Dice and Cubes

Dice and Cubes questions test your ability to visualize 3D objects. A standard die is a cube with 6 faces numbered 1 to 6.

1. Standard Die: The sum of opposite faces is always 7 (1-6, 2-5, 3-4).
2. General Die: The sum of opposite faces is not necessarily 7. Adjacent faces can never be opposite.

When a cube is unfolded, alternate faces (in a straight line of 3 or 4 squares) are opposite to each other. Faces that touch at corners or edges are adjacent.

For a cube of side N units cut into smaller cubes of side 1 unit:
- Total smaller cubes = N³
- 3 faces painted (Corner) = 8 (always fixed)
- 2 faces painted (Edge) = 12 × (N - 2)
- 1 face painted (Face) = 6 × (N - 2)²
- 0 faces painted (Inner) = (N - 2)³