Minesweeper strategy: from patterns to probability

Every number is a constraint

Minesweeper is not a game of luck with occasional logic. It is a constraint-solving puzzle with occasional luck. Almost every position that looks like a guess is a position where two numbers have not been compared with each other.

The method never changes:

1. Subtract the flags. A 3 with two flags beside it is a 1 for the squares that remain.
2. Check the obvious two cases. If the remaining count equals the remaining closed squares, all are mines. If the count is zero, all are safe.
3. Compare neighbours. Where two numbers share squares, the difference between them resolves the squares they do not share.

Level 1 — Basic: the two rules and the first patterns

The two rules that solve most squares

• All mines found — if a number's flags equal its value, every other neighbour is safe. Open them (or chord).
• All squares are mines — if a number's remaining closed neighbours equal its remaining count, they are all mines. Flag them.

Alternating between these two, square by square along the frontier, clears the great majority of any board.

The 1-1 pattern

Where two 1s sit side by side against an edge or a cleared region, the left-hand 1 already accounts for the mine in the shared squares — so the extra square next to the second 1 is safe. This is the first pattern worth learning by sight.

The 1-2-1 pattern

Three numbers in a row reading 1-2-1, with three closed squares beneath: the mines are under the two 1s and the square under the 2 is safe. Learn it as a picture, not as a derivation.

Open the corners first

Corners and edges have fewer neighbours, so the numbers there are more constrained and easier to resolve. Starting a new board at a corner gives you more usable information per click than starting in the middle.

Level 2 — Intermediate: subtraction, patterns and counting

The subtraction method

This is the technique that turns "I think it's a guess" into a solved square. Take two numbers whose neighbourhoods overlap. Let A be the number of mines still needed by the first, B by the second. If the first number's unshared squares are all accounted for by the difference, the shared squares resolve.

The everyday version: a 2 next to a 1 that share two squares means the 2's extra square is a mine. Once you see it this way, most "impossible" positions dissolve.

The 1-2-2-1 pattern

Count the mines globally

The mine counter is information you are probably ignoring. Late in a board, the number of mines remaining constrains the whole frontier at once:

• If three mines remain and three squares are unresolved, they are all mines.
• If one mine remains and the frontier already accounts for it, every other closed square on the board is safe — including the big interior region you have not touched.

The endgame of a hard board is usually solved by counting, not by patterns.

Flag only what pays

Flagging a mine has no value in itself. Flag when it lets you chord a number, and skip it otherwise. Efficient players flag perhaps half the mines on a board.

Level 3 — Advanced: probability, guess minimisation and efficiency

When it really is a guess, guess well

Some positions are genuinely undetermined. Then the question is not "which square is safe?" but "which square is least likely to be a mine, and which guess tells me the most if I survive?"

• Compute the frontier probabilities. Enumerate the consistent mine arrangements along the frontier and count how often each square is a mine. Squares that appear in fewer arrangements are safer.
• Compare with the interior. Remaining mines not forced onto the frontier are spread over the untouched squares. If the frontier squares are all 50/50 but the interior has 40 squares and 3 mines, click the interior — it is a far better bet.
• Prefer informative guesses. Between two squares of equal risk, click the one whose result resolves more of the board.

Guess early, not late

A forced guess costs you the board whenever it fails. Guessing early, when little work has been invested, is cheaper than guessing at the end of a nearly-cleared expert field. Where you have a choice of which region to develop, resolve the risky area first.

Efficiency and 3BV

If you are chasing times rather than completions:

• Chord aggressively. One chord can replace five clicks.
• Flag only to enable chords. Every other flag is a wasted click.
• Open large empty regions first — they clear the most squares per click and create long frontiers to work along.
• Learn the patterns visually. Expert players do not derive 1-2-1; they see it.

The honest ceiling

Because the general problem is NP-complete, there is no method that makes hard boards easy. What improves is your speed at the easy 95% and your judgement in the remaining 5% — and that is precisely where all the difference between a 200-second expert board and a 60-second one lies.

Where boards are lost

1. Chording on a wrong flag. The single most common catastrophic mistake.
2. Clicking blind into the interior when the frontier still has solvable squares.
3. Ignoring the mine counter in the endgame, when it often resolves everything.
4. Failing to subtract flags and misreading a 3 that is already half-solved.
5. Guessing late after investing several minutes, when an earlier guess was available.
6. Flagging everything — it feels tidy and it costs a lot of time.

Common questions