Let us assume that there are no points which are 1 unit apart and have the same colour.

Now, if I choose two points on the plane which are √3 units apart (let’s call them P and Q, and say that P is coloured red), there are two points R and S which lie in between P and Q and such that both PRS and QRS are equilateral triangles with sides of length 1 unit.

We have assumed that any points 1 unit apart are of different colours, therefore in the triangle PRS, R and S must be one blue and one green. This then means that in the triangle QRS, as R and S are one blue and one green, Q must be red.

Using this argument shows that any point that is a distance √3 units from P must also be red.

Therefore, there must be a circle of red points with centre P and radius √3 units.

But if we have a circle all of red points with a radius greater than 1, it must be possible to find many pairs of points that are 1 unit apart and are both red.

We have contradicted what we originally assumed, which means that the original assumption must be incorrect.

Hence, it must be possible to find two points on the plane which are 1 unit apart and have the same colour.

Challenge 340: A Colouring Challenge