Sometimes the opposition is the whole game: whoever has to move must let the other king through. If it is your move in that position, you need to lose one. Your king can walk round a triangle of three squares and come back to where it started, while the other king, stuck guarding the same squares, can only go back and forth between two. You reach the same position with the other side to move, and it has to give way.
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.
