There’s an equation that describes many situations in the Minesweeper game where you can reveal or flag a group of squares with complete confidence, if those squares already neighbor an uncovered square.

Definitions

  • By beside a square \(s\), we mean the \(8\) squares around the square \(s\).
  • By a flagged square, we refer to a square that we are completely certain hides a bomb underneath it.
  • By a revealed square, we refer to a square that we are completely certain does not hide a bomb. In addition, it displays a number. Remember that revealed squares with no number on them are implicitly numbered \(0\).

The Theorem

  • Consider any two revealed squares \(a\) and \(b\).
  • Let the set of unflagged but unrevealed squares around \(a\) be \(A\), and the set of unflagged but unrevealed squares around \(b\) be \(B\).
  • In addition, let the set of unflagged but unrevealed squares around both be \(U=A\cap B\) and let \(A’=A\setminus B\). Likewise, let \(B’=B\setminus A\). Then \(A’\) is the set of unflagged but unrevealed squares beside
  • Let \(a\) and \(b\) also represent the number on those respective uncovered squares.
  • Let \(f_a\) be the number of flagged squares around \(a\) and let \(f_b\) be the number of flagged squares around \(b\).
  • Let \(n_a=a-f_a\) be the number of bombs in unflagged but unrevealed squares beside \(A\) but not beside \(B\). Likewise, let \(n_b=b-f_b\).

Then, without loss of generality, let \(a\geq b\). If \[n_a-n_b=\left|A’\right|,\] then all the squares in \(A’\) contain bombs and all the squares in \(B’\) do not contain bombs.