Cats in cubes

TL;DR

This paper determines the maximum number of times the word 'CAT' can appear in a filled d-dimensional cube and characterizes conditions for this maximum, extending to other words.

Contribution

It provides a precise upper bound for the occurrence of 'CAT' in d-dimensional cubes and characterizes when this bound is achieved, extending previous questions.

Findings

Maximum occurrences of 'CAT' are bounded by (3^{d-1}/2)n^d.

Equality conditions for the maximum are characterized.

Results extend to words other than 'CAT'.

Abstract

Answering a recent question of Patchell and Spiro, we show that when a $d$-dimensional cube of side length $n$ is filled with letters, the word $\mathsf{CAT}$ can appear contiguously at most $(3^{d-1}/2)n^d$ times (allowing diagonals); we also characterize when equality occurs and extend our results to words other than $\mathsf{CAT}$.