Cube and Dice
About 6 min · Reasoning Ability · English
Time worked out from the lesson's 416 words at 150 words a minute, plus a minute for each question.
Cube and Dice problems test your spatial reasoning and ability to visualize three-dimensional objects mentally. They require you to determine opposite faces, identify correct dice configurations, or count visible/painted faces after a larger cube is cut into smaller pieces.
Key ideas
- Standard Dice Rule: On a standard dice, opposite faces always sum to 7. The pairs are (1,6), (2,5), and (3,4).
- Non-Standard Dice: Any dice where opposite faces do not follow the sum-of-7 rule. Questions may ask you to identify whether a dice is standard or non-standard.
- Cube Net (Unfolded Cube): A 2D pattern of six connected squares that folds into a cube. There are exactly 11 distinct cube nets. Understanding nets helps identify opposite faces.
- Opposite Face Identification: In any cube net, faces that are separated by exactly one square (with a gap square between them in a straight line) are opposite to each other.
- Adjacent Faces: Faces that share an edge when the cube is folded. If two faces touch in the net (share a side), they are adjacent, not opposite.
Watch out for
- Assuming adjacent in net means opposite in cube → Wrong! If two squares share a side in the net, they are adjacent faces, not opposite. Opposite faces have one square gap between them.
- Forgetting corner cubes in edge-cube count → Edge cubes with 2 painted faces exclude corners. Always use 12 x (n-2), not 12 x n.
- Applying standard dice rule to all dice → Not all dice are standard. Only apply the sum-of-7 rule when the question confirms it is a standard dice.
Quick reference
- Standard dice: Opposite faces sum to 7 → (1,6), (2,5), (3,4)
- 3 painted faces = 8 corner cubes (always, regardless of n)
- 2 painted faces = 12 x (n - 2) edge cubes
- 1 painted face = 6 x (n - 2)² face-center cubes
- 0 painted faces = (n - 2)³ internal cubes
Check yourself: 3 questions
1. A cube is painted red on all faces and then cut into 64 smaller identical cubes. How many smaller cubes will have exactly two faces painted?
2. Three different positions of a dice are shown. Find the number opposite to 6. Position 1: Visible faces are 1, 2, 3 Position 2: Visible faces are 2, 4, 5 Position 3: Visible faces are 1, 5, 6
3. A standard dice is thrown twice. The sum of the numbers on the top face is 9. What is the sum of the numbers on the bottom faces in the two throws?
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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