Clickomania is Hard, Even with Two Colors and Columns

Abstract

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: 1. Elimination variant: The player wins by removing all the tiles. 2. Score variant: The player gains points from each click based on the number of tiles removed with that click (typically, this relation is quadratic: removing x tiles awards approximately x 2 points) and wins by achieving at least a specified score. For each variant, we study how hard it is for a computer to determine whether it is possible to win from a given initial configuration of the board. Specifically, we show that Clickomania is computationally hard (" NP-complete ") even for two-color patterns occupying just two columns (and thus also hard for more than two colors or columns). This result is best possible: one-color Clickomania puzzles are trivial to solve with just a single click, and one-column Clickomania puzzles are computationally easy to solve [BDD + 02]. Indeed, this paper completes a quest started 15 years ago when Biedl et al. [BDD + 02] proved Clickomania computationally hard for two columns and five colors, or …

11 Figures and Tables

Showing 1-8 of 8 references

Samegame (swell foop) Google Play app

  • 2015

Morisuke " Moribe. Chain-shot! http://www.asahi-net.or.jp/ ∼ ky6k-mrb/chainsht

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Rows Colors Solved Open Copies Open Open Table 1: Three out of four combinations of restriction types remain open