Question 47
Let $n$ be a positive 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.
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.