Cubes and Dice
About 6 min · Logical Reasoning · English
Time worked out from the lesson's 369 words at 150 words a minute, plus a minute for each question.
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.
Key ideas
- 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.
Quick reference
- Standard die: Opposite faces sum to 7 (1–6, 2–5, 3–4)
- Corner cubes: Always 8, each with 3 painted faces
- Edge cubes (not corners): 12(n − 2) with 2 painted faces
- Face-center cubes: 6(n − 2)² with 1 painted face
- Internal cubes: (n − 2)³ with 0 painted 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?
2. 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?
3. 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?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 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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