A cube of side 5 cm is painted on all its faces. If it is sliced into 1 cm3…

Concept

For a cube of edge n (in cm) that is painted on the outside and cut into 1 cm3 unit cubes, every small cube falls into exactly one of four categories based on how many of its faces were on the original surface: 3 painted faces (corners), 2 painted faces (edges, excluding corners), 1 painted face (face centres, excluding edges), or 0 painted faces (strictly interior). On each face of the big cube, the cubes with exactly one painted face form the interior (n − 2) × (n − 2) grid of that face, so across all 6 faces there are 6 × (n − 2)2 such cubes.

Step-by-step solution

1. The cube is cut into 1 cm3 pieces, so the number of unit cubes along each edge is n = 5.

2. Cubes with exactly one painted face lie in the (n − 2) × (n − 2) inner grid of each face, away from every edge: (n − 2)2 = (5 − 2)2 = 32 = 9 per face.

3. A cube has 6 faces, so the total is 6 × 9 = 54.

Cross-check

The four categories must account for all n3 unit cubes: interior cubes (0 painted faces) = (n − 2)3 = 27; edge cubes (2 painted faces) = 12(n − 2) = 36; corner cubes (3 painted faces) = 8 (fixed, independent of n); and face-centre cubes (1 painted face) = 54. Their sum, 27 + 36 + 8 + 54 = 125, equals 53, the total number of unit cubes — confirming the count is consistent.

So 54 unit cubes have exactly one face painted.