On the Complexity of Slide-and-Merge Games

TL;DR

This paper analyzes the computational complexity of popular slide-and-merge games like 2048 and Threes, proving NP-hardness for 2048 and discussing similar results for Threes.

Contribution

It introduces decision problems for generalized slide-and-merge games and proves NP-hardness for 2048, providing new insights into their computational difficulty.

Findings

2048 is NP-hard via reduction from 3SAT

Decision problems for generalized slide-and-merge games are computationally complex

Discussion and conjecture on similar NP-hardness for Threes

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.