# Mrs. Perkins's Quilts

Mrs. Perkins's Quilts

"For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches." — Henry E. Dudeney

Define a quilt as a square made from smaller integer squares. The order of a quilt is the number of smaller squares. The size is the edge length of the full quilt. Given all that, try to solve Dudeney's original problem, and divide a 13×13 square into 11 squares. It is impossible to dissect the 13×13 square into fewer than 11 squares, so this dissection is optimal. An additional constraint is that the side lengths cannot have a common factor. Thus, an 8×8 square divided into four 4×4 squares is not considered optimal. Any optimal quilt is a Mrs. Perkins's quilt.

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One method for finding new quilts is to replace a corner square with a square as big as the original quilt, then to add two squares. A second method adds three squares the same size as the original quilt around one corner. A third method doubles all squares and divides an odd-sided corner. A quilt that uses none of these methods is called primitive. If all the squares have different sizes, the dissection is called perfect. When no two squares of the same size share a full edge, the dissection is semiperfect.

Problem C3 in R. K. Guy's Unsolved Problems in Geometry involves finding optimal quilts. In 2003, author Ed Pegg found some new best solutions with Lance Gay, and sent them to Guy, who then found even more solutions. In 2010, Guy suggested that we compile all our findings and methods for a paper, which this Demonstration strives to do. A number of new discoveries were made in the compilation process.

Above size 200, it is extremely difficult to prove that a quilt is optimal. As such, many of these findings should be considered best known. The authors welcome any improvements.