Painted Cube Cut into Smaller Cubes
Painted Cube को छोटे cubes में काटना
Painted Cube Cut into Smaller Cubes
- Dice & Cubes
- Painted Cube Cut into Smaller Cubes
Count how many small cubes have 3, 2, 1 or 0 painted faces when a painted cube is cut into n×n×n pieces.
🎯 Learning Objective
Count how many small cubes have 3, 2, 1 or 0 painted faces when a painted cube is cut into n×n×n pieces.
💡 Concept
- Paint all 6 faces of a cube, then cut it into n×n×n = n³ equal small cubes
- 3 painted faces = the CORNER cubes = always 8 (a cube has 8 corners)
- 2 painted faces = the EDGE cubes (not corners) = 12 × (n − 2)
- 1 painted face = the FACE-centre cubes = 6 × (n − 2)²
- 0 painted faces = the hidden INNER cubes = (n − 2)³
- Check: 8 + 12(n−2) + 6(n−2)² + (n−2)³ always equals n³
🧮 Key Formulas
3 faces painted = 8
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2 faces painted = 12 × (n − 2)
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1 face painted = 6 × (n − 2)²
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0 faces painted = (n − 2)³
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Total = n³
✏️ Easy Example
Q. A painted cube is cut into 27 equal small cubes (n = 3). How many small cubes have exactly 3 painted faces?
- 3 painted faces = the corner cubes
- A cube always has 8 corners
Answer: 8
🇮🇳 Real-Life Example
A 3×3×3 Rubik's cube is exactly this: 8 corner pieces show 3 colours, 12 edge pieces show 2, 6 centre pieces show 1, and 1 hidden core shows none.
📝 Exam-Level Example
Q. A painted cube is cut into 64 equal small cubes (n = 4). How many have exactly 2 painted faces?
- 2 painted faces = 12 × (n − 2)
- = 12 × (4 − 2) = 12 × 2
Answer: 24
📝 Exam-Level Example
Q. For the same cube cut into 64 pieces (n = 4), how many small cubes have NO face painted?
- 0 painted faces = (n − 2)³
- = (4 − 2)³ = 2³
Answer: 8
📝 Exam-Level Example
Q. A painted cube is cut into 125 equal small cubes (n = 5). How many have exactly 1 painted face?
- 1 painted face = 6 × (n − 2)²
- = 6 × (5 − 2)² = 6 × 9
Answer: 54
🪄 Memory Trick
Memorise the four counts by position: corners 8, edges 12(n−2), faces 6(n−2)², core (n−2)³. Every single one uses (n − 2), never n.
⚠️ Common Mistakes
- ❌ Using n instead of (n − 2) in the edge, face and core formulas
- ❌ Mixing up edges (2 faces) with face-centres (1 face)
- ❌ Thinking corner cubes change with n — they are always 8
🏆 Exam Tips
- ✅ Always write n first, then (n − 2), before plugging into the formulas
- ✅ Sanity check: your four counts must add back up to n³
📌 Summary
- 3 faces → 8 corners (fixed)
- 2 faces → 12(n−2) edges
- 1 face → 6(n−2)² face-centres
- 0 faces → (n−2)³ core; all four sum to n³