from some vertex and then place one pebble at an adjacent vertex. The question of interest is to determine if it is possible to get one pebble to a specified vertex by repeatedly using the pebbling steps from any starting distribution of 2 pebbles. This question is answered affirmatively by proving several stronger and more general results. AMS(MOS) subject classification. 05C 1. Introduction. An n-dimensional cube, or n-cube for short, consists of 2 vertices labelled by (0,)-tuples of length n. Two vertices are adjacent if their labels are different in exactly one entry. Because of its highly parallel structure, the n-cube possesses many nice properties and is an ideal model for games of various distributive types. In this paper we investigate the following game that was first proposed by Lagarias and Saks [4], [7]. Suppose 2 pebbles are distributed onto vertices of an n-cube. A pebbling step consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. We say a pebble can be moved to a vertex if we can apply pebbling steps repeatedly (if necessary) so that in the resulting configuration the vertex has one pebble. The question of interest is to determine if it is always possible to move one pebble to a specified vertex from any starting distribution of 2 pebbles. In this paper we answer this problem affirmatively. Independently, Guzman also