Words with Repeated Letters in a Grid

TL;DR

This paper investigates the maximum occurrences of a word in a multi-dimensional grid along various directions, extending previous work to more complex cases and exploring connections to combinatorial problems like the n-queens puzzle.

Contribution

It generalizes prior results on word appearances in grids to include words with repeated letters and higher dimensions, revealing greater complexity and new combinatorial links.

Findings

Extended analysis to words with repeated letters.

Identified increased complexity in higher dimensions.

Connected the problem to the n-queens problem.

Abstract

Given a word $w$, what is the maximum possible number of appearances of $w$ reading contiguously along any of the directions in $\{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$ in a large $d$-dimensional grid (as in a word search)? Patchell and Spiro first posed a version of this question, which Alon and Kravitz completely answered for a large class of "well-behaved" words, including those with no repeated letters. We study the general case, which exhibits greater variety and is often more complicated (even for $d=1$). We also discuss some connections to other problems in combinatorics, including the storied $n$-queens problem.