Cubes and dice appear in every ASO paper: dice positions (2012, 2018, 2021, 2025), unfolded nets (2015, 2018, 2021), cube counting (2015, 2018, 2021) and painted cubes (2012, 2021, 2025). Most of these can be solved with three rules.
Dice: opposite faces
- Numbers seen together on one view are adjacent, never opposite.
- If one number is seen next to four others, the sixth number is opposite it.
- On a standard dice, opposite faces add up to 7.
Nets (unfolded cube)
- In a straight row, faces with exactly one square between them are opposite.
- Squares that share an edge are never opposite.
- For an L or zig-zag shape, fold one square at a time in your mind.
Painted cube cut into n × n × n
| Painted faces | Number of cubes |
|---|---|
| 3 (corners) | 8 |
| 2 (edges) | 12(n − 2) |
| 1 (face centres) | 6(n − 2)² |
| 0 (inside) | (n − 2)³ |
n = side of the big cube ÷ side of the small cube. Example: 18 cm cut into 3 cm cubes → n = 6 → two faces painted: 12 × 4 = 48 (ASO 2025).
Counting cubes in a stack
Count column by column from the back, and include hidden cubes that support the visible ones.
Minimum colours
Three colours are enough, because each pair of opposite faces can share a colour.