Dice and Cube Cutting
Dice and Cube Cutting
Dice questions ask which face is opposite a given face, from two or more views of the same die. The key fact: a face is never seen together with its opposite face, so the face that never appears beside a given face is opposite it. Cube-cutting questions ask how many small cubes have 3, 2, 1 or 0 painted faces when a painted cube is cut.
1. Dice Rules
| Situation | Rule |
|---|---|
| Two views with one common face in the same position | The other faces, read in the same order, are opposite in pairs |
| Several views | List each face's neighbours; the one face never seen with it is opposite |
| Standard die | Opposite faces add to 7: 1-6, 2-5, 3-4 |
2. Cube-Cutting Counts (painted cube cut into n × n × n small cubes)
| Painted faces | Number of small cubes |
|---|---|
| 3 faces (corners) | 8 |
| 2 faces (edges) | 12(n − 2) |
| 1 face (face centres) | 6(n − 2)² |
| 0 faces (hidden core) | (n − 2)³ |
Check: the four counts add to n³
Cuboid l × b × h: 2 faces = 4[(l − 2) + (b − 2) + (h − 2)], 1 face = 2[(l − 2)(b − 2) + (b − 2)(h − 2) + (h − 2)(l − 2)], 0 faces = (l − 2)(b − 2)(h − 2)
Minimum cuts to make n³ equal cubes: 3(n − 1)
✗ Two views show 1 next to 2 and 1 next to 4, so 2 and 4 are adjacent | ✓ They may be opposite; adjacency must be seen in a single view
✗ A 3 × 3 × 3 painted cube has 6 cubes with two faces painted | ✓ Two-face cubes sit on the 12 edges: 12 × (3 − 2) = 12
हिंदी नोट: पासे में जो अंक किसी अंक के साथ कभी नहीं दिखता, वही उसके विपरीत होता है। मानक पासे में विपरीत फलकों का योग 7 होता है। रंगे घन को n × n × n में काटने पर: 3 फलक 8, 2 फलक 12(n − 2), 1 फलक 6(n − 2)², कोई नहीं (n − 2)³।
Exam Pointer: Verified NTPC pattern: two views of one die, finding the face opposite a number (December 2020 CBT-1). Standard-die, cube-cutting and cuboid-cutting questions had no verified NTPC shift in our check and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Opposite faces from views of a die
[PYQ: NTPC CBT-1 30-Dec-2020 Shift-2]
EXAM LEVEL
Q. Two views of a die show faces (1, 2, 3) and (1, 4, 5), each read clockwise starting from the common face 1, with 1 in the same position. Which face is opposite 3?
With the common face fixed, the faces in matching positions are opposite: 2 is opposite 4 and 3 is opposite 5. The remaining face 6 is opposite 1.
Answer: 5
EXAMATLAS LEVEL
Q. Three views of a die show the faces (1, 2, 3), (1, 5, 6) and (3, 4, 6). Which faces are opposite 1, 3 and 2?
Neighbours of 1 seen: 2, 3, 5, 6, so 4 is opposite 1. Neighbours of 3 seen: 1, 2, 4, 6, so 5 is opposite 3. The remaining pair is 2 and 6.
Answer: 1-4, 3-5, 2-6
Pattern 2: Standard dice
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. On a standard die, 3 is on top. What number is at the bottom?
Opposite faces add to 7: 7 − 3 = 4.
Answer: 4
EXAMATLAS LEVEL
Q. Two standard dice are thrown and the top faces show 2 and 5. What is the sum of the bottom faces, and of all the hidden side faces of the first die?
Bottoms are 5 and 2, a sum of 7. A die's faces total 21; for the first die the top and bottom (2 and 5) total 7, so the four side faces total 14.
Answer: 7; 14
Pattern 3: Painted cube cut into small cubes
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A cube painted on all faces is cut into 27 equal small cubes. How many have exactly two faces painted?
n = 3: 12 × (3 − 2) = 12.
Answer: 12
EXAMATLAS LEVEL
Q. A cube painted on all faces is cut into 125 equal small cubes. How many have exactly one face painted, and how many have no face painted?
n = 5: one face 6 × 3² = 54; none 3³ = 27. (Check: 8 + 36 + 54 + 27 = 125.)
Answer: 54; 27
Pattern 4: Cuts and painted cuboids
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. What is the minimum number of cuts needed to divide a cube into 64 identical small cubes?
n = 4 needs 3 cuts along each of the three directions: 3 × 3 = 9.
Answer: 9
EXAMATLAS LEVEL
Q. A cuboid 4 cm × 3 cm × 2 cm is painted on all faces and cut into 1 cm cubes. How many cubes have exactly two faces painted, and how many have exactly one?
Two faces: 4[(4 − 2) + (3 − 2) + (2 − 2)] = 4 × 3 = 12. One face: 2[(2 × 1) + (1 × 0) + (0 × 2)] = 4. With 8 corners and none hidden, 8 + 12 + 4 = 24 cubes in all.
Answer: 12; 4
60-Second Revision
- The face never seen beside a face is its opposite.
- Standard die: opposite faces add to 7.
- n³ cubes: 8, 12(n − 2), 6(n − 2)², (n − 2)³.
- Minimum cuts for n³ cubes: 3(n − 1).