A cube is coloured blue on all faces. It is cut into 64 small cubes of equal…

Concept: For a cube painted on all six outer faces and cut into n3 equal unit cubes, every small cube falls into exactly one of four position categories: corner cubes have 3 painted faces, edge cubes (excluding corners) have 2 painted faces, face-centre cubes (excluding edges and corners) have exactly 1 painted face, and fully interior cubes have 0 painted faces. The face-centre count on each face is (n − 2)2, so across all 6 faces, the cubes with exactly one painted face total 6(n − 2)2.

Applying this to the question:

1. 64 small cubes means n3 = 64, so n = 4 — the big cube is a 4×4×4 arrangement.

2. Cubes with exactly one painted face on a single face are the interior (n − 2) × (n − 2) block of that face, away from every edge: (4 − 2)2 = 22 = 4.

3. There are 6 faces on the cube, and by symmetry this face-centre count is the same on each face.

4. Total = 6 × 4 = 24.

Cross-check: counting every position category confirms they sum to 64 (the total number of small cubes): corners = 8 (always, for n ≥ 2, three faces each); edges = 12(n − 2) = 12×2 = 24 (two faces each); face-centres = 6(n − 2)2 = 24 (one face each — this is the count we want); fully interior = (n − 2)3 = 23 = 8 (zero faces each). Sum: 8 + 24 + 24 + 8 = 64, which matches the total, confirming the count.

Answer: 24 small cubes have exactly one face coloured.