Cubes and Dice

Cube colouring, opposite faces, dice and counting cubes problems.

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Study Notes: Cubes and Dice (SOF NSO)

Overview

Cubes and Dice problems test your spatial reasoning and visualization skills—the ability to mentally manipulate 3D objects. These questions appear regularly in the Logical Reasoning section of NSO and other competitive exams. You must quickly determine properties like opposite faces, painted surfaces, or dice patterns without physically handling the object.

This topic matters because it directly tests mental rotation and pattern recognition, skills essential for scientific thinking. Expect 2–4 questions covering cube painting problems (how many cubes have 0, 1, 2, or 3 faces painted), dice configurations (identifying opposite faces or predicting rotations), and counting hidden cubes in 3D stacks. Mastering the standard formulas and visualization techniques ensures quick, accurate solutions under exam pressure.

The key is recognizing that despite appearing different, most problems follow predictable patterns. Learn the cube-painting formulas, memorize the opposite-face rule for standard dice, and practice mentally unfolding or rotating cubes. With focused practice, these become formula-based questions rather than guessing games.

Key Concepts

**Standard dice configuration**: A normal die has opposite faces summing to 7 (1–6, 2–5, 3–4). This rule helps identify opposite faces when two or three views are given.

**Painted cube formula**: When a large cube of side *n* units is painted and cut into unit cubes, specific formulas count cubes by number of painted faces—essential for rapid calculation without drawing.

**Adjacent vs opposite faces**: Two faces sharing an edge are adjacent; two faces never visible together in any view are opposite. Tracking adjacency helps reconstruct the entire cube from partial views.

**Rotation and reflection**: A cube shown in different orientations is still the same cube. Learn to mentally rotate cubes clockwise/counterclockwise to match given configurations.

**Counting cubes in stacks**: For 3D assemblies built from unit cubes, count visible cubes from one view, then add hidden cubes by analyzing the structure layer-by-layer.

**Open vs closed dice**: Open dice problems show a flat net (unfolded cube)—you fold it mentally to identify which faces meet. Closed dice show the assembled cube from multiple angles.

**Corner, edge, and face cubes**: In painted cube problems, cubes at corners have 3 painted faces, edge cubes have 2, face-center cubes have 1, and internal cubes have 0.

Formulas / Key Facts

**Painted Cube Formulas** (cube of side *n* units cut into unit cubes):

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Q1 · Cubes and Dice · EASY
A cube is painted red on all six faces. It is then cut into 64 smaller identical cubes. How many of the smaller cubes have exactly two faces painted?
Q2 · Cubes and Dice · EASY
A standard dice has dots 1, 2, 3, 4, 5, 6 on its faces. If the faces with 1 and 6 dots are opposite to each other, and 2 and 5 are opposite to each other, which number is opposite to 3?
Q3 · Cubes and Dice · MEDIUM
Two positions of the same dice are shown below. When 3 is at the bottom, which number will be on the top? Position 1: Front face shows 2, top face shows 3, right face shows 5. Position 2: Front face shows 4, top face shows 2, right face shows 3.
Q4 · Cubes and Dice · MEDIUM
A cube is painted with three different colours: red on two opposite faces, blue on two opposite faces, and green on the remaining two opposite faces. The cube is then cut into 27 smaller identical cubes. How many smaller cubes will have all three colours on them?
Q5 · Cubes and Dice · HARD
A cube of edge 10 cm is painted on all faces and then cut into smaller cubes of edge 2 cm each. How many smaller cubes will have no face painted?

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Notes generated on 10 May 2026

Cubes and Dice

Study Notes: Cubes and Dice (SOF NSO)

Overview

Key Concepts

Formulas / Key Facts

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