"Minesweeper" and spectrum of discrete Laplacians

Abstract

The paper is devoted to a problem inspired by the "Minesweeper" computer game. It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of "Minesweeper" but does not require a computer.Comment: We add consideration of tables based on the triangle tiling of the plane. Its paper version encounters situations typical for the computer "Minesweeper" gam