  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.
