324 Answers- TESSELLATED PAVEMENTS
The illustration shows how the square space may be covered with twenty-
nine square tiles by laying down seventeen whole and cutting each of the re-
maining twelve tiles in two parts. Two parts having a similar number form a
whole tile.
- SQUARE OF SQUARES
There is, we believe, only one solution to this puzzle, here shown. The few-
est pieces must be 11, the portions must be of the sizes given, the three largest
pieces must be arranged as shown, and the remaining group of eight squares
may be "reflected" but cannot be differently arranged.
[For a discussion of the general problem, still unsolved, of dividing a square
lattice of any size, along lattice lines, into the minimum number of smaller
squares, see J. H. Conway, "Mrs. Perkins's Quilt," Proceedings of the Cam-
bridge Philosophical Society, VoL 60, 1964, pp. 363-368; G. B. Trustrum's
paper of the same title, in the same journal, VoL 61, 1965, pages 7-11; and my
Scientific American column for September 1966. The corresponding problem
on a triangular lattice has not, to my knowledge, yet been investigated.
Although this puzzle also appears in Dudeney's earlier book, Amusements
in Mathematics (1917), under the title "Mrs. Perkins's Quilt" (Problem 173,