Clickomania is Hard, Even with Two Colors and Columns

Clickomania is a classic computer puzzle game (also known as SameGame, Chain-Shot!, and Swell-Foop, among other names). Originally developed by Kuniaki “Morisuke” Moribe under the name Chain-Shot! for the Fujitsu FM-8, and announced in the November 1985 issue of ASCII Monthly magazine, it has since been made available for a variety of digital platforms [Mor03]. Figure 1 shows some examples. Although rules and objectives vary slightly among different versions, the basic premise is always the same: you are presented with a two-dimensional grid of colored square tiles and asked to clear those tiles by removing a contiguous like-colored group at each step (or click). After each click, any tiles suspended above empty space will fall, and empty columns will contract, so that the remaining tiles always form a contiguous group. Though the number of clicks is unlimited, if a contiguous like-colored group has only one tile (which we refer to as a singleton), then it cannot be clicked. To remove it, one must first connect it to at least one other tile of the same color. The game ends when no further clicks are possible, either because all tiles have been eliminated or because all remaining tiles are singletons. See Figure 2 for a sample Clickomania board and the results of a few clicks. There are two main variants of the game, which differ in the objective of the player: