Geometrical optics and chessboard pebbling

Recently Chung, Graham, Morrison and Odlyzko [1] studied some combinatorial and asymptotic enumeration aspects of chessboard pebbling. In this problem, we start with a single pebble, placed at the origin (0,0) of an infinite chessboard. At each step we remove a pebble from (i,j) and replace it with two pebbles at positions (i + 1,j) and (i,j + 1), provided the latter are unoccupied. After m steps there will be m + 1 pebbles on the board, in various configurations. Some subsets of the lattice first quadrant are unavoidable, as they must always contain at least one pebble. We study asymptotically the number f(k) of minimal unavoidable sets that consist of k lattice points, as k → ∞. We also analyze a related double sequence f(k,r), using various asymptotic approaches, including the ray method of geometrical optics.