I did mean to exclude those cases. (But maybe there's no need to, since they wouldn't have the two exceptions mentioned.) Also, even with all the technical nonsense, I neglected to specify that the N regular polygons must be congruent! So to summarize, I hope, more clearly: ********************************************************* * PUZZLE: There is a closed metric surface S that, for * * exactly eight values of N in the range 1 <= N <= 10, * * can be tiled by N congruent regular k-gons * * (where k may depend on N). * * * * Find a surface S satisfying this condition and * * prove that it works. * * * ********************************************************* --Dan Rich wrote: << Are you excluding the case of cutting the surface of a sphere into orange wedges? I'd view an orange wedge as a two-sided regular polygon, but opinions may vary. << Quoting Dan Asimov <dasimov@earthlink.net>: . . . ____________________________________________________________

Notes: Closed means of finite extent and without boundary. A tiling means S is the union of regular polygons such that if P and Q are distinct polygons, then their intersection is either empty or a union of common vertices and/or edges. Also, some of one polygon's own edges may coincide pairwise. A regular polygon Q of k sides inherits its metric from S such that the isometry group of Q has 2k elements.

--Dan

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_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele