Logical Reasoning
Cubes and dice
Rotate a 3D die to understand opposite faces and painted-cube counting.
On a standard die, opposite faces add up to 7: 1–6, 2–5, 3–4.
A cube painted on all faces and cut into n × n × n small cubes has:
- 8 cubes with three faces painted (corners)
- 12(n − 2) with two faces painted
- 6(n − 2)² with one face painted
- (n − 2)³ with no face painted
📌 Remember for the exam
- Opposite faces of a standard die sum to 7
- Faces adjacent to the same face cannot be opposite each other
🎯 Learning objective: Find theoretical probability and compare it with experiment.
💬 Questions and answers
1What am I looking at?
Answer: A 3D die that you can roll many times while a bar chart records results.
2What are its parts? (click a part to identify it)
Answer: The die, its six faces and the frequency chart.
3How does it work?
Answer: Each face is equally likely, so the theoretical probability of each is 1/6. With more rolls, the experimental results move closer to 1/6.
4Why is it important?
Answer: Probability is used in weather forecasts, games, insurance and data science.
5What happens if I change something? (use the sliders)
Answer: Roll 10, 100 or 1,000 times and watch the bars even out.
6What should I remember for the exam?
Answer: P(E) = favourable outcomes ÷ total outcomes; 0 ≤ P(E) ≤ 1; P(E) + P(not E) = 1. Opposite faces of a die add up to 7.