A generalization of magic squares with applications to digital halftoning
A semimagic square of order n is an n x n matrix containing the integers 0, ..., n(2) - 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 x 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 kxk 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=k(m), 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.
magic square digital halftoning matrix discrepancy latin square