The Maximum Number of Appearances of a Word in a Grid

TL;DR

This paper investigates optimal arrangements of letters in grids to maximize the occurrences of a specific word, providing exact solutions for certain cases and asymptotic results for higher dimensions.

Contribution

It offers exact solutions for specific word arrangements in small grids and asymptotic analysis for larger, higher-dimensional grids.

Findings

Exact solutions for certain word patterns in 3x3 grids.

Asymptotic results for maximizing word appearances in higher dimensions.

Methodologies for optimizing word frequency in grid arrangements.

Abstract

How can you fill a $3\times 3$ grid with the letters A and M so that the word ``AMM'' appears as many times as possible in the grid? More generally, given a word $w$ of length $n$, how can you fill an $n\times n$ grid so that $w$ appears as many times as possible? We solve this problem exactly for several families of words, and we asymptotically solve this problem in higher-dimensional grids.