The Complexity of Clickomania

TL;DR

This paper analyzes the computational complexity of Clickomania, showing that solving certain configurations is efficient, while deciding solvability in others is NP-complete, highlighting the problem's computational difficulty.

Contribution

It provides complexity results for Clickomania, including polynomial-time algorithms for some cases and NP-completeness proofs for others.

Findings

Linear time solution for two colors in one column

Polynomial time for multiple colors in one column

NP-completeness for specific multi-column and multi-color cases

Abstract

We study a popular puzzle game known variously as Clickomania and Same Game. Basically, a rectangular grid of blocks is initially colored with some number of colors, and the player repeatedly removes a chosen connected monochromatic group of at least two square blocks, and any blocks above it fall down. We show that one-column puzzles can be solved, i.e., the maximum possible number of blocks can be removed, in linear time for two colors, and in polynomial time for an arbitrary number of colors. On the other hand, deciding whether a puzzle is solvable (all blocks can be removed) is NP-complete for two columns and five colors, or five columns and three colors.