Cubes and Dice
CSAT · Logical Reasoning
Introduction
Cube and dice questions test spatial visualisation. Painted-cube questions have exact formulas, while dice questions are solved by noting which faces can be seen together.
Key concepts
Painted cubes
A large cube painted on all faces and cut into n × n × n small cubes:
| Painted faces | Count | Where |
|---|---|---|
| 3 | 8 | corners |
| 2 | 12(n − 2) | edges, excluding corners |
| 1 | 6(n − 2)² | face centres |
| 0 | (n − 2)³ | hidden inside |
The four counts add to n³.
Dice
- A die has 6 faces; faces seen together are adjacent, never opposite.
- On a standard die, opposite faces add to 7 (1–6, 2–5, 3–4).
- If two views share a common face in the same position, the faces in matching positions are the same faces.
- If a face appears in two views, the faces not seen with it across both views include its opposite.
Nets
A cube net has six squares. In a straight row of three or four squares, alternate squares are opposite each other. In the common 'cross' net, the square two away in a line is the opposite face.
Method for two views
List faces adjacent to the number in question from both views (up to four distinct numbers). The remaining number is its opposite.
- Corners 8; edges 12(n − 2); faces 6(n − 2)²; core (n − 2)³
- Standard die: opposite faces sum to 7
- Adjacent faces never opposite
Solved examples
Example 1
A cube painted on all faces is cut into 27 identical small cubes (3 × 3 × 3). How many small cubes have exactly one face painted?
(a) 1 (b) 12 (c) 8 (d) 6
- Count with n = 3: 6(n − 2)² = 6 × 1 = 6.
- Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Answer: (d) 6
Example 2
A cube painted on all faces is cut into 125 identical small cubes (5 × 5 × 5). How many small cubes have three faces painted?
(a) 36 (b) 8 (c) 54 (d) 27
- Count with n = 5: 8 (the corners) = 8.
- Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Answer: (b) 8
Example 3
Two views of the same die are shown. View 1: top F, front B, right E. View 2: top A, front E, right C. Which face is opposite E?
(a) F (b) A (c) D (d) C
- Faces seen together in one view are adjacent, never opposite.
- Faces adjacent to E: A, B, C, F (from the views).
- Checking every possible labelling consistent with both views, the only face that can be opposite E is D.
Answer: (c) D
Example 4
Two views of the same die are shown. View 1: top 4, front 3, right 6. View 2: top 4, front 5, right 1. Which face is opposite 3?
(a) 6 (b) 5 (c) 2 (d) 1
- Faces seen together in one view are adjacent, never opposite.
- Faces adjacent to 3: 4, 6 (from the views).
- Checking every possible labelling consistent with both views, the only face that can be opposite 3 is 5.
Answer: (b) 5
Example 5
A cube painted on all faces is cut into 27 identical small cubes (3 × 3 × 3). How many small cubes have three faces painted?
(a) 6 (b) 12 (c) 1 (d) 8
- Count with n = 3: 8 (the corners) = 8.
- Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Answer: (d) 8
Example 6
A cube painted on all faces is cut into 343 identical small cubes (7 × 7 × 7). How many small cubes have exactly one face painted?
(a) 125 (b) 60 (c) 150 (d) 8
- Count with n = 7: 6(n − 2)² = 6 × 25 = 150.
- Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Answer: (c) 150
Example 7
Two views of the same die are shown. View 1: top F, front C, right E. View 2: top E, front A, right B. Which face is opposite A?
(a) F (b) D (c) B (d) C
- Faces seen together in one view are adjacent, never opposite.
- Faces adjacent to A: B, E (from the views).
- Checking every possible labelling consistent with both views, the only face that can be opposite A is F.
Answer: (a) F
Example 8
A cube painted on all faces is cut into 216 identical small cubes (6 × 6 × 6). How many small cubes have exactly one face painted?
(a) 8 (b) 96 (c) 64 (d) 48
- Count with n = 6: 6(n − 2)² = 6 × 16 = 96.
- Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Answer: (b) 96
- Check that the four painted-cube counts add up to n³.
- In a net, alternate squares in a straight line are opposite.
- Collect all neighbours of the face from every view – the face never seen with it is the opposite.
