📑 Topics on this page (Jump to Topic):
📌 1. Dice Rules
- Standard die: opposite faces sum to 7 (1-6, 2-5, 3-4).
- The three faces seen in one position are adjacent — never opposite.
- Two positions with one common face: read clockwise from the common face; the remaining faces are opposite in order.
- If 4 adjacent faces of a face are known, the remaining one is its opposite.
⚠️ Exam Traps:
- If two positions share two faces, the remaining two faces are opposite each other.
📌 2. Cutting a Painted Cube
| Small cubes (n × n × n) | Count |
|---|---|
| Total | n³ |
| 3 faces painted (corners) | 8 |
| 2 faces painted (edges) | 12(n−2) |
| 1 face painted | 6(n−2)² |
| No face painted | (n−2)³ |
💡 Memory Tip: Check: 8 + 12(n−2) + 6(n−2)² + (n−2)³ = n³. For n = 3: 8 + 12 + 6 + 1 = 27.
📌 3. Cube Nets
- In a straight row of a net, faces one apart are opposite.
- A cube has 11 distinct nets; shapes where four faces meet at one point cannot fold into a cube.
📌 4. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. On a standard die, which number is opposite 2? Practice
Correct answer: (B)
Opposite faces sum to 7 ⇒ 7 − 2 = 5.
Opposite faces sum to 7 ⇒ 7 − 2 = 5.
Q2. Two positions of a die show (1, 2, 3) and (1, 4, 5). Which number is opposite 1? Practice
Correct answer: (C)
1 is adjacent to 2, 3, 4, 5 ⇒ the remaining 6 is opposite.
1 is adjacent to 2, 3, 4, 5 ⇒ the remaining 6 is opposite.
Q3. A 3 × 3 × 3 cube painted on all sides is cut into 27 small cubes. How many have exactly 2 faces painted? Practice
Correct answer: (D)
12(n − 2) = 12 × 1 = 12.
12(n − 2) = 12 × 1 = 12.
Q4. A 4 × 4 × 4 painted cube is cut into 64. How many small cubes have no face painted? Practice
Correct answer: (A)
(4 − 2)³ = 8.
(4 − 2)³ = 8.
Q5. In a painted 4 × 4 × 4 cube, how many small cubes have exactly 1 face painted? Practice
Correct answer: (B)
6(4 − 2)² = 24.
6(4 − 2)² = 24.
Q6. In any painted cube (n ≥ 2), how many small cubes have 3 faces painted? Practice
Correct answer: (C)
Only the 8 corner cubes.
Only the 8 corner cubes.
Q7. In a painted 5 × 5 × 5 cube, how many small cubes have exactly 2 faces painted? Practice
Correct answer: (D)
12(5 − 2) = 36.
12(5 − 2) = 36.
Q8. Two positions of a die show (A, B, C) and (A, B, D). What is opposite C? Practice
Correct answer: (A)
Two faces common ⇒ the remaining C and D are opposite.
Two faces common ⇒ the remaining C and D are opposite.
Q9. What is true about two faces seen in the same position? Practice
Correct answer: (B)
Faces seen together are never opposite.
Faces seen together are never opposite.
Q10. How many edges does a cube have? Practice
Correct answer: (C)
Cube: 6 faces, 8 vertices, 12 edges.
Cube: 6 faces, 8 vertices, 12 edges.
Q11. In a painted 3 × 3 × 3 cube, how many small cubes have no face painted? Practice
Correct answer: (D)
(3 − 2)³ = 1 (the centre one).
(3 − 2)³ = 1 (the centre one).
Q12. On a standard die, 1, 2, 3 are visible together. Sum of the three hidden faces? Practice
Correct answer: (A)
Hidden = 6, 5, 4 ⇒ 15 (total 21 − 6).
Hidden = 6, 5, 4 ⇒ 15 (total 21 − 6).
Q13. In a cube net, faces P, Q, R, S lie in a straight row in that order. What is opposite P? Practice
Correct answer: (B)
In a straight row, alternate faces are opposite ⇒ P-R, Q-S.
In a straight row, alternate faces are opposite ⇒ P-R, Q-S.
Q14. A cube is painted red on two opposite faces, blue on two and green on two, then cut into 27. How many small cubes have both red and blue? Practice
Correct answer: (C)
Red-blue edges = 2 × 2 = 4; 3 cubes on each edge ⇒ 12 (including the 8 corners, which also have green).
Red-blue edges = 2 × 2 = 4; 3 cubes on each edge ⇒ 12 (including the 8 corners, which also have green).
Q15. Of 64 small cubes from a painted 4 × 4 × 4 cube, how many have at least one face painted? Practice
Correct answer: (D)
64 − 8 = 56.
64 − 8 = 56.
Q16. Three positions show 1 adjacent to 2, 3, 5 and 6. What is opposite 1? Practice
Correct answer: (A)
Four neighbours of 1 are known ⇒ the remaining 4 is opposite.
Four neighbours of 1 are known ⇒ the remaining 4 is opposite.
Q17. How many distinct nets does a cube have? Practice
Correct answer: (B)
A cube has 11 distinct nets.
A cube has 11 distinct nets.
Q18. A painted cube is cut into 125 equal small cubes. How many have exactly 1 face painted? Practice
Correct answer: (C)
n = 5 ⇒ 6(3)² = 54.
n = 5 ⇒ 6(3)² = 54.
Q19. A standard die shows 4 on top. Which number is at the bottom? Practice
Correct answer: (D)
7 − 4 = 3.
7 − 4 = 3.
Q20. A cube is cut with 3 cuts (one in each direction). How many pieces at most? Practice
Correct answer: (A)
One cut per direction ⇒ 2 × 2 × 2 = 8.
One cut per direction ⇒ 2 × 2 × 2 = 8.