Puzzle:

Let T denote the torus obtained by identifying each pair of opposite edges of a regular hexagon.

For which positive integers N is it possible to cut T into N congruent regular hexagons?

--Dan

Let S be the set of distances between lattice points of a regular triangular lattice of side 1. For all N in { s^2 : s in S } you can cut T into N congruent regular hexagons, and I don't know how to get any other N. Thus possible N's include 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, ... I suspect there's a nice algebraic characterization, but I don't know it. - Scott P.S.- For the analogous problem with a torus obtained by identifying opposite sides of a square, use distances in a square lattice. The algebraic characterization is sums of two squares.