Cutting a cube · Almost Obvious

A solid \(3\times3\times3\) cube consists of 27 unit cubes joined together, like an idealized Rubik’s Cube. What is the minimum number of straight planar cuts needed to separate all 27 cubes?

You may rearrange and stack the pieces between cuts, but each unit cube must remain intact.

A cube with each visible face divided into a three-by-three grid, representing 27 joined unit cubes.

Hint

Which unit cube is hardest to free?

Solution

Look at the middle cube. It has six neighbours, one sharing each of its six faces. To free it, all six shared faces must be cut.

A straight planar cut can run along at most one of these faces, since no two lie in the same plane. Rearranging or stacking the pieces does not change this: the middle cube still needs a separate cut along each face. Thus, at least six cuts are necessary.

Six cuts also suffice: make two parallel cuts in each of the three directions, along the unit-cube boundaries. This separates the original cube into all 27 unit cubes.

The minimum is therefore \(\boxed{6}\).

The middle cube needs six cuts

0 / 6 cuts

An exploded view of the golden middle cube and its six green face-sharing neighbours. Each of its faces needs a separate cut.

Middle cubeSix neighbours

Exploded view · gaps added for clarity.

All six neighbours still share a face with the middle cube.

Info

Next cut follows two parallel cuts in each of the three directions. The dashed plane marks the latest cut. Switch to Whole cube to see how these same cuts make 27 pieces.

The middle-cube view separates the cubes for visibility. Pale neighbours have already been disconnected; each cut frees just one face of the golden cube.