Dice/Cubes
Problems and solutions related to dice and cubes
Learning Content
🎲 Diagonal Basics
Full Explanation of Dice and Cube Puzzles in TNPSC Exam!
1️⃣ What is Baguita?
Dice Features:
- A cube with six faces
- 1 to 6 points on each face
- Sum of opposite faces =7
- A three-dimensional (3D) figure
Opposite Faces:
| 1 opposite | 6 |
| 2 opposite | 5 |
| 3 opposite | 4 |
Rule: Opposite faces + = 7
2️⃣ Types of division
| Type | Explanation | question pattern |
|---|---|---|
| Fixed cluster | 1-6 numbers in fixed place | Opposite face, side face detection |
| Open compartment | Expanded Format (Net) | How about a fold? |
| Spinning wheel | Rotated at different angles | What is the hidden face? |
| A painted cube | A large cube is cut | How many faces were painted? |
3️⃣ Views of Pakuda
Top View
Face on top of umbrella
Front View
Front face of umbrella
Side View
Right/Left Face of Diagonal
4️⃣ Open Dice Net
Expanded format:
Unfolding the dice yields 6 connected squares
Total Net Patterns = 11
Common Net Formats:
□ □ □ □ □ □
□ □ □ □ □ □
5️⃣ Painted cube
Cube Cutting:
Question Type:An n × n × n cube is painted on all six faces and then cut into smaller 1 × 1 × 1 cubes.
| Cubic type | Place | Painted faces |
|---|---|---|
| Corner cubes | 8 corners | 3 faces |
| Edge cubes | 12 edges | 2 faces |
| Face-centered cubes | Center of 6 faces | 1 face |
| Cubes inside | Interior | 0 faces |
6️⃣ Divisive Formulas (n × n × n)
| Detail | formula | For 3×3×3 |
|---|---|---|
| Aggregate small cubes | n³ | 27 |
| 3 faces painted | 8 (always) | 8 |
| 2 faces painted | 12(n-2) | 12 |
| 1 Face paint | 6(n-2)² | 6 |
| 0 face paint | (n-2)³ | 1 |
7️⃣ dice puzzle types
📌 Type 1: Opposite face
What number is opposite the given face?
📌 Type 2: Side face
What number is next to the given face?
📌 Type 3: Rotation
What is the hidden face when you rotate the dice?
📌 Type 4: Net folding
How about folding the unfolded shape?
📌 Type 5: Cutting
How many smaller cubes have n faces painted?
📌 Type 6: Collateral division
Compare the two groups and choose the correct option?
8️⃣ Positions
| condition | Explanation | vision |
|---|---|---|
| Level 1 | 1 up, 2 ahead | Basic level |
| Stage 2 | pre-rotated | Up-down dynamic |
| Level 3 | Side rotated | Left-right dynamic |
| Level 4 | Rotated up | Front-back dynamic |
9️⃣ Solution steps
Step 1: Find common faces
Mark the numbers common to two or three groups
Step 2: Mark the adjacent faces
List what numbers are next to each number
Step 3: Calculate the opposite faces
Find the opposite sides using the sum = 7 rule
Step 4: Verification
Check all conditions again
🔟 Important rules
| fate | Explanation |
|---|---|
| Rule 1 | The sum of opposite sides is always 7 |
| Rule 2 | A face has 4 adjacent faces |
| Rule 3 | Opposite boxes in Net should not be in a straight line |
| Rule 4 | Corner cubes are always 8 (regardless of n value). |
| Rule 5 | 1-2-3 faces meet at a corner (right hand rule) |
💡 Things to remember:
- A dice has a total of 6 faces, 12 edges, and 8 corners
- Opposite faces: 1-6, 2-5, 3-4
- 4 consecutive numbers for each number
- A total of n³ smaller cubes in an n×n×n cube
- The net shape is folded to get the correct quotient
📐 Fraction formulas
n × n × n Cubic Intersection Formulas!
🎲 Basic formulas
| calculation | formula |
|---|---|
| Aggregate small cubes | n³ |
| 3 face paint (corner) | 8(always) |
| 2 faces painted (margin) | 12(n-2) |
| 1 Face Paint (Center of Face) | 6(n-2)² |
| 0 Face Paint (Internal) | (n-2)³ |
📊 Quick values
| n | Total | 3 face | 2 face | 1 face | 0 face |
|---|---|---|---|---|---|
| 2 | 8 | 8 | 0 | 0 | 0 |
| 3 | 27 | 8 | 12 | 6 | 1 |
| 4 | 64 | 8 | 24 | 24 | 8 |
| 5 | 125 | 8 | 36 | 54 | 27 |
🎯 Opposite face rule
1↔6, 2↔5, 3↔4
📚 Solved Questions – 10
Question 1: Opposite face
What number is opposite 3 in standard dice?
3 + ? = 7 → ? =4
Question 2: 3×3×3 cube
How many smaller cubes in a 3×3×3 cube have 2 faces painted?
Question 3: 4×4×4 cube
How many uncolored cubes are there in a 4×4×4 cube?
Question 4: Corner cubes
How many cubes are there in a 5×5×5 cube with 3 faces painted?
Question 5: 1 Face paint
How many 4×4×4 cubes have only 1 face painted?
Question 6: Total
How many small cubes are there in a 5×5×5 cube?
Question 7: Margin
2 faces painted in a 6×6×6 cube?
Question 8: Internal Cubes
How many are uncolored in a 5×5×5 cube?
Question 9: Validation
Check everything is correct in 3×3×3 cube
3 face = 8, 2 face = 12, 1 face = 6, 0 face = 1
Total = 8+12+6+1 = 27 = 3³ ✓
Question 10: Net folding
1 is followed by 2,3,4,5 in Net. Against 1?
The rest6should be opposite.
⚡ Speed shortcuts
🎯 Opposite Face Velocity Law
The sum is always 7!
📊 Values to remember
- 3 face paint = 8(for any n!)
- 3×3×3:27 = 8+12+6+1
- 4×4×4:64 = 8+24+24+8
⚡ Speed formulas
| corner (3 face) | Always 8 |
| edge (2 face) | 12 × (n-2) |
| Face center (1 face) | 6 × (n-2)² |
| internal (0 face) | (n-2)³ |
🎲 Net rule
• If 4 page numbers are known for a number in Net
• Remaining number = opposite face
💡 Exam Tips
- Corner = 8 does not change
- If n=2 then all are corner only
- Sum of opposite faces = 7
- Verification: If all add up to n³
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