PK 2020 Q21
21. A large cube is built using 64 smaller identical cubes. Three of the faces of the large cube are painted. What is the maximum possible number of small cubes that can have exactly one face painted?
- A 27
- B 28
- C 32
- D 34
- E 40
Pink Kangaroo 2020, Question 21. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 32
The large cube is 4 × 4 × 4. There are only two possible configurations of the three painted faces: either they all share a common vertex, or they don’t.
When the three faces share a common vertex, the edges that they share will have more than one face painted, leaving 9 cubes on each of the three faces with exactly one face painted, as shown in the diagram on the left. This is 27 cubes altogether.
When the three faces don’t share a common vertex, then each of the cubes on the common edges will have more than one face painted. This leaves two of the large faces having 12 cubes with one face painted, and the middle large face having 8 cubes, giving a total of 32 cubes, shown in the diagram on the right.
Solution © UK Mathematics Trust, from the PK 2020 solutions.