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TR14-116 | 6th September 2014 15:07
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#### 2048 is (PSPACE) Hard, but Sometimes Easy

**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.