In the given figure, how many small cubes are such that each face of the cube…

A small cube inside a larger cube built from n × n × n unit cubes has all six of its faces touching neighbouring cubes only if it does not lie anywhere on the outer surface of the larger cube. Removing the outermost layer of cubes along every one of the three dimensions (length, breadth, height) leaves a smaller (n − 2) × (n − 2) × (n − 2) block, and only the cubes inside this reduced block are fully surrounded on every face.

1. From the figure, the large cube is built from 4 unit cubes along each edge, i.e., a 4 × 4 × 4 arrangement.

2. Any cube lying in the outermost layer along the length, breadth, or height has at least one face exposed to the outside, so it cannot be fully surrounded.

3. Removing one layer of cubes from each of the two opposite faces along every axis reduces each edge length by 2, i.e., 4 − 2 = 2.

4. The fully surrounded cubes therefore form an inner block of size 2 × 2 × 2.

5. Counting the cubes in this inner block: 2 × 2 × 2 = 8.

Cross-check layer by layer: of the 4 layers along the height, only the middle 2 layers can possibly contain hidden cubes; within each such layer, only the inner 2 × 2 block (excluding its outer ring) does not touch an outer face — 2 layers × 4 cubes per layer = 8, the same result.

Hence, 8 small cubes have all six faces touching neighbouring cubes.