Cube and Dice — Study Notes for HSSC CET
Overview
Cube and Dice problems test your spatial reasoning and ability to visualize three-dimensional objects mentally. They require you to determine opposite faces, identify correct dice configurations, or count visible/painted faces after a larger cube is cut into smaller pieces.
This topic rewards systematic learning over guesswork. Once you master the standard rules for identifying opposite faces and the formula for counting painted cubes, you can solve most questions in under 60 seconds. The good news: the question types are limited and predictable, making this a high-scoring area if you prepare the fundamentals well.
Students who struggle here usually skip the visualization step or forget the standard opposite-face rules. Build a mental model of how a cube unfolds into a net and how dice rotate, and this topic becomes straightforward.
Key Concepts
- Standard Dice Rule: On a standard dice, opposite faces always sum to 7. The pairs are (1,6), (2,5), and (3,4).
- Non-Standard Dice: Any dice where opposite faces do not follow the sum-of-7 rule. Questions may ask you to identify whether a dice is standard or non-standard.
- Cube Net (Unfolded Cube): A 2D pattern of six connected squares that folds into a cube. There are exactly 11 distinct cube nets. Understanding nets helps identify opposite faces.
- Opposite Face Identification: In any cube net, faces that are separated by exactly one square (with a gap square between them in a straight line) are opposite to each other.
- Adjacent Faces: Faces that share an edge when the cube is folded. If two faces touch in the net (share a side), they are adjacent, not opposite.
- Rotation Principle: When a cube is rotated, opposite faces remain opposite. Only adjacent faces change their relative positions during rotation.
- Painted Cube Problems: A larger cube is painted on all faces, then cut into smaller identical cubes. You must count cubes with 0, 1, 2, or 3 painted faces.
- Corner, Edge, Face, and Core Cubes: In a painted cube cut into n x n x n smaller cubes — corners have 3 painted faces, edges have 2, face-centers have 1, and internal cubes have 0.
Formulas / Key Facts
For a cube of side n units cut into 1-unit smaller cubes:
| Painted Faces | Position | Count Formula |
|---|---|---|
| 3 faces painted | Corner cubes | Always 8 |
| 2 faces painted | Edge cubes (excluding corners) | 12 x (n - 2) |
| 1 face painted | Face-center cubes | 6 x (n - 2)² |
| 0 faces painted | Internal cubes | (n - 2)³ |
| Total small cubes | — | n³ |
Standard Dice Opposite Pairs: 1-6, 2-5, 3-4 (each pair sums to 7)
Net Rule for Opposite Faces: In a cross-shaped or L-shaped net, count squares in a straight line — if there is exactly one square between two faces, they are opposite.
Same Dice Verification: Two positions show the same dice if rotating one can produce the other. Different opposite-face pairs mean different dice.
Worked Examples
Example 1: Finding Opposite Face from Dice Positions
Problem: A dice shows the following in two positions:
- Position 1: Top = 3, Front = 5, Right = 2
- Position 2: Top = 3, Front = 4, Right = 5
What is opposite to face 2?
Solution:
- In Position 1: Face 3 is on top, Face 5 is in front, Face 2 is on right
- In Position 2: Face 3 is still on top (same), but now Face 4 is in front and Face 5 is on right
- Since Top remains 3, the cube has rotated horizontally
- In Position 1, Face 5 was front; in Position 2, Face 5 moved to right, and Face 4 came to front
- This means Face 4 and Face 5 are adjacent (they swap positions during horizontal rotation)
- Face 2 was on right in Position 1; Face 5 is on right in Position 2
- Since Face 2 and Face 5 occupy the same position (right) in different orientations where top is constant, Face 2 is opposite to Face 4
Example 2: Painted Cube Counting
Problem: A cube of side 5 cm is painted red on all faces and then cut into cubes of side 1 cm. How many smaller cubes have exactly 2 faces painted?
Solution:
- Here n = 5
- Cubes with 2 painted faces lie on edges (excluding corners)
- Each edge has (n - 2) = 5 - 2 = 3 such cubes
- A cube has 12 edges
- Total = 12 x 3 = 36 cubes
Example 3: Identifying Opposite from a Net
Problem: In a cube net shaped like a cross (one central square with four squares attached to its four sides, plus one square attached to the bottom of the lower arm), which face is opposite to the top face?
Solution:
- In a cross net: the central square and any square that has exactly one square between it and the central square are opposite
- The top of the cross is adjacent to the center (they share an edge)
- The bottom of the lower arm has one square (the lower arm square) between it and the center
- The face at the bottom of the lower arm is opposite to the center face
Common Mistakes
- Assuming adjacent in net means opposite in cube → Wrong! If two squares share a side in the net, they are adjacent faces, not opposite. Opposite faces have one square gap between them.
- Forgetting corner cubes in edge-cube count → Edge cubes with 2 painted faces exclude corners. Always use 12 x (n-2), not 12 x n.
- Applying standard dice rule to all dice → Not all dice are standard. Only apply the sum-of-7 rule when the question confirms it is a standard dice.
- Confusing rotation with flipping → A dice rotated keeps opposite faces opposite. Mentally track which faces stay fixed during rotation before concluding.
- Miscounting internal cubes → Internal cubes exist only when n > 2. For n = 2, there are no internal cubes (0 painted faces = 0 cubes).
Quick Reference
- Standard dice: Opposite faces sum to 7 → (1,6), (2,5), (3,4)
- 3 painted faces = 8 corner cubes (always, regardless of n)
- 2 painted faces = 12 x (n - 2) edge cubes
- 1 painted face = 6 x (n - 2)² face-center cubes
- 0 painted faces = (n - 2)³ internal cubes
- In a net, one-square-gap in a straight line = opposite faces