Guides

Minesweeper probability

When logic runs out, a good guess is a number you can calculate.

The global density

Subtract only mines you have correctly identified, then divide by the covered cells that are still unknown, excluding those confirmed mines. This is a baseline, not the exact chance for every cell. Initial densities are 10/81 (about 12%), 40/256 (about 16%) and 99/480 (about 21%); the remaining density changes as you open cells.

Local chances from a number

Subtract confirmed neighboring mines from the displayed number. If the remaining count is 1 and there are four interchangeable unknown neighbors with no other constraints, each has a 1 in 4 chance. With two such neighbors it is 1 in 2. Compare with the current remaining density, rather than the starting density. A far-away cell is not automatically safer.

Worked example

Suppose one 1 touches unknown cells A, B and C, and another 1 touches B, C and D, with all other neighbors already resolved. The local possibilities are a mine in B, a mine in C, or mines in both A and D. They are not necessarily equally likely: each must be weighted by how many complete board layouts match the remaining mine count and all other numbers. Counting these three local possibilities alone does not establish the safest cell.

Practical rules

• Compare the local chance with the global density. If every number-based guess is worse than a random far-away cell, open the far-away cell.
• Prefer a guess that, if safe, tells you more, such as a cell that is likely to be a zero.
• A forced 50/50 should be left until the end, because opening other cells may still resolve it.