Pebbling a Chessboard

Abstract

After k steps the board will have k + 1 pebbles on it. We call such configurations of pebbles reachable configurations. We will denote by R(k) the set of reachable configurations with k pebbles, and we set R = U k 2 1R(k). In Figure 2, we show the eight possible reachable configurations with at most four pebbles. A little experimentation convinces one that in any reachable configuration, some pebble must occupy a cell having coordinates (i, j) with i + j < 3. This fact first seems to have been noted by M. Kontsevich [9]. We give the "book" proof of this in the next section. If L(k) denotes the set (or "level") {(i, j): i + j = k} then