- Using n instead of n − 2 for inner cubes.
- Assuming every die is standard when the views show otherwise.
- Treating diagonally placed squares in a net as opposite.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. Two views of the same die are shown. View 1: top C, front E, right A. View 2: top F, front B, right C. Which face is opposite E?
(a) D (b) B (c) F (d) C
Q2. A cube painted on all faces is cut into 64 identical small cubes (4 × 4 × 4). How many small cubes have three faces painted?
(a) 8 (b) 24 (c) 12 (d) 9
Q3. Two views of the same die are shown. View 1: top 6, front 1, right 3. View 2: top 5, front 3, right 4. Which face is opposite 5?
(a) 1 (b) 2 (c) 4 (d) 3
Q4. A cube painted on all faces is cut into 27 identical small cubes (3 × 3 × 3). How many small cubes have exactly two faces painted?
(a) 6 (b) 12 (c) 1 (d) 8
Q5. Two views of the same die are shown. View 1: top C, front E, right F. View 2: top D, front C, right B. Which face is opposite F?
(a) B (b) E (c) C (d) D
Q6. Two views of the same die are shown. View 1: top 5, front 3, right 1. View 2: top 6, front 5, right 2. Which face is opposite 1?
(a) 2 (b) 3 (c) 4 (d) 6
Q7. Two views of the same die are shown. View 1: top C, front A, right E. View 2: top F, front C, right B. Which face is opposite D?
(a) B (b) E (c) C (d) A
Q8. Two views of the same die are shown. View 1: top 1, front 4, right 5. View 2: top 3, front 5, right 2. Which face is opposite 6?
(a) 1 (b) 2 (c) 4 (d) 5
Q9. A cube painted on all faces is cut into 343 identical small cubes (7 × 7 × 7). How many small cubes have three faces painted?
(a) 125 (b) 8 (c) 60 (d) 150
Q10. Two views of the same die are shown. View 1: top A, front D, right E. View 2: top E, front C, right B. Which face is opposite A?
(a) D (b) B (c) C (d) F
Q11. Two views of the same die are shown. View 1: top F, front A, right C. View 2: top F, front E, right D. Which face is opposite C?
(a) B (b) D (c) F (d) E
Q12. Two views of the same die are shown. View 1: top D, front B, right F. View 2: top B, front A, right E. Which face is opposite A?
(a) F (b) C (c) E (d) D
Q1 – (c) F. Faces seen together in one view are adjacent, never opposite. Faces adjacent to E: A, C (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite E is F.
Q2 – (a) 8. Count with n = 4: 8 (the corners) = 8. Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Q3 – (a) 1. Faces seen together in one view are adjacent, never opposite. Faces adjacent to 5: 3, 4 (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite 5 is 1.
Q4 – (b) 12. Count with n = 3: 12(n − 2) = 12 × 1 = 12. Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Q5 – (d) D. Faces seen together in one view are adjacent, never opposite. Faces adjacent to F: C, E (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite F is D.
Q6 – (d) 6. Faces seen together in one view are adjacent, never opposite. Faces adjacent to 1: 3, 5 (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite 1 is 6.
Q7 – (c) C. Faces seen together in one view are adjacent, never opposite. Faces adjacent to D: (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite D is C.
Q8 – (d) 5. Faces seen together in one view are adjacent, never opposite. Faces adjacent to 6: (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite 6 is 5.
Q9 – (b) 8. Count with n = 7: 8 (the corners) = 8. Corners: 3 painted faces; edges (not corners): 2; face centres: 1; the hidden core: 0.
Q10 – (c) C. Faces seen together in one view are adjacent, never opposite. Faces adjacent to A: D, E (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite A is C.
Q11 – (b) D. Faces seen together in one view are adjacent, never opposite. Faces adjacent to C: A, F (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite C is D.
Q12 – (a) F. Faces seen together in one view are adjacent, never opposite. Faces adjacent to A: B, E (from the views). Checking every possible labelling consistent with both views, the only face that can be opposite A is F.
- 8, 12(n−2), 6(n−2)², (n−2)³
- Seen together ⇒ adjacent
- Net: alternate squares opposite
- Standard die sums to 7
