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Chapter: Cubes & Dice
Non-verbal Reasoning – Foundation Level

🎯 What You’ll Learn Today

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

Identify opposite and adjacent faces of a dice within seconds.

Unfold a dice net in your mind without paper.

Instantly count painted faces when a cube is cut.

Spot hidden cubes that examiners love to hide.

Dodge the trickiest traps in cubes & dice questions.

🔥 Hook – The Magician’s Dice

A street magician throws three dice on the table. The top faces show 2, 4, 6. Without flipping them, he instantly tells you what’s on the bottom. Is it magic? No, it’s reasoning. And if you can master this, you’ll crack 3–5 marks in every competitive exam faster than you can say “standard dice.” Ready to become that magician? Let’s roll.

😰 Why Do Students Stumble Here?

Most aspirants treat Cubes & Dice as guesswork. They stare at twisted cubes, panic, and mark a random option. Here’s the truth: Every dice problem follows a tiny set of logical rules. Once you lock those rules in your brain, you’ll solve them faster than cutting a birthday cake.

🎲 Part 1: Inside a Dice – The 1-to-6 Universe

1.1 Two Types of Dice

Standard Dice – Opposite faces always add up to 7.
(1 opposite 6, 2 opposite 5, 3 opposite 4)

Non-standard Dice – No sum rule. You must find opposites from the given views.

🧠 Quick Recall:
What is the opposite of 4 in a standard dice? (Pause, think, then read: 3, because 4+3=7)

1.2 Finding Opposite Faces When Two Views Are Given

Rule: The Common-Face Clockwise Trick
If the same face appears in two views, keep that face still in your mind. Move clockwise from the common face in both views. The first neighbouring face in view-1 becomes opposite to the first neighbouring face in view-2.

Example:
View-1: 1 on top, 2 in front, 3 on right.
View-2: 1 on top, 4 in front, 5 on right.
Common face = 1. Clockwise from 1: in View-1 → 2,3; in View-2 → 4,5.
Thus, 2 ↔ 4 and 3 ↔ 5. Opposite pairs: (2,4) and (3,5). Remaining: 1 opposite 6.

🧠 Quick Recall:
If the common face appears on top in both views, do you still use clockwise? Yes! Imagine looking down at the top face and move to the next face in clockwise order.

📦 Part 2: The Open Dice (Net) – Folding in Your Head

An open dice is a flat cutout that folds into a cube. Rule: Opposites Never Touch. In a cross-shaped net, faces that are one apart in the same row/column are opposites.

Mnemonic: O-CAN – Opposite faces Cannot be Adjacent, Never.

Fast trick for standard cross net:

Top row’s 2nd and 4th faces are opposites.

Middle row’s 1st and 3rd, and 2nd with the bottom row’s middle face, etc.
Better yet: just visualise folding. 4 adjacent to 2? If after folding they sit on opposite sides, then 4 and 2 are opposites.

🧠 Quick Recall:
In a T-shaped net, are the two faces at the ends of the T's arms opposite? No, they become adjacent. Fold the vertical stem upward and the arms come together.

🧊 Part 3: The Painted Cube That Gets Sliced

A large cube is painted on all faces and cut into smaller equal cubes. Counting coloured faces is a goldmine for examiners.

Formula Chart (side = n smaller cubes per edge):

Total small cubes =
𝑛
3
n
3

3 faces painted (corners) = 8 (always)

2 faces painted (edges minus corners) =
12
×
(
𝑛
−
2
)
12×(n−2)

1 face painted (face centres) =
6
×
(
𝑛
−
2
)
2
6×(n−2)
2

No face painted (inner core) =
(
𝑛
−
2
)
3
(n−2)
3
(if n ≥ 3)

🧠 Quick Recall:
What’s the number of cubes with at least 2 faces painted when n=4?
2-face = 12×(2)=24; 3-face = 8; total = 32. Remember “at least 2” includes both 2 and 3.

🧩 Part 4: Counting Cubes in a Solid – The Hidden Gems

An irregular solid made of small cubes is shown. You need to count how many cubes are used.

Two foolproof methods:

Column method (top view): Look from the top, write the height in each square cell, sum them.

Layer method (bottom-up): Start from the lowest visible layer, count cubes, move up. Remember cubes can hide behind others; a taller column may cast a “shadow” of missing cubes in the front view? No, draw the top view grid and mark heights.

🧠 Quick Recall:
If a front view shows a stack of 3 cubes, but top view reveals that column is in the second row, behind a 2-high column, is the hidden cube counted? Yes, the top view height accounts for all cubes in that column, even if hidden from front.

🧠 Mnemonic That Sticks: 3-2-1-0: The Paint Song

Sing in your head:
“Three on corners, eight my friend,
Two on edges, formula trend,
One on faces, six flat squares,
Zero inside, nobody cares.”
(Okay, the last line is for fun, but the sequence 3-2-1-0 helps recall the order from outside in.)

⚠️ Caution: Examiner’s Trap

Trap 1: “Only 2 faces” vs “At least 2 faces”
When the question says “only 2 faces painted,” do not add the 3-face cubes. But if it says “at least 2 faces painted,” include both 2 and 3. One word changes the answer completely.

Trap 2: The Invisible Back Row
In a solid cube arrangement, the rear columns are not visible from the front. If you assume they are empty, you lose marks. Always rely on the top view or a side view to confirm heights.

Trap 3: Non-Standard Dice Assumption
Just because a dice looks like a standard cube, don’t assume opposite sum is 7 unless the question explicitly states “standard dice.” In non-standard dice, apply the common-face rule; never guess.

📊 Summary Table – Your Quick Revision Card

🔁 Review This After 2 Days

When you come back:

Close your eyes and explain the clockwise trick to an imaginary friend.

Draw a painted cube with n=5 and quickly find cubes with 2 faces painted (12×3=36) and no face painted (27).

Pick one open dice net from a previous paper and fold it mentally to find 3 opposite pairs.

Recall the “only” vs “at least” trap – what would you answer if n=4 and they ask “only 2 faces painted”? (Answer: 24, not 32.)

Keep rolling, future officer. This chapter is now your reasoning superpower.