I see now that Torsten Sillke's problems collection includes the page http://www.mathematik.uni-bielefeld.de/~sillke/PROBLEMS/dissection which includes the citation to the aforementioned Gardner column (that would have saved me a bunch of time...) as well as the following: --------- Question A: =========== For which k does there exist a dissection of the n-dimensional hypercube into k conguent connected parts which are not boxes? In other words: For which k do there exist subsets A_1, ..., A_n of the unit hypercube s.t. i) all A_i open, connected ii) the A_i are pairwise disjoined and congruent iii) the closure of the union of the A_i is the hypercube If 1<k<=n, there is such a dissection. If in addition k is a prime, there are infinitely many dissections. --------- I'm not sure why the "k is prime" is there -- it seems to me that the n-dim hypercube is acted on by a symmetry of order k for all k<=n -- just cycle the first k coordinates -- and any fundamental domain for that action, of which there are plenty if k>2, is the requested dissection. I'm just recycling the argument for trisection of the cube in my last message, of course. Torsten's page also includes, further down, a reference to the following note, whose abstract (but not full text, I think) I can get to on line: --------- Polyominoes of order 3 do not exist I. N. Stewart and A. Wormstein Journal of Combinatorial Theory, Series A Volume 61, Issue 1 , September 1992, Pages 130-136 The order of a polyomino is the minimum number of congruent copies that can tile a rectangle. It is an open question whether any polyomino can have an odd order greater than one. Klarner has conjectured that no polyomino of order three exists. We prove Klarner's conjecture by showing that if three congruent copies of a polyomino tile a rectangle then the polyomino itself is rectangular. The proof uses simple observations about the topology of a hypothetical tiling, and symmetry arguments play a key role --------- So you can't do it with a polyomino. --Michael Kleber