KA-0089Building and counting with unit cubes
4 points, difficulty 2 of 3Level 3–4about 75sA cube 3 units on each side is built from unit cubes and painted all over the outside. How many of the small cubes have no paint on them at all?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Which cubes touch no outside face?
2The strategy
A cube has no paint only if it is strictly inside — not on any face, edge or corner. Peel one layer off every side of the block and see what is left.
3The full solution
Removing one layer from all six sides of a 3 by 3 by 3 block leaves a 1 by 1 by 1 block: a single cube.
Solution
The reliable way
An unpainted cube must not touch any outside face, so strip one layer from each of the six sides. That takes the 3 by 3 by 3 block down to 1 by 1 by 1 — a single cube, right in the middle. The transferable idea: the unpainted core of an n by n by n block is an (n - 2) cube, which is why a 3-cube has exactly one.
The elegant way
The hidden core is (3 - 2) cubed = 1.
Why this is on the test: It builds the peel-a-layer picture that every painted-cube question uses, including the ones asking for exactly one or exactly two painted faces.