# Generalized game

Sudoku (4×4)
Sudoku (9×9)
Sudoku (25×25)
Generalized Sudoku includes puzzles of different sizes

In computational complexity theory, a generalized game is a game or puzzle that has been generalized so that it can be played on a board or grid of any size. For example, generalized chess is the game of chess played on an n×n board, with 2n pieces on each side. Generalized Sudoku includes Sudokus constructed on an n×n grid.

Complexity theory studies the asymptotic difficulty of problems, so generalizations of games are needed, as games on a fixed size of board are finite problems.

For many generalized games which last for a number of moves polynomial in the size of the board, the problem of determining if there is a win for the first player in a given position is PSPACE-complete. Generalized hex and reversi are PSPACE-complete.[citation needed]

For many generalized games which may last for a number of moves exponential in the size of the board, the problem of determining if there is a win for the first player in a given position is EXPTIME-complete. Generalized chess, go and checkers are EXPTIME-complete.[citation needed]