How to Solve: CUET Reasoning – Dice, Cube and Cuboid

How to Solve: CUET Reasoning – Dice, Cube and Cuboid

Introduction

"Imagine you’re handed a folded paper cube in your CUET exam—can you predict the hidden faces in 30 seconds? Master dice, cubes, and cuboids, and you’ll crack 5+ questions in under 10 minutes—guaranteed."

What You Need To Know First

1. Basic 3D shapes: Know what a cube, cuboid, and die look like (6 faces, 12 edges, 8 vertices).
2. Net of a cube: A 2D layout that folds into a cube (6 squares in specific arrangements).
3. Opposite faces rule: On a standard die, opposite faces always add up to 7 (1-6, 2-5, 3-4).

Key Vocabulary

Formulas To Know

1. Opposite faces of a standard die
2. Formula: Face₁ + Face₂ = 7
3. Variables:
   • Face₁ = Number on one face.
   • Face₂ = Number on the opposite face.

4. MEMORISE THIS (not given on exam sheet).

5. Number of painted faces on a cuboid

6. Formula:
   • 3 faces painted: 8 corners (always).
   • 2 faces painted: 4 × (L + B + H – 6) edges.
   • 1 face painted: 2 × (LB + BH + LH – 4L – 4B – 4H + 12) faces.
   • 0 faces painted: (L–2)(B–2)(H–2) inner cubes.
7. Variables:
   • L = Length in units, B = Breadth, H = Height.

8. Given on exam sheet (but practice deriving it).

9. Surface area of a cube/cuboid

10. Cube: 6 × side²
11. Cuboid: 2(LB + BH + LH)
12. MEMORISE THIS (used for volume/painting problems).

Step-by-Step Method

For Dice Problems (Standard/Non-Standard)

Step 1: Identify the type of die - Standard die: Opposite faces add to 7 (1-6, 2-5, 3-4). - Non-standard die: Opposite faces are given or must be deduced.

Step 2: Note the visible faces - Write down the numbers/colors you see and their positions (top, front, right).

Step 3: Apply the opposite faces rule - For standard dice, subtract each visible face from 7 to find its opposite. - For non-standard dice, use the given opposite pairs.

Step 4: Eliminate impossible options - Adjacent faces cannot be opposite. Cross out options that violate this.

Step 5: Visualize or sketch the die - Draw a net or imagine rotating the die to confirm hidden faces.

Step 6: Answer the question - Predict the hidden face, opposite face, or next view based on the above.

For Cube/Cuboid Problems (Painting/Unfolding)

Step 1: Understand the question - Is it about painting (how many cubes have 1/2/3 faces painted)? - Or unfolding (which net matches the cube)?

Step 2: For painting problems - 3 faces painted: Always the 8 corners. - 2 faces painted: Cubes on edges (not corners). Formula: 4 × (L + B + H – 6). - 1 face painted: Cubes on faces (not edges). Formula: 2 × (LB + BH + LH – 4L – 4B – 4H + 12). - 0 faces painted: Inner cubes. Formula: (L–2)(B–2)(H–2).

Step 3: For unfolding problems - Count the squares in the net (must be 6 for a cube). - Identify the "base" face and fold mentally to match the 3D view. - Check adjacency: Faces sharing an edge in the net must be adjacent in 3D.

Step 4: Verify with options - For painting: Plug numbers into formulas. - For unfolding: Compare with given nets/3D views.

Worked Example (Dice Problem)

Question: A die is rolled. The top face shows 3, and the front face shows 5. What is the number on the face opposite to 3?

Step 1: Standard die (opposite faces add to 7). Step 2: Visible faces: Top = 3, Front = 5. Step 3: Opposite of 3 = 7 – 3 = 4. Step 4: Check adjacency: 3 and 5 are adjacent (valid). Step 5: Sketch:

    [Top: 3]
[Left] [Front: 5] [Right]

    [Bottom: ?]

Opposite of 3 is bottom = 4. Step 6: Answer = 4.

What we did and why: Used the standard die rule to find the opposite face quickly. Adjacency check ensures no contradictions.

Worked Examples

Example 1 - Basic (Standard Die)

Question: A die shows 2 on the top face. Which number is on the bottom face? Solution: 1. Standard die: Opposite faces add to 7. 2. Top = 2 → Bottom = 7 – 2 = 5. What we did and why: Direct application of the opposite faces rule.

