Cube and Dice

 Cubes and Dice problems test your ability to visualize and interpret 3D objects based on 2D representations or verbal clues. These questions appear in SSC, Railways, Banking, and other exams.

• A cube has 6 faces, 12 edges, and 8 corners.

• A dice is a cube with numbers/symbols/letters marked on its faces.

🔍 Important Concepts

1️⃣ Structure of a Cube

• Faces: 6

• Edges: 12

• Vertices (corners): 8

• Opposite faces: 3 pairs

• Adjacent faces: Share a common edge

• Touching faces: Meet at an edge or corner

• Non-touching faces: Opposite to each other

2️⃣ Types of Cube & Dice Questions

Type	Description
Type 1: Counting cubes	Questions based on cutting a cube into smaller cubes
Type 2: Open Dice (Net)	Based on folding or unfolding of dice
Type 3: Opposite Face Identification	Based on visible faces to determine opposite ones
Type 4: Rotation of Dice	Position changes without changing the die itself

🖼️ Common Cube Net Diagrams

Here are common 2D dice nets that fold into cubes:

Pattern A:

     +-----+
     |  1  |
+----+-----+----+
| 2  |  3  | 4  |
+----+-----+----+
     |  5  |
     +-----+
     |  6  |
     +-----+

✅ Opposite faces:

• 1 ↔ 6

• 2 ↔ 4

• 3 ↔ 5

📘 Important Methods & Rules

🔹 Rule 1: Opposite Face Rule (Open Dice)

In a standard cube net, opposite faces never appear together in a single view.

If three adjacent faces are shown, the remaining three faces are on the opposite side of each.

🔹 Rule 2: Common Face Method (Rotation Dice)

When two dice are shown with two common faces, observe the relative position of the third:

➡️ If common faces are in the same position, the third faces are opposite.

➡️ If common faces rotate, then positions may differ.

🔹 Rule 3: Painted Cubes / Cutting Cubes

Used when a larger cube is painted and cut into smaller cubes.

Formulae:

• Total smaller cubes:
  = (n³), if cube of side n is divided equally on all sides

• Cubes with 3 faces painted: Always 8 (at corners)

• Cubes with 2 faces painted:
  = 12(n−2) for edges (excluding corners)

• Cubes with 1 face painted:
  = 6(n−2)² (face centres)

• Cubes with no face painted:
  = (n−2)³ (inner cubes)

✅ Examples with Solution

🔢 Example 1: Opposite Faces

Q: Two positions of a dice are shown:

Dice 1: 1 on top, 2 on front, 3 on right  
Dice 2: 1 on top, 3 on front, 5 on right

Find the face opposite to 2.

🧠 Solution:

• Dice 1: Top = 1, Front = 2, Right = 3

• Dice 2: Top = 1, Front = 3, Right = 5

Common face = 1
In Dice 1: 2 is front, in Dice 2: 3 is front → so positions change
⇒ 2 is opposite to 5 ✅

🔢 Example 2: Painted Cube

Q: A cube of side 3 cm is painted on all faces and then cut into 27 small cubes of 1 cm each. Find the number of cubes with:

a) 3 faces painted
b) 2 faces painted
c) 1 face painted
d) No face painted

🧠 Solution:

• Total cubes: 3³ = 27

• a) 3 faces painted: 8 (all corners)

• b) 2 faces painted: 12(3−2) = 12

• c) 1 face painted: 6(3−2)² = 6

• d) No face painted: (3−2)³ = 1 ✅

🔢 Example 3: Open Dice (Net)

Q: Which number is opposite to 3?

Given net:

    +---+
    | 1 |
+---+---+---+
| 2 | 3 | 4 |
+---+---+---+
    | 5 |
    +---+

🧠 Solution:

From the net:

• 3 is adjacent to 2, 1, 4, 5

• So opposite face = not adjacent = 6 (hidden in this net) ✅

🧠 Tips & Tricks

Concept	Trick
Opposite Faces	Never visible together in a single view
Rotation Dice	Use common face comparison
Net to Cube	Opposite faces never adjacent
Painted Cubes	Memorize the corner/edge/face formulas
Visualizing	Practice folding cube nets mentally

🎯Practice Questions – Cube and Dice

🔢 Q1.

A cube has six faces labelled A, B, C, D, E, and F.
If A is opposite to D, B is opposite to E, and C is opposite to F,
then which of the following face pairs cannot be adjacent?

a) A & B
b) C & D
c) A & D
d) B & C

🟩 Answer: c) A & D
Explanation: Opposite faces can’t be adjacent.

🔢 Q2.

In a cube, numbers 1 to 6 are marked.
Two adjacent faces show 1 next to 2, and 2 is next to 3.
Which number cannot be on the face opposite to 1?

a) 2
b) 3
c) 4
d) 6

🟩 Answer: a) 2
Explanation: 1 and 2 are adjacent, so they cannot be opposite.

🔢 Q3.

Two positions of a dice are shown:

• First: 1 on top, 2 on front, 3 on right

• Second: 3 on top, 1 on front, 5 on right
  Which number is opposite to 2?

🟩 Answer: 5
Explanation: Use common face (1), and track face shifts.

🔢 Q4.

A cube of side 3 cm is painted on all six faces and cut into smaller cubes of 1 cm each.
How many cubes will have exactly 2 faces painted?

🟩 Answer: 12
Explanation: Edges, excluding corners → 12 × (n – 2) = 12 × 1 = 12

🔢 Q5.

A cube has colours on its faces: Red, Blue, Green, White, Yellow, and Black.
If opposite of Red is White and opposite of Blue is Yellow,
then what is the opposite of Green?

🟩 Answer: Black
Explanation: Remaining pair of opposite faces.

🔢 Q6.

A cube is painted red on all faces and cut into 64 smaller cubes.
How many cubes will have 3 faces painted?

🟩 Answer: 8
Explanation: Corners → always 8 cubes have 3 faces painted.

🔢 Q7.

A dice is rolled twice. First roll shows 5 at the top, 3 at the front.
Second roll shows 5 at the top, 2 at the front.
What is the number on the face opposite to 3?

🟩 Answer: 2
Explanation: 5 is common. Positions of 3 and 2 are different; hence opposite.

🔢 Q8.

From a cube of side 4 cm painted on all sides, small cubes of side 1 cm are cut.
How many cubes will have no face painted?

🟩 Answer: 8
Explanation: Internal unpainted cubes = (4 – 2)³ = 8

🔢 Q9.

In a dice, 6 is opposite to 1, 3 is opposite to 4, and 2 is opposite to 5.
Which number will be adjacent to all others except 6?

🟩 Answer: 3
Explanation: 3 is not opposite to any other answer choices, so can be adjacent.

🔢 Q10.

In the net of a cube, three faces visible are 2, 3, and 6.
Which of the following cannot be opposite to 2?

a) 3
b) 4
c) 5
d) 6

🟩 Answer: a) 3
Explanation: If 2 and 3 appear together on the net, they can't be opposite.