Cubes and Dice
About 6 min · General Intelligence and Reasoning · English
Time worked out from the lesson's 437 words at 150 words a minute, plus a minute for each question.
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.
Key ideas
- 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.
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.
Check yourself: 3 questions
1. 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 red?
2. A standard dice is thrown once. What is the probability that the number on the top face is either a prime number or an even number?
3. A cube is painted blue on two opposite faces, red on two other opposite faces, and green on the remaining two opposite faces. The cube is then cut into 8 equal smaller cubes. How many of the smaller cubes will have all three colours on them?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 18 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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