Analysis of the Game "2048" and its Generalization in Higher Dimensions

TL;DR

This paper extends the analysis of the game 2048 to higher dimensions, establishing maximum achievable values and strategies for both computer and human players in these generalized settings.

Contribution

It provides the first theoretical analysis of 2048 in n-dimensional space, including maximum values and strategic insights.

Findings

Maximum value in 2D is 2^{n_1 n_2 + 1}

Maximum value in d-dimensions is 2^{(∏_{i=1}^d n_i) + 1}

Strategies for computer and human players are proposed under certain conditions.

Abstract

We theoretically analyze the popular mobile app game `2048' for the first time in $n$-dimensional space. We show that one can reach the maximum value $2^{n_1n_2+1}$ and $2^{\left({\prod_{i=1}^{d} n_i}\right)+1}$ for the two dimensional $n_1\times n_2$ board and $d$ dimensional $n_1\times n_2\times \ldots \times n_d$ board respectively. We also present a strategy for the computer and a winning strategy for the human player in certain conditions.