I love the idea of multiplicative magic squares. Using n^2 unknown but distinct vectors in R^n that are indexed by (i,j) for 1 <= i,j <= n, it's the same as finding nonnegative integer coordinates for these vectors such that all vectors (i, j0) for 1 <= i <= n sum to the same as all vectors (i0,j) for 1 <= j <= n (for all choices of 1 <= i0,j0 <= n). (Or did someone already mention this?) I'm interested in how magic these can be. I find the old rule of having all rows & columns plus the 2 diagonals sum to the same sum as rather dorky: the diagonals are kind of random. 1) Better here would be all rows & columns and broken diagonals (i.e., diagonals on the torus) multiply to the same product. 2) Best of all would be for simply all affine lines on Z_n x Z_n multiply to the same product. (This is when you repeat the same step (K,L) on the torus and first come back to your starting point with exactly n steps. (There are 4x4 additive magic squares with this property; I'm not sure if there are for any other n.) Are 1) or 2) possible for a multiplicative magic square (MMS)? (And is there a better name for an MMS than multiplicative magic square?) --Dan