What is the smallest number of pieces that you can cut an n-sided regular
polygon into so that the pieces can be rearranged to form a rectangle? Call it
r(n). The rectangle may have any proportions you wish, as long as it is a
rectangle. The rules are the same as for the classical problem where the
rearranged pieces must form a square. Let s(n) denote the minimum number of
pieces for that problem. For both problems the pieces may be turned over and
the cuts must be simple curves. The conjectured values of s(n), 3 <= n <= 12,
are 4, 1, 6, 5, 7, 5, 9, 7, 10, 6. However, only s(4)=1 is known for certain.
The problem of finding r(n) has received less attention. In this paper we give
constructions showing that r(n) for 3 <= n <= 12 is at most 2, 1, 4, 3, 5, 4,
7, 4, 9, 5, improving on the bounds for s(n) in every case except n=4. For the
10-gon our construction uses three fewer pieces than the bound for s(10). Only
r(3) and r(4) are known for certain. We also briefly discuss q(n), the minimum
number of pieces needed to dissect a regular n-gon into a monotile.Comment: 26 pages, one table, 41 figures, 14 reference