Variation: Dissect a disk into congruent pieces, so that a small region around the center is within one of the pieces. -- Rich

This is also included in the problem from UPIG that I previously mentioned. From my 1991 edition, problem C6 "Cutting up squares, circles and polygons", at the bottom of page 87: "Stein asks whether it is possible to partition the unit circle into congruent pieces so that the center is in the interior of one of the pieces?[sic] It need not be on the boundary of all the pieces, as Figure C8 [on page 89] shows. Is it true that the pieces must have a diameter of at least one? The first question is also of interest for the regular n-gon, n >= 5 ." Actually, it's easy to do for the regular hexagon, so the last sentence should probably read "n = 5 and n >= 7". For the equilateral triangle and square, there are such dissections into 4 and 3 congruent pieces, respectively. What's the fewest number of pieces in such a dissection of the regular hexagon? Is there any update on any of these in a later edition of the book? Michael Reid