GK 2020 Q14
14. Andrew buys 27 identical small cubes, each with two adjacent faces painted red. He then uses all of these cubes to build a large cube. What is the largest number of completely red faces that the large cube can have?
- A 2
- B 3
- C 4
- D 5
- E 6
Grey Kangaroo 2020, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 4
Since no small cubes have three faces painted red, it is impossible to build a large cube with three faces that meet at a vertex painted red. Therefore no more than four of the faces can be red and, without loss of generality, let us consider that they are the front, back and sides of the cube. We can construct a large cube with four faces completely red by arranging that the 12 small cubes along the four vertical edges of the large cube have two red faces showing and the 12 small cubes down the centre of the four vertical faces have one red face showing. Hence it is possible to build a large cube with four completely red faces.
Solution © UK Mathematics Trust, from the GK 2020 solutions.