Black Path Game

This elegant little game is playable on a rectangular grid with square Trax-like tiles, each of which has 2 black lines connecting pairs of sides (either opposite pairs of sides, or adjacent pairs of sides). Players alternate placing tiles, each tile extending the black path from a designated start arrow at the side of the board. If a player connects the path to the edge of the board, they lose.

The game is analyzed and solved in the combinatorial game theory book Winning Ways for Your Mathematical Plays (pages 746-747, volume 3 of A K Peters 2003 edition). The first player can win if the grid has an even number squares, and the second player can win if the grid has an odd number of squares.