Will the pawn queen?

The rule, in one sentence

Count how many moves the pawn needs to reach its promotion square, draw a square of that many ranks and files with the pawn and the promotion square as opposite corners, and check whether the defending king can step inside that square on its own move. If it can, the king catches the pawn no matter how the race plays out from there. If it can't, nothing stops the pawn — it queens by force, and calculating the race move by move is unnecessary.

Why counting works at all

A king moves one square in any direction, so the number of moves it needs to reach a given square is just the larger of the rank-distance and file-distance to it (a king moving diagonally covers both at once). The pawn, meanwhile, needs exactly as many moves as the ranks remaining to promotion (one extra move if it hasn't moved yet, since it can advance two squares the first time). The "square" is simply the geometric shorthand for "can the king's fastest path get there before the pawn's fixed path finishes" — it turns a move-by-move calculation into something you can see at a glance.

A worked example

A pawn on e5 needs three moves to promote (e6, e7, e8 — no head start, since it's already moved). Its square is therefore three ranks deep. A defending king sitting anywhere three king-moves or fewer from e8 — with the right side to move — is inside that square and draws; a king any further away is outside it, and the pawn runs home untouched. Push the same pawn back to its starting square and the count changes (the first move can cover two squares), which is exactly the wrinkle that catches players who memorise a diagram instead of the counting rule behind it.

Drag the king above and watch the live squares update — the fastest way to internalise the rule is to see it redraw itself as the position changes, rather than recompute it by hand every time.