UP Police Constable · Bite-size lessons
Dice & Cube
About 6 min · Mental Aptitude / IQ / Reasoning · English
Time worked out from the lesson's 446 words at 150 words a minute, plus a minute for each question.
Dice and cube problems are a reasoning question type in competitive exams. These questions assess your spatial visualization ability — the skill to mentally rotate objects and track changes in orientation. You'll encounter standard six-faced dice (with dots numbered 1–6), numbered/lettered cubes, and painted cube scenarios.
Key ideas
- Standard die numbering: On a normal die, opposite faces always sum to 7. So 1 is opposite 6, 2 is opposite 5, and 3 is opposite 4. This is the fundamental property used in most dice problems.
- Rotation preserves adjacency: When a die rolls, the top and bottom faces change, but the four faces forming the "belt" around the middle maintain their circular order. If you know two adjacent faces, you can deduce others through rotation.
- Two views are often sufficient: Most exam questions show a die in two or three positions. By comparing these views and noting which faces appear together, you can deduce the complete configuration and identify opposite faces.
- Painted cube problems: A large cube is painted and cut into smaller unit cubes. The number of cubes with 0, 1, 2, or 3 painted faces follows fixed formulas based on the cube's dimensions (n × n × n).
- Corner, edge, and face cubes: In a painted cube of side n, corner cubes have 3 painted faces (always 8), edge cubes have 2 painted faces, face-center cubes have 1 painted face, and internal cubes have 0 painted faces.
Quick reference
- Standard die: opposite faces sum to 7 (1-6, 2-5, 3-4).
- Two views usually suffice to deduce all opposite pairs — compare adjacent faces across views.
- Painted cube n×n×n: 3 faces = 8 cubes; 2 faces = 12(n–2); 1 face = 6(n–2)²; 0 faces = (n–2)³.
- Rotation cycles top→front→bottom→back→top; sides rotate as a ring.
- In a net, opposite faces are never adjacent — count one face gap in between.
Check yourself: 3 questions
1. A standard dice is thrown twice. In the first throw, the face showing 3 dots is on top and the face showing 5 dots is facing you. The dice is then rotated 90 degrees forward (towards you). Which number will be on top after this rotation?
2. A cube is painted red on all faces. It is then cut into 64 smaller identical cubes. How many of the smaller cubes have exactly two faces painted red?
3. Three different positions of a dice are shown below: Position 1: 1 is on top, 2 is facing you Position 2: 2 is on top, 3 is facing you Position 3: 3 is on top, 1 is facing you Based on these positions, when 6 is on top, which number will be at the bottom?
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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