GK 2019 Q18
18. Natasha has many sticks of length 1. Each stick is coloured blue, red, yellow or green. She wants to make a 3 × 3 grid, as shown, so that each 1 × 1 square in the grid has four sides of different colours. What is the smallest number of green sticks that she could use?
- A 3
- B 4
- C 5
- D 6
- E 7
Grey Kangaroo 2019, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 5
Since each 1 × 1 square has four sides of different colours, there is a green stick along the side of each square. Also, since any stick is part of at most two squares four green sticks could only contribute to at most eight squares. Therefore at least five green sticks are needed. The diagram on the right shows that such an arrangement is possible with five green sticks. Hence the smallest number of green sticks she could use is five.
Solution © UK Mathematics Trust, from the GK 2019 solutions.