Cube Dissection

A cube can be divided into subcubes for only , 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and (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 (Gardner 1992, pp. 297-298).

If and are in the sequence, so is , since -dissecting one cube in an -dissection gives an -dissection. The numbers 1, 8, 20, 38, 49, 51, 54 are in the sequence because of dissections corresponding to the equations

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

Combining these facts gives the remaining terms in the sequence, and all numbers , and it has been shown that no other numbers occur.

It is not possible to cut a cube into subcubes that are all different sizes (Gardner 1961, p. 208; Gardner 1992, p. 298).

The seven pieces used to construct the cube dissection known as the Soma cube are one 3-polycube and six 4-polycubes (), illustrated above.

Another cube dissection due to Steinhaus (1999) uses three 5-polycubes and three 4-polycubes (), illustrated above. There are two solutions.

It is possible to cut a rectangle into two identical pieces which will form a cube (without overlapping) when folded and joined. In fact, an infinite number of solutions to this problem were discovered by C. L. Baker (Hunter and Madachy 1975).

Lonke (2000) has considered the number of -dimensional faces of a random -dimensional central section of the -cube , and gives the special result

(8)

where is the -dimensional Gaussian probability measure.