Being allowed to move and stack the pieces changes the count completely. But finding an efficient procedure is not enough: we must also prove that no shorter one can work.
Twenty-Seven Cubes
Riddle statement
You have a 3 × 3 × 3 wooden cube made from 27 identical small cubes.
You want to separate all 27 cubes completely using a saw.
Each cut consists of a single straight plane. After every cut, you may freely separate, move, rotate, align, and stack the pieces obtained. A single cut may pass through every piece in a correctly aligned stack at once.
Ignore the thickness of the saw blade and any loss of material.
What is the minimum number of cuts required?
Show solution
Solution
A construction using six cuts
Make two parallel cuts, one-third and two-thirds of the way through the cube. This produces three 3 × 3 × 1 slabs.
Stack the three slabs with their faces and edges exactly aligned. Make two more parallel cuts, perpendicular to the first pair and passing through the entire stack. Each slab is divided into three bars, giving nine 3 × 1 × 1 bars.
Align and stack the nine bars. Make two final transverse cuts, one and two units from their ends. Both planes pass through every bar and produce the 27 unit cubes.
The total is:
2 + 2 + 2 = 6 cuts.
Why five cuts cannot be enough
Follow the material that will become the cube originally at the center. None of its six final faces was exposed initially: each separated it from one of its six neighbors.
Each face must be created by a different cutting plane. Until the central cube is fully separated, the material that will form it belongs to one physical piece. A single planar cut intersects that piece in only one plane and can therefore create at most one of the six distinct faces of the central cube.
Stacking multiplies the work done on other pieces, but it cannot make one cut create two different faces of the unique central cube.
At least six cuts are therefore necessary. Since the procedure above uses exactly six, the minimum is 6.