Steinhaus dissection

A cube can be divided into n subcubes for only n = 1, 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and n>=48 (OEIS A014544; Hadwiger 1946; Scott 1947; Gardner 1992, p. 297). This sequence provides the solution to the so-called Hadwiger problem, which asks for the largest number of subcubes (not necessarily different) into which a cube cannot be divided by plane cuts, and has the answer 47. If m and n are in the sequence, so is m + n - 1, since n-dissecting one cube in an m-dissection gives an (m + n - 1)-dissection.

Conway puzzle | dissection | Hadwiger problem | polycube | Slothouber-Graatsma puzzle | soma cube