A Generalization of Magic Squares with Applications to Digital Halftoning

Abstract A semimagic square of order n is an n×n matrix containing the integers 0,…,n2−1 arranged in such a way that each row and column add up to the same value. We generalize this notion to that of a zero k×k-discrepancy matrix by replacing the requirement that the sum of each row and each column be the same by that of requiring that the sum of the entries in each k×k square contiguous submatrix be the same. We show that such matrices exist if k and n are both even, and do not if k and n are relatively prime. Further, the existence is also guaranteed whenever n=km, for some integers k,m≥2. We present a space-efficient algorithm for constructing such a matrix. Another class that we call constant-gap matrices arises in this construction. We give a characterization of such matrices. An application to digital halftoning is also mentioned.