To see whether your king can catch a passed pawn, picture a square with the pawn in one corner and its queening square in the next. If your king can step inside that square on its move, it catches the pawn. If not, the pawn promotes. A pawn on its starting rank can move two squares, so count its square from one rank further up. Even a king outside the square can sometimes draw: if it heads for the enemy pawn and toward its own pawn at the same time, it gains the moves it needs.
Every move is checked with an endgame tablebase, which knows the exact result of every position with up to seven pieces. The computer defends perfectly, and if a move throws the result away, the trainer stops and shows you a better one.
