Cube division by planes is the problem of determining the regions into which planes divide a cube. For the average below, draw 
planes independently
from the motion-invariant measure on planes,
restricted to those intersecting the cube and normalized
to a probability measure. In terms of a plane's distance 
from a fixed origin and unit
normal vector 
,
the measure is proportional to 
, where 
is spherical area measure
(Santaló 1945). The average number of regions is then
 |
(Finch 2003, p. 482).
The maximum number of regions is the same as for space division by planes, namely
 |
(Yaglom and Yaglom 1987, pp. 102-106). An arrangement of planes in general position
can be scaled and translated so that every region intersects the cube.
For 
,
2, ... planes, the maximum has the values 2, 4, 8, 15,
26, 42, ... (OEIS A000125), a sequence
whose values are sometimes called cake numbers because
of their relation to the cake cutting problem.
planos arbitrarios." Rev. Unión Mat. Argentina 10, 101-108,
1945.