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Cubes & Dice

Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Identify the net of a cube and tell which faces will be opposite each other.
2. Solve standard dice and ordinary dice problems in seconds.
3. Find the number on the hidden face when two views of the same dice are given.
4. Count painted small cubes (faces painted, unpainted, etc.) after cutting a big cube.
5. Spot the examiner's favourite traps before they trap you.

The Hook: One Cube, Six Secrets

Imagine a magician hands you a dice and says, "I can see the bottom face without lifting it." Sounds impossible? It isn't. In competitive exams, the dice is that magician — and you are about to learn the trick. Cubes and dice questions are among the quickest to solve in the reasoning section: students who know the rules finish them in under 30 seconds while others keep rotating imaginary cubes in their heads.

Let's make you the fast one.

The Common Struggle (You're Not Alone)

Most students struggle here because:

• They try to visualise 3D rotation in their head — and the brain drops the cube halfway.
• They memorise formulas without knowing when to apply which one.
• They mix up standard dice and ordinary dice rules.

The fix? Don't rotate the cube in your mind. Use rules — small, mechanical, foolproof rules. That's exactly what this chapter gives you.

Part 1: The Dice — Basics First

A dice is a cube with numbers 1 to 6 on its six faces.

Two types of dice:

1. Standard Dice: Sum of opposite faces is always 7.

• 1 ↔ 6, 2 ↔ 5, 3 ↔ 4

2. Ordinary (Non-standard) Dice: Opposite faces do not necessarily add to 7. You must find opposites from the given views.

🧠 Active Recall: In a standard dice, what is opposite to 3? … (Answer: 4, because 3 + 4 = 7.)

Part 2: The Golden Rule for Two Views of One Dice

When two views of the same dice are shown and one face is common in both views (and in the same position):

> Rule: The remaining two faces of each view are opposite to each other, pair by pair.

Example:

• View 1: 1 (top), 2 (front), 3 (side)
• View 2: 1 (top), 4 (front), 5 (side)

Face 1 is common and on top in both. So: 2 ↔ 4 and 3 ↔ 5. The leftover number 6 is opposite to 1.

When two faces are common in both views, the remaining two faces are opposite to each other directly.

🧠 Active Recall: Two views show faces (2, 3, 5) and (2, 3, 6). What is opposite to 5? … (Answer: 6 — two faces 2 and 3 are common, so the remaining faces 5 and 6 are opposite.)

Part 3: Nets of a Cube

A net is a flat shape that folds into a cube. There are exactly 11 valid nets of a cube.

Rule to find opposite faces in a net:

> In a straight row/column of 4 squares, the 1st is opposite the 3rd, and the 2nd is opposite the 4th. The two leftover squares are opposite each other.

Quick check for an invalid net: If four squares meet at one point (a 2×2 block of four with extras attached wrongly), or if folding makes two faces overlap — it can never form a cube.

🧠 Active Recall: In a row of four squares numbered 1-2-3-4, which face is opposite 1? … (Answer: 3.)

Part 4: Painted Cube Problems (The Cutting Chapter)

A big cube of side n units is painted on all faces and cut into n³ small cubes.

Memory trick (mnemonic): "8-12-6-1, the cube's family" — 8 corners, 12 edges, 6 faces, 1 inner body. The powers go 0, 1, 2, 3 with (n−2): 8·(n−2)⁰, 12·(n−2)¹, 6·(n−2)², 1·(n−2)³.

Example: n = 4 → 3-face: 8; 2-face: 12×2 = 24; 1-face: 6×4 = 24; 0-face: 8. Total = 64 = 4³ ✓

🧠 Active Recall: A cube is cut into 27 small cubes (n = 3). How many have no paint? … (Answer: (3−2)³ = 1.)

Part 5: Dice Rolled on a Path

When a dice rolls along squares on the floor, track only the top and bottom:

• Rolling forward: top → front side, back side → top.
• The face that was opposite the direction of rolling comes on top.

Practice with a real dice or eraser once — this topic becomes permanent in memory.

Summary Table

📅 Review this chapter after 2 days. Redo the Active Recall questions without looking — if you answer all instantly, the chapter is yours forever.

⚠️ Caution: Examiner's Trap

Trap 1 — Blurry figures on screen: In CBT exams, dice figures may look unclear. Never guess from a hazy image — re-read the numbers written on faces carefully. One misread number flips the whole answer.

Trap 2 — Look-alike views with a hidden difference: Two dice views may appear identical, but one number is changed (e.g., 4 instead of 5). The examiner hides the difference so you apply the "common face" rule wrongly. Always list the three numbers of each view before applying any rule.

Trap 3 — Rotation vs. mirror image: A rotated dice is the same dice; a mirror image is a different dice. If a question asks "which dice cannot be formed," the mirror arrangement of a valid view is usually the answer. Check the clockwise order of three faces meeting at a corner — if the order is reversed, it's a mirror trap.