Pebblings

The analysis of chessboard pebbling by Andrew Odlyzko is strengthened and generalized, rst to higher dimension and then to arbitrary posets. 1 The pebbling game The pebbling game of Kontsevich is played on the grid points of the rst quadrant. One starts with a single pebble on the origin and a move consists of replacing any pebble with two pebbles, one above and one to the right of the vanishing pebble: f p p-f f p. However, only one pebble is allowed on each grid point. The original problem, posed by Kontsevich in 1981, was to show that the ten grid-points closest to the origin, f(i; j) j i + j 3g, form an unavoidable set, meaning that every game position has at least one pebble in this set. The intended proof was the following. To a pebble at (i; j) assign the weight 2 ?i?j. That makes the total weight of the pebbles equal to 1 in all positions, for each move splits a pebble into two, half as heavy, and the total weight was 1 to start with. With at most one pebble on each point, all grid points outside the ten-point triangle can carry at most P i;j0 2 ?i?j ? (1 + 2 2 + 3 4 + 4 8) = 3 4 , so some pebble must be left in the triangle. Shortly afterwards, A. Khodulev 10] made the surprising observation, that already the ve-point set s s s s s p is unavoidable. The rst complete proof appeared thirteen years later in the American Mathematical Monthly 3], in which Chung, Graham, Morrison and Odlyzko also gave new enumerative results. The purpose of this paper is to extend these results to the higher dimension analogues of the pebbling game and to a more general poset version.