DISJOINT ODD INTEGER SUBSETS HAVING A CONSTANT ODD SUM

Abstract

We prove that for positive k, n and m, the set {1, 3,...,2n - 1} of odd integers contains k disjoint subsets having a constant odd sum m if and only if 9(k - 1) less-than-or-equal-to m less-than-or-equal-to 2n - 1, or 9k less-than-or-equal-to m less-than-or-equal-to n2/k and n2 - mk not-equal 2.