Cubes and Dice — Study Notes

Overview

Cubes and dice questions are a staple in the General Intelligence and Reasoning section of SSC CHSL Tier 1, typically contributing 1–2 questions per paper. These problems test spatial visualization and logical reasoning without requiring heavy mathematics. A standard die has six faces numbered 1 to 6, and the sum of opposite faces always equals 7 (1 opposite 6, 2 opposite 5, 3 opposite 4). Cube problems involve painting or cutting a larger cube into smaller units, then counting cubes with specific colour patterns. Mastery requires understanding standard dice configurations, developing mental rotation skills, and applying systematic counting methods for painted cubes. Students who practice visualization techniques can solve these questions in under 60 seconds, making this topic a reliable score-booster.

The three main question types are: (1) identifying opposite faces on a dice using multiple views, (2) determining the number shown on a hidden face when three faces are visible, and (3) counting cubes with 0, 1, 2 or 3 painted faces when a large cube is cut into smaller identical cubes. Each type follows predictable patterns once you learn the underlying logic.

Key Concepts

• **Standard dice rule**: On a normal die, opposite faces always sum to 7. Face 1 is opposite face 6, face 2 opposite face 5, and face 3 opposite face 4. This is the foundation for most dice problems.

• **Adjacent faces**: When you know two opposite pairs, the third pair is determined automatically. If you see three faces meeting at a corner, those three faces can never be opposite to each other.

• **Common adjacent rule**: If two numbers appear adjacent (touching edges) in one view, they cannot be opposite faces. Use this to eliminate wrong answer choices.

• **Rotation vs reflection**: When a die is rotated, the relative positions of numbers remain consistent. Learn to mentally rotate dice clockwise/anticlockwise to match different views.

• **Cube painting formula**: When an n×n×n cube is painted on all faces and cut into unit cubes, corner cubes have 3 faces painted (always 8 cubes), edge cubes have 2 faces painted, face-center cubes have 1 face painted, and internal cubes have 0 faces painted.

• **Counting systematically**: For painted cubes, use formulas rather than manual counting. For a 4×4×4 cube cut into 64 smaller cubes, calculate each category separately using position logic.

• **Dice probability basics**: When rolling a fair die, each face has probability 1/6. For multiple dice, multiply probabilities for independent events. The question may ask for specific sum outcomes or matching numbers.

• **Net diagrams**: Some questions show an unfolded cube (net). Identify which faces will be opposite when the net is folded. Faces separated by one face in the net become opposite faces in the cube.

Formulas / Key Facts

**Standard Dice:**

• Opposite face sum = 7 always (1↔6, 2↔5, 3↔4)
• Three faces meeting at any corner are mutually adjacent, never opposite

**Painted Cube (n×n×n cut into unit cubes):**

• Total unit cubes = n³
• 3 faces painted (corners) = 8 (always, regardless of n)
• 2 faces painted (edges, not corners) = 12(n−2)
• 1 face painted (face centers) = 6(n−2)²
• 0 faces painted (internal) = (n−2)³

**Dice Probability:**

• Probability of any specific number = 1/6
• Probability of sum = 7 with two dice = 6/36 = 1/6
• Probability of doubles (same number on both dice) = 6/36 = 1/6

**Quick position checks:**

• If face A is on top and face B is facing you, use right-hand rule to determine which face is on the right
• Two faces adjacent in one view must appear adjacent or perpendicular in all valid views

Worked Examples

**Example 1: Opposite Face Identification**

*Problem:* Two positions of a dice are shown. In position I, face 2 is on top and face 4 faces you. In position II, face 5 is on top and face 3 faces you. Which number is opposite to 2?

