A Mrs. Perkins's quilt is a dissection of a square of side
into a number of smaller squares. The name "Mrs. Perkins's Quilt" comes
from a problem in one of Dudeney's books, where he gives a solution for . Unlike a perfect
square dissection, however, the smaller squares need
not be all different sizes. In addition, only prime dissections are considered so
that patterns which can be dissected into lower-order squares
are not permitted.
The smallest numbers of squares needed to create relatively prime dissections of an quilt for , 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11,
12, ... (OEIS A005670), the first few of which
are illustrated above.
On October 9-10, L. Gay (pers. comm. to E. Pegg, Jr.) discovered 18-square quilts for side lengths 88, 89, and 90, thus beating all previous records. The following
table summarizes the smallest numbers of squares known to be needed for various side lengths , with those for (and possibly 16) proved minimal by J. H. Conway
(pers. comm., Oct. 10, 2003). More specifically, Conway (1964) gave an upper
bound of order
for general ,
which Trustrum (1965) improved to order . However, the coefficient of the upper bound does not appear
to be known.