🎙 Concept / आधार
Cubes & Dice (घन व पासा) में मुख्य रूप से opposite faces, adjacent faces और
hidden faces की पहचान पूछी जाती है।
Delhi Police में 1 या 2 direct, easy questions आते हैं।
✔ Important Rules / मुख्य नियम
Dice Rules (पासा):
➤ Opposite faces का sum = 7 (Standard Dice)
➤ Example: (1–6), (2–5), (3–4)
Cube Rules (घन):
➤ Opposite faces कभी adjacent नहीं होते
➤ Adjacent faces हमेशा एक-दूसरे को छूते हैं
➤ Folding/Unfolding से positions समझें
Color Cube:
➤ Painted cube → cut into small cubes
➤ 3-face painted → corners
➤ 2-face painted → edges
➤ 1-face painted → face-centre cubes
➤ 0-face painted → inner cubes
✔ Solved Examples (Delhi Police Level)
Example 1 – Dice Opposites:
If 3 is opposite 4, and 1 is opposite 6, then opposite of 2?
Remaining pair = 2–5
Example 2 – Rotation:
If 5 is on top and 2 in front, what is at bottom?
Top = 5 → Bottom = 2 (if standard dice logic reversed)
Final = 2
Example 3 – Cube Painting:
A cube painted on all sides is cut into 27 small cubes.
How many cubes have 3 faces painted?
Corners = 8
Ans: 8
Example 4 – 1 Face Painted:
Face-centre cubes = edges/centre count = 6 faces × 1 each = 6
Ans: 6
Example 5 – Opposite Faces (Diagram Logic):
If in unfolded cube A adjacent to B, B adjacent to C,
and A opposite D → D never touches A.
Opposite: A–D
✔ Practice MCQs
- In standard dice, opposite of 1?
Ans: 6
- If 2 opposite 5, 3 opposite 4. What is opposite of 1?
Ans: 6
- Cube painted on 6 faces → cut into 64 cubes.
How many 3-face painted?
Ans: 8
- How many cubes have exactly 2 faces painted from 64 cubes?
Ans: 24
- How many have 1 face painted?
Ans: 24
- Unpainted cubes?
Ans: 8
- If 3 on top and 6 in front → opposite of 3?
Ans: 4
- If 4 adjacent to 5 and 5 adjacent to 2 → opposite of 4?
Ans: 3
- Number of corners of a cube?
Ans: 8
- Edges in a cube?
Ans: 12
🔥 Challenge Question
Cube painted on all sides → cut into 125 small cubes.
How many cubes have 0 faces painted?
Inner cubes = (n−2)³ = (5−2)³ = 27
Answer: 27
Motivation: Cubes & Dice आसान है — बस rules याद रखो और diagram मन में visualize करो। 🚓🔥