Non-Verbal Reasoning • Topic 5 of 5

Cubes & Dice

Dice: on a standard die opposite faces sum to 7 (1-6, 2-5, 3-4). Painted cube: a cube painted on all faces and cut into n×n×n small cubes gives 8 corner cubes (3 faces), 12(n−2) edge cubes (2 faces), 6(n−2)² face-centre cubes (1 face) and (n−2)³ inner cubes (0 faces).

Dice — opposite faces sum to 7

On a standard die the three pairs are 1-6, 2-5, 3-4. So the face opposite any number is simply 7 minus that number. From two views of a die you can deduce hidden faces by elimination.

The painted cube

Imagine a big cube painted on all six faces, then sliced into n×n×n small cubes. A small cube's painted-face count depends only on where it sat:

corner edge centre
On a 3×3×3 cube: corners (pink) = 3 painted faces, edges (blue) = 2, face-centre (green) = 1, and the hidden inside cube = 0.
Painted facesPositionCount (n×n×n)
3cornersalways 8
2edges12(n−2)
1face centres6(n−2)²
0inner(n−2)³
Check with a 3×3×3 (n=3): corners 8, edges 12(1)=12, faces 6(1)²=6, inner (1)³=1. Total = 8+12+6+1 = 27 = 3³. The four counts must always add up to n³ — use that to verify your answer.

✅ Solved examples

1. On a standard die, the face opposite 2 is?
7 − 2 = 5.
2. A cube painted and cut into 3×3×3. How many small cubes have exactly 3 painted faces?
The 8 corners.
3. Same cube — how many have 0 painted faces?
(3−2)³ = 1.
4. Same 3×3×3 — how many have exactly 1 painted face?
6(3−2)² = 6.

✏️ Practice — try these, take hints as needed

1. Opposite of 3 on a standard die?
Sum 7.
—
—
4
2. 3×3×3 cube: how many have exactly 2 painted faces?
12(n−2).
12×1.
—
12
3. 4×4×4 cube: cubes with 0 painted faces?
(n−2)³.
2³.
—
8
4. 4×4×4: cubes with 3 painted faces?
Corners.
—
—
8
5. Opposite of 1 on a die?
7−1.
—
—
6

📝 Topic test — 8 questions

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