L game

Description

The L game uses a 4x4 board. Two players each control a single L-shaped piece that covers three squares in a 2x3 area. Two small neutral pieces, each covering one square, also sit on the board.

A turn begins with the player moving their own L piece. After that, they may choose to move one of the neutral pieces. The goal is to make it impossible for the opponent to place their L piece in a new spot. No piece can overlap another or extend off the board.

When moving an L piece, the player lifts it and sets it down on any empty squares. It can be rotated or flipped, but it must end up covering at least one square it did not cover before. Moving a neutral piece is simpler: just pick it up and put it on any empty square.

Strategy

A key tactic involves using one neutral piece and your own L piece to seal off a 3x3 area in a corner, while the other neutral piece stops your opponent from flipping their L into a mirror image. Another common approach is to shift your L piece so it blocks half the board, then use the neutral pieces to cut off the opponent's other possible placements.

These setups often become possible after you leave a neutral piece on one of the eight killer spaces—these are the perimeter squares that are not corners. On your following turn, you either incorporate that killer piece into your own 3x3 block, or you use it to block a spot on the edge. Then, with your own L and the other neutral piece, you complete either a corner square or a half-board blockade.

Analysis

The L game is fully solvable, and if two perfect players face off, the result is always a draw. The board has 2,296 distinct valid arrangements, ignoring rotations, mirrors, and treating the two neutral pieces as identical. Every one of these positions can occur during play, regardless of whose turn it is. In 15 of these arrangements, the player whose turn it is has already lost; in each case, that player’s L-shaped piece touches a corner.

A perfect opponent will force a loss within a few moves from 14 more positions. From the remaining 2,267 arrangements, a player can always force at least a draw by playing indefinitely without ever losing. Even without perfect play, cautious players can avoid moving a neutral piece into a losing spot, allowing the game to continue forever. In that cautious style, a sudden-death variant lets a player move both neutral pieces after their own move.