King and Pawn versus King
The foundation of all pawn endings — when one extra pawn wins, when it draws, and the geometric concepts of key squares and the rule of the square that decide every case in three rules of thumb.
Every endgame technique reduces, eventually, to king-and-pawn versus king. Whether the stronger side wins is the most-asked question in endgame analysis, the recurring question hidden inside opposite-colour bishops and queen-and-pawn endings and rook endgames that have run out of rooks. The honest answer fits in three sentences, and once you have read it, the entire foundation of endgame theory is in place.
The fundamental question
A king and a pawn against a lone king is either a win for the stronger side or a draw. There are no intermediate results. The stronger side wins if her king can support the pawn to the queening square; she draws if she cannot. The whole science of K+P vs K is the geometry of that question — which squares the king must reach, which the defender can block, and how to use the move-tempo to force the issue.
Three rules summarise the answer for non-rook pawns:
The stronger side wins if her king reaches any of the three squares immediately in front of the pawn — for a pawn on d3, those are c5, d5, and e5. We call these the key squares.
If the defender’s king can occupy any key square first, the defender draws by opposition (we cover this below).
The stronger side has the move-tempo advantage by default, but losing the move at the right moment (triangulation) can switch the burden onto the defender.
These three rules cover every K+P vs K position with a non-rook pawn. They have been the canonical formulation since the late nineteenth century.
Key squares
The key squares are the squares the stronger side’s king is racing to reach. For a pawn on the second through fourth rank, the key squares are the three squares two ranks in front of the pawn: pawn on d4, key squares c6, d6, e6. For a pawn on the fifth rank, the key squares are the three squares immediately in front of the pawn: pawn on d5, key squares c6, d6, e6. For a pawn on the sixth or seventh rank, every square in front of the pawn is a key square — the position is winning if the king is anywhere reasonable in front of the pawn.
(The position above is not a legal FEN — the diagram is illustrative. The principle stands: the white king on c6, d6, or e6 in front of the white pawn on d5 wins regardless of where Black’s king is, provided the position is reached with Black to move.)
The proof that key squares win is the technique of opposition. The attacker uses her king to drive the defender’s king back, gaining the opposition on each key square, until the pawn can advance with support. The classical phrasing is: the king goes first, the pawn follows.
The rule of the square
The second question in K+P vs K is whether the defender’s king can reach the pawn at all — whether she has time to catch it before promotion. The rule-of-the-square answers this without any calculation.
Imagine a square drawn on the board with one corner on the pawn and the diagonal corner reaching forward to the queening rank. If the defender’s king is inside that square (or can step inside on her move), she catches the pawn. If she is outside, the pawn promotes without help.
The rule of the square is purely geometric. If the king is on the diagonal corner of the square at the moment it is the defender’s turn to move, she catches the pawn — barely. One square farther and she is too late. The rule applies before the pawn moves; if it is the attacker’s move and the pawn is still on the second rank, the defender gets one tempo grace from the pawn not yet starting its run, but only for pawns on the second rank.
The rook-pawn exception
For rook pawns — a- and h-file — the win-conditions break down. The defender’s king can reach the corner ahead of the pawn and refuse to leave; the attacker cannot drive her out, because there is no space on the side of the pawn for the king to manoeuvre. Even with the attacker’s king already on the queening square, the result is a draw by stalemate or by perpetual king moves.
The rule: K+P vs K with a rook pawn is drawn if the defender’s king reaches the queening corner. If the defender is one tempo slow, the attacker wins by the standard method; if she is in time, the position holds. The reason matters more than the position: the rook pawn lacks the second flank that gives every other pawn its winning potential.
Every more elaborate endgame eventually reduces to this case. A pawn ending with several pawns on each side, after exchanges, becomes K+P vs K with one or two pawns; if the resulting K+P vs K is a draw, every preceding pawn move was bluff. The lesson is unromantic but absolute: pawn endings are won and lost three or four moves before the pieces come off the board, by the player who saw the K+P vs K position six moves earlier and made his moves accordingly.