TITLE: Colouring 10 dots to form equilateral triangles
QUESTION [9 upvotes]: Suppose we have 10 dots arranged in as a triangle as shown below.

Colour the dots so that some are red and the rest are blue. Show that regardless of how we colour the dots, an equilateral triangle can be formed from three dots of the same colour.
Any advice or clues is greatly appreciated.

REPLY [4 votes]: The six dots around the center dot form two equilateral triangles. Clearly no five dots can be of the same colour. If both colours occur exactly three times in these six dots then we get an equilateral triangle of one colour by considering the colour of the center dot if necessary. Hence one colour occurs four times and the other twice, and the center colour is the other colour. So the colours form three bands in some direction, and now any colouring of the corner dot yields an equilateral triangle of one colour.