Example 2 - Medium (Non-Standard Die)

Question: A die has the following opposite faces: 1-4, 2-6, 3-5. If the top face is 2 and the front face is 3, what is the right face? Solution: 1. Non-standard die: Given opposite pairs (1-4, 2-6, 3-5). 2. Top = 2 → Bottom = 6 (opposite). 3. Front = 3 → Back = 5 (opposite). 4. Remaining faces: 1 and 4 must be left/right. 5. Right face cannot be 6 (bottom) or 5 (back) or 2 (top) or 3 (front). 6. Right face must be 1 or 4. Since 1 and 4 are opposite, only one is possible. 7. Answer: 4 (if 1 is left) or 1 (if 4 is left). The question implies a single answer, so pick one (e.g., 4). What we did and why: Used given opposite pairs to eliminate impossible options. Adjacency rules narrow it down to two choices.

Example 3 - Exam Style (Cuboid Painting)

Question: A cuboid of dimensions 4×3×2 is painted red on all faces and then cut into 1×1×1 cubes. How many small cubes have exactly 2 faces painted? Solution: 1. Use the formula for 2 faces painted: 4 × (L + B + H – 6). 2. Plug in L=4, B=3, H=2:
4 × (4 + 3 + 2 – 6) = 4 × (3) = 12. 3. Verify:
- Edges (not corners): 4 edges of length 4 (4–2=2 cubes each), 4 edges of length 3 (3–2=1 cube each), 4 edges of length 2 (2–2=0 cubes).
- Total: (4×2) + (4×1) + (4×0) = 8 + 4 + 0 = 12. What we did and why: Applied the formula first, then verified by counting edges to avoid mistakes.

Common Mistakes

1. Mistake: Assuming all dice are standard.
   WHY IT HAPPENS: Students memorize 1-6 opposite pairs and forget non-standard dice exist.
   CORRECT APPROACH: Check if the question specifies opposite pairs. If not, assume standard.

2. Mistake: Counting corners for 2 faces painted.
   WHY IT HAPPENS: Corners have 3 painted faces, but students include them in "2 faces painted."
   CORRECT APPROACH: 2 faces painted = edges only (exclude corners).

3. Mistake: Miscounting net squares.
   WHY IT HAPPENS: Students forget a cube’s net must have exactly 6 squares.
   CORRECT APPROACH: Count squares before folding. If ≠6, it’s not a cube net.

4. Mistake: Ignoring adjacency in dice problems.
   WHY IT HAPPENS: Students pick opposite faces without checking if they’re adjacent in the view.
   CORRECT APPROACH: Adjacent faces share an edge; opposite faces never do.

5. Mistake: Using the wrong formula for cuboid painting.
   WHY IT HAPPENS: Mixing up formulas for 1, 2, or 3 faces painted.
   CORRECT APPROACH: Memorize:

6. 3 faces: 8 corners.
7. 2 faces: 4 × (L + B + H – 6).
8. 1 face: 2 × (LB + BH + LH – 4L – 4B – 4H + 12).

Exam Traps

1. Trap: Non-standard dice with hidden opposite pairs.
   How to Spot it: The question shows a die with numbers/colors but doesn’t state opposite pairs.
   How to Avoid it: Look for clues in the question (e.g., "the die has 1 opposite 4"). If none, assume standard.

2. Trap: Cuboid dimensions in different units.
   How to Spot it: The question gives dimensions like 4 cm × 3 m × 2 mm.
   How to Avoid it: Convert all dimensions to the same unit before applying formulas.

3. Trap: "Which of these cannot be a net of a cube?"
   How to Spot it: The options include nets with 5 or 7 squares.
   How to Avoid it: Remember: A cube net must have exactly 6 squares. Eliminate options with ≠6 squares immediately.

1-Minute Recap

"Alright, CUET warriors—last-minute dice, cube, and cuboid recap. Here’s what you must remember:

1. Standard die: Opposite faces add to 7. Top=3? Bottom=4. Done.
2. Non-standard die: Use the given opposite pairs. No pairs? Assume standard.
3. Adjacent faces: They share an edge. Opposite faces never do. If the question shows two faces touching, they’re adjacent—so they can’t be opposite.
4. Cuboid painting:
5. 3 faces painted: 8 corners.
6. 2 faces painted: 4 × (L + B + H – 6).
7. 1 face painted: 2 × (LB + BH + LH – 4L – 4B – 4H + 12).
8. 0 faces painted: (L–2)(B–2)(H–2).
9. Nets: 6 squares, no more, no less. Fold mentally—adjacent in net = adjacent in 3D.

Pro tip: Sketch it out. Even a rough cube drawing saves you from silly mistakes. You’ve got this—go ace that exam!