MathDB

Let

n \ge 2

be an integer. Consider the following game: Initially,

k

stones are distributed among the

n^2

squares of an

n\times n

chessboard. A move consists of choosing a square containing at least as many stones as the number of its adjacent squares (two squares are adjacent if they share a common edge) and moving one stone from this square to each of its adjacent squares. Determine all positive integers

k

such that: (a) There is an initial configuration with

k

stones such that no move is possible. (b) There is an initial configuration with

k

stones such that an infinite sequence of moves is possible.

Problem Statement