Minesweeper is a game of logical deduction under uncertainty that connects to deep topics in probability theory, Bayesian reasoning, and computational complexity. Originally included with Microsoft Windows 3.1 in 1992, Minesweeper became one of the most played computer games in history, introducing millions of people to probabilistic reasoning without them even realizing it.

The core reasoning in Minesweeper is a form of constraint satisfaction. Each revealed number constrains the configuration of mines in its neighboring cells. When a cell shows "3," exactly 3 of its up-to-8 neighbors contain mines. Players must combine multiple overlapping constraints to deduce which cells are safe and which contain mines. Simple cases involve single-constraint reasoning: if a "1" cell has only one unrevealed neighbor, that neighbor must be a mine. Advanced play requires multi-constraint reasoning, where information from several numbered cells must be combined simultaneously. This intersection logic is equivalent to solving a system of linear equations over binary variables (mine or no mine), a formulation that connects Minesweeper directly to mathematical optimization.

The Minesweeper Consistency Problem, determining whether a given partially revealed board has a valid mine configuration, was proven NP-complete by Richard Kaye in 2000. This landmark result means that Minesweeper is, in a formal computational sense, as hard as the most difficult problems in computer science. Kaye showed that Minesweeper boards can simulate logic circuits, effectively making Minesweeper a universal computer. This NP-completeness result implies that no efficient algorithm exists (assuming P does not equal NP) to determine whether an arbitrary Minesweeper position is solvable without guessing.

When pure logical deduction reaches its limits, skilled Minesweeper players employ Bayesian probability estimation. If multiple valid mine configurations exist for the unrevealed cells, the probability of a specific cell containing a mine equals the fraction of valid configurations in which that cell is mined. Computing these probabilities exactly is computationally expensive (it is a #P-hard counting problem), but players develop intuitive approximations. The global mine count constraint, knowing the total number of remaining mines, provides additional information that significantly affects cell-by-cell probabilities, especially in endgame positions where few cells remain.

Mine counting strategies leverage the total mine count displayed in the game interface. As mines are flagged and cells revealed, the remaining mine count constrains the possible configurations of unflagged mines. In endgame scenarios, combining the remaining mine count with local constraints often resolves positions that would otherwise require guessing. For example, if 2 mines remain and a region of 5 unrevealed cells must contain exactly 2 mines based on surrounding numbers, the probabilities can be computed precisely. Expert players mentally track these global constraints alongside local deductions, achieving win rates exceeding 90% on intermediate difficulty and 30 to 40% on expert, where some boards are provably unsolvable without guessing.