A cube can be divided into subcubes for only , 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46,
(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).
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
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.
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
is the -dimensional
Gaussian probability measure.