On the Complexity of Slide-and-Merge Games

Abstract: We study the complexity of a particular class of board games, which we call `slide and merge' games. Namely, we consider 2048 and Threes, which are among the most popular games of their type. In both games, the player is required to slide all rows or columns of the board in one direction to create a high value tile by merging pairs of equal tiles into one with the sum of their values. This combines features from both block pushing and tile matching puzzles, like Push and Bejeweled, respectively. We define a number of natural decision problems on a suitable generalization of these games and prove NP-hardness for 2048 by reducing from 3SAT. Finally, we discuss the adaptation of our reduction to Threes and conjecture a similar result.