Summary: Most people are familiar with the computer game minesweeper. It is a game of logic and luck whose goal is to locate the “mines” on the game board by gaining and using information about the neighborhoods of non-mine spaces on the game board. The game board is a particular graph and the information is cardinalities of the intersections of neighborhoods and the set of mines. Of course the game could be played on an arbitrary graph. We consider the problem of removing luck as a factor in winning the game.