6. The circles in the diagram are to be coloured so that any two circles connected by a line segment have different colours.
What is the smallest number of colours required?
A 2
B 3
C 4
D 5
E 6
Solution
B
Each pair of the circles labelled $P, Q$ and $R$ in the figure on the right is connected by a line segment. Therefore these three circles must be coloured using different colours. So at least three colours are needed.
The figure on the right (with the circles enlarged for the sake of clarity) shows one way to colour the circles using three colours so that circles connected by a line segment have different colours.

Therefore 3 is the smallest number of colours required.