**Title:** Tiling with dominoes and monomers and an equivalent (0; 1)-matrix existence theorem.

**Abstract:** Question: Suppose you are given two lists of numbers: *R* = (*r*_{1}, *r*_{2}, …, *r*_{7}), *S* = (*s*_{1}, *s*_{2}, …, *s*_{6}). You are then asked to tile a 7 × 6 checkerboard with vertical dominoes (2 × 1 blocks) and monomers (1 × 1 blocks) so that there are exactly dominoes whose top half is in row *i* and exactly dominoes occur in column *j*. For example, if *R* = (3, 1, 2, 2,1, 2, 0) and *S* = (2, 1, 2, 2, 1, 3), a tiling would be:

If *R* = (1, 2, 2, 2, 2, 3, 0) and *S* = (3, 1, 3, 1, 1, 3), can you successfully complete the tiling? What about when *R* = (1, 1, 3, 2, 2, 3, 0) and *S* = (3, 1, 3, 1, 1, 3)? This question is similar to the question answered by the classical, celebrated result known as the Gale-Ryser Theorem. The answer to the above questions, as well as the answer for the general case, will be given.