A solid cube of each side 8 cms has been painted red, blue and black on pairs…

When a cube is painted on all six outer faces and then cut into equal smaller cubes, every small cube's position inside the big cube decides how many painted faces it shows: a corner cube touches three outer faces, an edge cube (not a corner) touches two, a face-centre cube touches only one, and an interior cube touches none.

If each edge of the big cube is divided into n equal parts, these counts always follow fixed formulas: corners = 8, edge cubes = 12(n − 2), face-centre cubes = 6(n − 2)2, interior cubes = (n − 2)3 — no matter how many different colours were used on the outer faces, since colour only labels which face was painted, not how many faces a given small cube touches.

Applying this here:

1. Big cube side = 8 cm, small cube side = 2 cm, so the edge is divided into n = 8 ÷ 2 = 4 equal parts.

2. Total small cubes = n3 = 43 = 64.

3. Small cubes with exactly one painted face = 6(n − 2)2 = 6 × (4 − 2)2 = 6 × 22 = 6 × 4 = 24.

Cross-check against the total: corner cubes = 8, edge cubes = 12(4 − 2) = 24, face-centre cubes = 24 (from the step above), interior cubes = (4 − 2)3 = 8. Sum = 8 + 24 + 24 + 8 = 64, which matches the total number of small cubes, confirming the one-face count.

So 24 small cubes have exactly one face painted.