The number of chess games played so far exhausts just a small part of all the possibilities offered by the game. The factor of struggle dramatically reduces the number of positions which can actually appear on a chessboard. So a virtually infinite amount of possible positions can never be reached in a real game. It is exactly these positions that are the subject of the chess problem composers' explorations.
2345 problems: mate in 2, mate in 3, mate in 4, mate in 5
The most comprehensive anthology of Meredith (maximum 12 pieces) problems to date with detailed solutions.