Cubes and Dice
About 7 min · Logical Reasoning · English
Time worked out from the lesson's 552 words at 150 words a minute, plus a minute for each question.
Cubes and dice problems are part of logical reasoning sections in competitive exams, including SOF IMO. These questions test your spatial visualization skills—your ability to mentally manipulate three-dimensional objects without physically handling them. In the IMO context, you'll encounter three main problem types: cube colouring (how many small cubes have 0, 1, 2, or 3 faces painted?), opposite faces of dice (which number/symbol lies opposite to a given face?), and counting cubes (how many cubes are visible or hidden in a stack?).
Key ideas
- Standard dice convention: A standard die has faces numbered 1 to 6 such that opposite faces always sum to 7 (1-6, 2-5, 3-4). This is the most common assumption unless stated otherwise.
- Cube colouring structure: When a cube is painted and cut into smaller cubes, only the outermost layer gets paint. Interior cubes remain unpainted. The number of painted faces on a small cube depends on its position: corner (3 faces), edge (2 faces), face-centre (1 face), or interior (0 faces).
- Corner-edge-face formula: For an n×n×n cube, there are 8 corners (always), 12(n−2) edge cubes, 6(n−2)² face-centre cubes, and (n−2)³ interior cubes. These formulas let you count without drawing.
- Dice net/unfolding: When a dice is shown in different orientations, track one face and use the right-hand rule or elimination to deduce opposite faces. Two views of the same dice always maintain consistent internal relationships.
- Visibility principle in stacks: In a stack of cubes, only outer layer cubes are visible from any single viewpoint. Hidden cubes must be inferred from the structure—count total cubes in the stack, then subtract visible ones if needed.
Watch out for
- Mistake: Assuming all dice follow the opposite-sum-to-7 rule → Fix: Always check if the problem states "standard dice" or provides specific face information. Non-standard dice exist in tricky problems—work from given views only.
- Mistake: Forgetting corner cubes are shared by three faces → Fix: When counting painted cubes, remember corners touch 3 faces, edges touch 2 faces, face-centres touch 1 face. Don't count a corner cube as an edge cube.
- Mistake: Miscounting interior cubes in large n×n×n cubes → Fix: Interior cubes only exist if n≥3. For n=2, there are zero interior cubes. Use (n−2)³ carefully and check n value first.
Quick reference
- Standard dice: opposite faces sum to 7 (1-6, 2-5, 3-4)
- 3-face painted cubes = 8 corners (always)
- 2-face painted = 12(n−2) edge cubes
- 1-face painted = 6(n−2)² face-centre cubes
- 0-face painted = (n−2)³ interior cubes
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 these smaller cubes have exactly two faces painted red?
2. A standard dice has dots from 1 to 6 on its faces. The sum of dots on opposite faces is always 7. If the dice is placed such that face with 2 dots is on top and face with 3 dots is facing you, which number is on the bottom face?
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 by making two cuts parallel to each pair of opposite faces. How many of the smaller cubes 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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