Chocolate Numbers

TL;DR

This paper studies the enumeration of ways to break a rectangular chocolate bar into individual squares, introduces new related sequences, and proves divisibility properties of these sequences.

Contribution

It introduces the concept of chocolate numbers, derives their enumeration, and presents four new sequences with proven divisibility properties.

Findings

Enumerated the number of ways to break an m x n chocolate bar.

Introduced four new sequences related to chocolate numbers.

Proved divisibility results for these sequences.

Abstract

In this paper, we consider a game played on a rectangular $m \times n$ gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along existing grid lines, until just $mn$ individual squares remain. This paper enumerates the number of ways to break an $m \times n$ bar, which we call chocolate numbers, and introduces four new sequences related to these numbers. Using various techniques, we prove interesting divisibility results regarding these sequences.