Minesweeper 1-2-1 and 1-2-2-1 patterns explained

In a straight 1-2-1 boundary, the two outer covered squares are mines and the middle square is safe. In a straight 1-2-2-1 boundary, the two middle covered squares are mines and the outer two are safe, provided these are the clues' only unresolved neighbours.

Check the geometry before using the pattern

The numbers alone are insufficient. In the diagrams below, letters are covered squares in a single straight row immediately above the numbers. Every other neighbour of the shown numbers is already revealed safe or lies outside the board. There are no additional covered squares around either end or below the clues.

A letter may be either safe or a mine. In the equations, its value is zero for safe and one for a mine. This lets us derive the result instead of relying on visual memory.

The 1-2-1 pattern

Covered row:   A B C
Revealed row:  1 2 1

The left 1 touches A and B. The middle 2 touches A, B and C. The right 1 touches B and C. Their unresolved-neighbour equations are:

Clue	Equation
Left 1	A + B = 1
Middle 2	A + B + C = 2
Right 1	B + C = 1

Subtract the left equation from the middle equation. C must be a mine. The right equation then makes B safe, and the left equation makes A a mine.

Result:       mine safe mine

Notice that the centre square is safe even though it is next to the largest number. A large clue describes its entire neighbourhood; it is not a danger rating for the square directly above it.

The 1-2-2-1 pattern

Covered row:   A B C D
Revealed row:  1 2 2 1

Here the first 1 touches A and B; the first 2 touches A, B and C; the second 2 touches B, C and D; the final 1 touches C and D.

Subtract A+B=1 from A+B+C=2 to prove C is a mine. Since C+D=1, D is safe. Subtract C+D=1 from B+C+D=2 to prove B is a mine. Finally A+B=1 makes A safe.

Covered square	Result
A	Safe
B	Mine
C	Mine
D	Safe

The result is therefore safe, mine, mine, safe. Keep the covered row separate from the clue row when remembering which squares to act on.

Subtract proved mines first

Patterns can appear in reduced clues even when the displayed digits differ. Suppose a straight displayed 2-3-2 boundary has, for each clue, exactly one already-proved adjacent mine outside the unknown letter row. Subtracting those contributions produces a residual 1-2-1.

The same geometry requirements still apply. A displayed 3 with one proved neighbouring mine needs two mines among its remaining covered neighbours. Use that residual requirement in the equation, rather than mixing displayed and reduced numbers in one pattern.

Count each clue's proved mines separately. One flag might touch more than one clue, but it contributes once to each clue it touches. Never subtract it from a clue that is not actually adjacent.

When the shortcut fails

If an additional covered square E also touches the left 1, its equation becomes A+B+E=1. Subtracting it from the middle equation no longer gives C=1. The letters 1-2-1 may still be visible, but the deduction has changed.

A bend in the boundary can change which cells each number sees. So can a covered square diagonally around an end. Rotate or mirror the whole configuration if necessary, keeping all neighbour relationships intact; do not rotate just the digit sequence.

Before placing flags, list the unresolved neighbours of each end clue. If they are not exactly the required pair, return to equations and solve the actual position.

Practise the proof, then recognise the shape

Cover the result and derive it from the equations. Next, add an extra unresolved neighbour and explain why the original proof fails. That exercise teaches both recognition and restraint.

After opening a safe square, inspect the new numbers before continuing with a memorised sequence. Their information may produce simpler deductions elsewhere. Pattern knowledge is useful when it reduces work while preserving every condition.

Where to play

Sweeple offers five campaign difficulty tracks and explanatory hints that highlight the cells involved in a deduction. On iPhone and iPad, you can use the highlighted relationships to practise counting shared neighbours before applying a remembered pattern.