A cube is painted yellow on all its faces and then cut into 343 small cubes of the same size, as shown below. How many of the small cubes have no face painted?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 125
343 small cubes means each edge of the big cube is cut into \(n=7\) parts, since \(7^3=343\). 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.
Taking away the outer layer on every side leaves a cube of side \(n-2=5\)
Cubes with no paint: \(5^3=125\)
Why the other options are incorrect
Option (2) 294 — Incorrect. This counts all 49 cubes on each of the 6 faces, so cubes on the edges are counted more than once.
Option (3) 8 — Incorrect. This is the number of corner cubes, which have three painted faces.
Option (4) 150 — Incorrect. This is the number of cubes in the middle of the faces, which have one painted face.