No-guess Minesweeper

Why ordinary Minesweeper makes you guess

In classic Minesweeper the mines are placed at random. Random placement often creates positions that no amount of skill can solve: two closed squares, one mine between them, and every number that touches one of them touches the other in exactly the same way. Whichever you open, you have a 50% chance of losing. Players call this a 50/50.

How often does it happen? We generated plain random boards with a safe first click and ran them through our solver. About 91% of random Beginner boards happened to be solvable without a guess, about 65% of Intermediate boards, and only about 14% of Expert boards. So on Expert, a random board needs at least one guess most of the time — often near the end, after several minutes of careful play.

• Closed
• ✓Safe to open
• ⚑Must be a mine
• ⚑Flagged mine
• 1Look here
• ?Can only be guessed

How each board is made

Nothing is placed until you click. Your first square and its eight neighbours are kept free of mines, and the rest of the mines are scattered at random. Then a solver starts from your square and plays the board the way a careful player would — it never looks at where the mines actually are.

If the solver reaches a position where it cannot prove any square safe or any square a mine, the board would need a guess. Instead of throwing the whole board away, the generator moves a mine among the squares that are still undecided — never one the solver has already worked out — and the solver carries on. Squares already opened and mines already proven stay where they are, so everything deduced so far remains true. This repeats until the solver can open every safe square.

Finally the finished board is solved once more from scratch, from your first click, to confirm that it really needs no guess. Only then is it shown to you. All of this takes a few milliseconds — about 3 ms on average for an Expert board on a laptop — so the board appears the instant you click.

What the solver is allowed to know

The solver uses only what you can see: the numbers on open squares, the squares it has already proven to be mines, and the total number of mines. It reasons in three steps. First, single numbers: a number whose mines are all found makes its other neighbours safe, and a number with exactly as many closed neighbours as missing mines makes them all mines. Second, pairs of numbers: comparing two overlapping numbers, which is how patterns like 1-2-1 and 1-2-2-1 are solved. Third, full enumeration: for each connected stretch of the boundary it tries every possible arrangement of mines, combined with the count of mines left, and accepts a conclusion only if it holds in every arrangement.

That last step is complete — if it cannot decide any square, the position genuinely requires a guess. To make sure the solver never draws a wrong conclusion, it is covered by automated tests: on 400 random small positions its answers were compared, square by square, with a brute-force check of every possible mine layout, and they matched every time.

Same skill, no coin flips

No-guess does not mean easy. The boards keep the classic sizes and mine counts, and the solver is allowed to use deep reasoning, so you may need to compare several numbers or count the remaining mines to find the next move. What changes is the meaning of a loss: it is always a misread number or a slip of the mouse, never bad luck. And if a position looks like a 50/50, it is not — there is a deduction waiting somewhere on the board.

Questions