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A king outvalues two pawns. Sacrifice to crown.
A king moves and jumps in all four diagonals, so it controls twice the board of a man.
If giving up a pawn guarantees one of your pawns reaches the back row, do it — the resulting king usually wins the endgame.
Defensively: block your opponent's promotion squares. A pawn one step from crowning is worth more than a distant king.