Making Change in 2048

TL;DR

This paper analyzes the maximum achievable score in 2048-like games by establishing a min-max theorem linking game strategies to change-making algorithms, revealing how greedy strategies influence game length and termination.

Contribution

It introduces a formal framework connecting 2048 game strategies with change-making algorithms, providing new insights into optimal play and game termination conditions.

Findings

Max score characterized by a min-max theorem.

Greedy strategies align with optimal change-making in certain variants.

Game termination linked to gaps in tile value sequences.

Abstract

The 2048 game involves tiles labeled with powers of two that can be merged to form bigger powers of two; variants of the same puzzle involve similar merges of other tile values. We analyze the maximum score achievable in these games by proving a min-max theorem equating this maximum score (in an abstract generalized variation of 2048 that allows all the moves of the original game) with the minimum value that causes a greedy change-making algorithm to use a given number of coins. A widely-followed strategy in 2048 maintains tiles that represent the move number in binary notation, and a similar strategy in the Fibonacci number variant of the game (987) maintains the Zeckendorf representation of the move number as a sum of the fewest possible Fibonacci numbers; our analysis shows that the ability to follow these strategies is intimately connected with the fact that greedy change-making is optimal for binary and Fibonacci coinage. For variants of 2048 using tile values for which greedy change-making is suboptimal, it is the greedy strategy, not the optimal representation as sums of tile values, that controls the length of the game. In particular, the game will always terminate whenever the sequence of allowable tile values has arbitrarily large gaps between consecutive values.