Minesweeper

I will be using the Ubuntu version of Minesweeper, called "Mines," to demonstrate most strategies. For most of the screenshots below, light gray squares represent safe squares, dark gray squares represent unknown squares.


If some center point contains the number \(k,\) and there are \(k\) empty squares surrounding it, then all of those squares must contain mines. For example, the 2 is surrounded by 2 unrevealed squares, so both must be mines:


If a square contains \(k,\) and \(k\) mines have already been marked, then all other surrounding squares must be empty. For example, the 1 in the center square below is already surrounded by 1 mine, and thus, the unrevealed square must be empty.


In the puzzle below, there can't be a mine in both of the bottom left squares, because of the 1 above the bottom left square. But then the other unrevealed square attached to the 3 square must contain a mine. Sometimes, figuring out the possibilities for where a mine must be, tells us where some other mine must be.


In the puzzle below, one mine must belong to either the bottom unrevealed square, or the one above it. Wherever that mine is, it satisfies both the 1 and 2 squares. Thus, the uppermost unrevealed square must be empty. Sometimes, figuring out the possibilities for where a mine must be, tells us where some other mine can't be.


Identifying groups of possibilities can be used in the same way. Refer to the puzzle below. Exactly one of the two squares above the 1 must contain a mine. Also, exactly one of the two squares above the 4 must contain a mine. But then exactly 1 of the two squares to the left of the blue 2 must contain a mine. Thus, the yellow 2 will be touching two mines, and so all other squares surrounding the yellow 2 are safe.