# Mrs. Perkins' quilt

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.