Quoridor
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Quoridor is a modern two-player abstract strategy game. The rules are simple, but it provides an interesting and deep game.
Two players sit on either side of a 9-by-9 grid with a piece in the middle of the closest side to them:
H G F E D C B A
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+-----------------------------------+
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1--| + + + + + + + + |--1
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2--| + + + + + + + + |--2
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3--| + + + + + + + + |--3
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4--| + + + + + + + + |--4
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5--| + + + + + + + + |--5
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6--| + + + + + + + + |--6
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7--| + + + + + + + + |--7
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8--| + + + + + + + + |--8
| X |
+-----------------------------------+
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H G F E D C B A
Each is trying to get to the opposite side first. On one's turn, one may either move in any compass direction, or may place a length 2 wall between squares on the board. The main restriction is that you may not make it impossible for your opponent to reach the other side.
In the boxed edition of this game by Gigamic, each player has a limited number of wall pieces; in the play by email version available at Richard's Play By Email Server, there is no limit to the number of walls that can be placed, except that imposed by the rules of the game.