Magic rectangle sets of odd order

A magic rectangle set M= MRS(a, b; c) is a collection of c arrays (a×b) with entries 1, 2, . . . , abc, each appearing once, with all row sums in every rectangle equal to a constant ρ and all column sums in every rectangle equal to a constant σ. It was proved by the author [AKCE Int. J. Graphs Comb. 10 (2013), 119–127] that if an MRS(a, b; c) exists, then a ≡ b (mod 2). It was also proved there that if a ≡ b ≡ 0 (mod 2) and b ≥ 4, then an MRS(a, b; c) exists for every c, and if a ≡ b ≡ 1 (mod 2) and an MRS(a, b; c) exists, then c ≡ 1 (mod 2). For a, b, c not all relatively prime, the existence of an MRS(a, b; c) follows from Hagedorn’s construction of a 3-dimensional magic rectangle 3-MR(a, b, c) [T.R. Hagedorn, Discrete Math. 207 (1999), 53–63]. We prove that if a ≤ b and both a, b are odd, then an MRS(a, b; c) exists if and only if 3 ≤ a and c is any odd positive integer. This completely settles the existence of magic rectangle sets.