L game

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Starting positions
L game made of wood in starting position

The L game is a simple strategic game invented by Edward De Bono.

The L game is a two-player turn-based game played on a board of 4×4 squares. Each player has a 3×2 L-shaped piece, and there are two 1×1 neutral pieces. On each turn, a player first must move their L piece, and then may elect to move one of the single neutral pieces. Pieces cannot overlap or cover other pieces. On moving the L piece it is picked up then placed in empty squares anywhere on the board and the piece may be rotated or even flipped over in doing so, the only rule is it must be in a different position than the one it was picked up from, thus come to rest covering at least one grid square that it was not previously covering. On moving a neutral piece, which is optional, a player simply picks it up then places it in any empty square anywhere on the board, (even the same square although this is the same as not moving it). The loser is the first player unable to move their L piece to a new position.

One basic strategy is to use a neutral piece and one's own piece to block a 3×3 square in one corner, and use a neutral piece to prevent the opponent's L piece from swapping to a mirror-image position. Another basic strategy is to move an L piece to block a half of the board, and use the neutral pieces to prevent the opponent's possible alternate positions.

These positions can often be achieved once a neutral piece is left in one of the eight killer spaces on the perimeter of the board. The killer spaces are the spaces on the perimeter, but not in a corner. On the next move, one either makes the previously placed killer a part of one's square, or uses it to block a perimeter position, and makes a square or half-board block with one's own L and a moved neutral piece.

[edit] Analysis

All possible final positions, blue has won
All positions, red to move, where red will lose to a perfect blue, and maximum number of moves remaining for red. By looking ahead one move and ensuring one never ends up in any of the above positions, one can avoid losing.

In a game with two perfect players, neither will ever win or lose. The L game is small enough to be completely solvable. There are 2296 different possible valid ways the pieces can be arranged, not counting a rotation or mirror of an arrangement as a new arrangement, and considering the two neutral pieces to be identical. Any arrangement can be reached during the game, with it being any player's turn. Each player has lost in 15 of these arrangements, if it is that player's turn. The losing arrangements involve the losing player's L-piece touching a corner. Each player will also soon lose to a perfect player in an additional 14 arrangements. A player will be able to at least force a draw (by playing forever without losing) from the remaining 2267 positions.

Even if neither player plays perfectly, defensive play can continue indefinitely if the players are too cautious to move a neutral piece to the killer positions. If both players are at this level, a sudden-death variant of the rules permits one to move both neutral pieces after moving. A player who can look three moves ahead can defeat defensive play using the standard rules.

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