The object is to remove matching pairs of uncovered tiles until all tiles have been removed. NP-complete even if all tile positions are known. Then from a satisfying assignment we can produce an ordering of pair removals that frees one pair of tiles from each variable, freeing at least one pair of tiles from each clause, which in turn frees the keys and the remaining variable and clause tiles. Conversely any solution to the Shanghai puzzle produces a satisfying assignment from the original 3-SAT formula. Fun with Algorithms , Elba, June 1998.
Players alternate placing vertices on a square grid.
Each player's vertices may be connected with edges connecting pairs of vertices a knight's move apart. After placing a vertex, the player may remove some of his own edges or add more of them. No two edges may cross. NP-complete to determine whether a single set of vertices can support a connecting path. What this implies about the game itself is unclear but it can likely be turned into a proof that determining the correct outcome is NP-hard.