Optimality of 20-piece Mrs Perkins's quilts for sides 121, 123, 124, 125 and 127

A Mrs Perkins's quilt of side n is a tiling of the n x n square by integer squares whose sides have no common factor, and q(n) (OEIS A005670) is the least number of pieces. Wynn's enumeration of all quilts with at most 18 pieces, together with known constructions, determines q(n) for n <= 120 and gives q(122) = q(126) = 19. For n = 121, 123, 124, 125 and 127, quilts with 20 pieces were known, but it was open whether 19 pieces suffice. We show by an exhaustive computer search that none of these five sides has a quilt with at most 19 pieces. The known 20-piece quilts are therefore optimal, and 121 is the smallest square that needs 20 pieces. The search runs over reduced rectangle shapes, using the decomposition of the electrical network of a squared rectangle into its 3-connected components. It covers all sides up to 127, uses exact arithmetic, and is proved to be complete. The record contains the paper and a zip file with its LaTeX source and ancillary files: the search programs, verification scripts, outputs and logs of all runs, the five 20-piece quilts, SHA-256 hashes, and a README with the commands that reproduce every run.