Neighbour Sum Patterns : Chessboards to Toroidal Worlds

TL;DR

This paper investigates the neighbor-sum property on chessboards and related structures, characterizing when solutions exist, exploring various geometries including toroidal and infinite boards, and extending concepts to three dimensions with algebraic tools.

Contribution

It provides a complete characterization of $n\times n$ boards satisfying the neighbor-sum property and explores solutions on diverse geometries, connecting to discrete harmonic functions and cyclotomic polynomials.

Findings

An $n\times n$ board satisfies the property iff $n\equiv 5\pmod 6$.

Solutions exist on certain rectangular and toroidal boards, as well as on infinite boards.

Connections to discrete harmonic functions and cyclotomic polynomials are established.

Abstract

We say that a chessboard filled with integer entries satisfies the neighbour-sum property if the number appearing on each cell is the sum of entries in its neighbouring cells, where neighbours are cells sharing a common edge or vertex. We show that an $n\times n$ chessboard satisfies this property if and only if $n\equiv 5\pmod 6$. Existence of solutions is further investigated of rectangular, toroidal boards, as well as on Neumann neighbourhoods, including a nice connection to discrete harmonic functions. Construction of solutions on infinite boards are also presented. Finally, answers to three dimensional analogues of these boards are explored using properties of cyclotomic polynomials and relevant ideas conjectured.