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### Paper:

TR14-116 | 6th September 2014 15:07

#### 2048 is (PSPACE) Hard, but Sometimes Easy

TR14-116
Authors: Rahul Mehta
Publication: 6th September 2014 15:27
Keywords:

Abstract:

We prove that a variant of 2048, a popular online puzzle game, is PSPACE-Complete. Our hardness result
holds for a version of the problem where the player has oracle access to the computer player's moves.
Specifically, we show that for an $n \times n$ game board $G$, computing a
sequence of moves to reach a particular configuration $C$ from an initial
configuration $C_0$ is PSPACE-Complete.
Our reduction is from Nondeterministic Constraint Logic (NCL).

We also show that determining whether or not there exists a fixed sequence of moves of length $k$
that results in a winning configuration for an $n \times n$ game board is fixed-parameter tractable (FPT). We describe an algorithm
to solve this problem in $O(4^k n^2)$ time.

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