A cube is painted blue on all its faces and then cut into 216 small cubes of the same size, as shown below. How many of the small cubes have exactly one face painted?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 96
216 small cubes means each edge of the big cube is cut into \(n=6\) parts, since \(6^3=216\). How many painted faces a small cube has depends on where it sits: corner cubes have 3, the other cubes along an edge have 2, the cubes in the middle of a face have 1, and the cubes inside have none. One face of the big cube is shown below with the number of painted faces of each small cube.
On each face the middle square is \((n-2)\times(n-2)=4\times4=16\) cubes
The cube has 6 faces, so \(6\times16=96\) cubes
Why the other options are incorrect
Option (1) 48 — Incorrect. This is the number of cubes on the edges (corners left out), which have two painted faces.
Option (2) 72 — Incorrect. This counts all 6 cubes along each of the 12 edges, corners included, and so counts every corner more than once.
Option (4) 8 — Incorrect. This is the number of corner cubes, which have three painted faces.