Mrs. Perkins' quilt is a square dissection problem first posed by Henry Dudeney in his Amusements in Mathematics (1917):¹

 

It will be seen that in this case the square patchwork quilt is built up of 169 pieces. The puzzle is to find the smallest possible 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.

 

Dudeney's problem can be generalized to the dissection of a square of side n into a number S_n of smaller squares. Unlike a perfect squaring the square problem, the smaller squares needn't be all different sizes. In addition, only prime dissections are considered so that patterns that can be dissected on lower order squares aren't allowed. The smallest number of relatively prime dissections of an n × n quilt for n = 1, 2, ..., are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... ²

 

References

1. Dudeney, H. E. Amusements in Mathematics. New York: Dover, 1917. Reprinted Mineola, NY: Dover, 1958.
2. Conway, J. H. "Mrs. Perkins's Quilt." Proc. Cambridge Phil. Soc., 60: 363–368, 1964.