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Cram (game)


Cram is a mathematical game played on a sheet of graph paper. It is the impartial version of Domineering and the only difference in the rules is that each player may place their dominoes in either orientation, but it results in a very different game. It has been called by many names, including "plugg" by Geoffrey Mott-Smith, and "dots-and-pairs." Cram was popularized by Martin Gardner in Scientific American.

The game is played on a sheet of graph paper, with any set of designs traced out. It is most commonly played on rectangular board like a 6×6 square or a checkerboard, but it can also be played on an entirely irregular polygon or a cylindrical board.

Two players have a collection of dominoes which they place on the grid in turn. A player can place a domino either horizontally or vertically. Contrary to the related game of Domineering, the possible moves are the same for the two players, and Cram is then an impartial game.

As for all impartial games, there are two possible conventions for victory : in the normal game, the first player who cannot move loses, and on the contrary, in the misère version, the first player who cannot move wins.

The winning strategy for normal Cram is simple for even-by-even boards and even-by-odd boards. In the even-by-even case, the second player wins by symmetry play. This means that whichever move Player 1 makes, Player 2 has a corresponding symmetric move across the horizontal and vertical axes. In a sense, player 2 "mimics" the moves made by Player 1. If Player 2 follows this strategy, Player 2 will always make the last move, and thus win the game.

In the even-by-odd case, the first player wins by similar symmetry play. Player 1 places his first domino in the center two squares on the grid. Player 2 then makes his move, but Player 1 can play symmetrically thereafter, thus ensuring a win for Player 1.

It should be noted that symmetry play is a useless strategy in the misère version, because in that case it would only ensure the player that he loses.


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