*Solution:*

• Step 1: From position I, face 2 is on top, 4 faces forward. The bottom face is opposite to 2.
• Step 2: From position II, face 5 is on top, 3 faces forward. The bottom face is opposite to 5.
• Step 3: List visible faces from both positions: 2, 4 (position I) and 5, 3 (position II). That's four faces visible.
• Step 4: The two faces not shown in either view must be on the remaining positions. If standard die, faces are 1, 2, 3, 4, 5, 6. Missing faces are 1 and 6.
• Step 5: Apply rule: 2 is on top in position I, so its opposite is the bottom. Since 5 is on top in position II, opposite of 5 is bottom there. Check adjacency: 2 and 5 are not opposite each other (they appear as top faces in different views). Therefore, 2 is opposite to 5.

*Answer:* Face 5 is opposite to face 2.

**Example 2: Painted Cube Counting**

*Problem:* A cube of side 4 cm is painted red on all faces, then cut into 1 cm cubes. How many smaller cubes have exactly two faces painted?

*Solution:*

• Step 1: Original cube is 4×4×4, so n = 4.
• Step 2: Use formula for 2 faces painted (edge cubes excluding corners): 12(n−2).
• Step 3: Substitute n = 4: 12(4−2) = 12 × 2 = 24.

*Answer:* 24 cubes have exactly two faces painted.

**Example 3: Hidden Face Number**

*Problem:* Three faces of a dice are visible showing 1, 3 and 4. Which number is on the face opposite to 4?

*Solution:*

• Step 1: Apply standard dice rule: opposite faces sum to 7.
• Step 2: Opposite of 4 is 7 − 4 = 3.
• Step 3: But face 3 is visible! This means it's NOT a standard dice, or the question shows three faces meeting at a corner.
• Step 4: If 1, 3, 4 meet at a corner, they are mutually adjacent. Their opposites are the remaining three faces: 6, 4, 3 are opposite to 1, 4, 3 respectively. Wait—recheck: 1↔6, 2↔5, 3↔4.
• Step 5: Since 3 is visible, opposite of 4 cannot be 3. For standard die, opposite of 4 is 3 only if they are NOT both visible. If both are visible, assume non-standard or re-read question.
• Step 6: For standard die with visible 1, 3, 4, the opposite of 4 must be checked against actual positions. Typically, opposite of 4 is 3, but since 3 is visible, there's a contradiction unless it's a trick question or non-standard.

*Clarified Answer (standard die):* Opposite of 4 is always 3 on a standard die. If 3 is visible on another face, the question may involve a non-standard configuration.

Common Mistakes

**Mistake 1:** Assuming all dice follow the standard 1–6 opposite rule without checking. **Fix:** Some SSC questions use non-standard dice with letters or symbols. Always verify the rule applies before using opposite-sum = 7.

**Mistake 2:** Confusing "adjacent" with "opposite" when two faces touch at an edge or corner. **Fix:** Two faces sharing an edge or meeting at a corner are adjacent, NOT opposite. Opposite faces never touch.

**Mistake 3:** Manually counting painted cubes for large cubes (e.g., 5×5×5), leading to errors and time waste. **Fix:** Memorize and apply the formulas: corners = 8, edges = 12(n−2), faces = 6(n−2)², internal = (n−2)³. Practice substituting n quickly.

**Mistake 4:** Forgetting that corner cubes always number exactly 8, regardless of cube size. **Fix:** A cube has 8 corners—always. Even for a 10×10×10 cube, only 8 unit cubes have three painted faces.

**Mistake 5:** Not visualizing dice rotation, leading to wrong face identification when views change. **Fix:** Sketch a simple cube on rough paper and label faces. Rotate it mentally or on paper to see which face ends up where. Practice with physical dice if possible.

Quick Reference

• **Standard die opposite rule**: 1↔6, 2↔5, 3↔4 (sum of opposites = 7).
• **Three faces at a corner are always mutually adjacent**, never opposite each other.
• **Painted cube shortcuts**: Corners = 8, Edges = 12(n−2), Faces = 6(n−2)², Internal = (n−2)³.
• **No face painted in an n×n×n cube**: (n−2)³ cubes remain completely unpainted inside.
• **Adjacency test**: If two numbers touch in any view, they cannot be opposite faces.
• **Mental rotation**: Practice rotating dice clockwise/anticlockwise to verify answer choices quickly.