GK 2015 Q15
15. Luis wants to make a pattern by colouring the sides of the triangles shown in the diagram. He wants each triangle to have one red side, one green side and one blue side. Luis has already coloured some of the sides as shown. What colour can he use for the side marked \(x\)?
- A only green
- B only blue
- C only red
- D either blue or red
- E The task is impossible
Grey Kangaroo 2015, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A only green
Label the internal sides of the diagram \(a\), \(b\), \(c\), \(d\) and \(e\) as shown. The side labelled \(a\) is in a triangle with a green side and in a triangle with a blue side and so is to be coloured red. This is also the case for the side labelled \(e\). Hence, the side labelled \(b\) is in a triangle with a red side and a green side and so is to be coloured blue while the side labelled \(d\) is in a triangle with a red side and a blue side and so is to be coloured green. Finally, the side labelled \(c\) is in a triangle with a green side and in a triangle with a blue side and so is to be coloured red. Hence the side labelled \(x\) is in a triangle with a side that is to be coloured blue (side \(b\)) and with a side that is to be coloured red (side \(c\)). Therefore the side labelled \(x\) is to be coloured green.
Solution © UK Mathematics Trust, from the GK 2015 solutions.