Cutting a cube into 27
A solid wooden cube, 3 units on each side, is to be cut into 27 unit cubes with straight saw cuts. Between cuts you may rearrange the pieces and stack them, so that one cut goes through several pieces at once.
What is the smallest number of cuts needed?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Six cuts are enough without any rearranging: two in each direction. Can rearranging help?
Think about the unit cube in the very middle. How were its faces made?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The centre cube has six faces, none of them on the surface of the original cube, so each must be made by a cut. A flat cut makes a face in a single plane, and the six faces of a cube lie in six different planes. So no cut can make two of them.
Upper bound. Cutting twice in each of the three directions, without moving anything, produces the 27 unit cubes: 6 cuts.
Lower bound. Each face of the centre cube is created by some cut, and one cut through a piece produces new faces lying in one plane on that piece. Two different faces of the centre cube are never in the same plane, and they belong to the same piece until the centre cube is free, so they need different cuts. Six faces, six cuts.
So the answer is 6, and rearranging does not help.
When it does help. For a cube, cutting without rearranging takes cuts, but stacking halves cleverly brings it down to 6: each cut can then halve every piece in one direction.