Unpublished note, August 2006.
We determine all (m, n, k, l) for which it is possible to fill an m×n board with consecutive distinct integers so that every k×l region has the same sum, where the edges of the board are linked to the opposite side.
We consider Aronov et al.'s dither matrix problem: put n2 consecutive numbers on an n×n board in as “balanced” a way as possible in the sense that the numbers in each k×k region on the board add up to approximately the same sum. We construct a fully balanced configuration for some cases, completing the investigation to identify all (n, k) for which this is possible. For some other cases, we give a heuristic method attaining lower discrepancy than existing techniques. This method obtains the entries of the desired matrix by ranking the values of a continuous function at grid points.