Question

The plane is coloured red and blue in some configuration. Prove that there must be two points of the same colour exactly 1 inch apart.

Answer 1

I have no idea how to approach this one.

Answer 2

Is there a picture that goes with it?

Answer 3

Assume that the statement is false

Answer 4

So there are no two points of the same colour which are 1 inch apart

Answer 5

Take any random blue point on the plane (could be blue or red doesn't matter) and draw a circle of radius one inch around it
All the points of this circle have to be red for our assumption to hold

Answer 6

Next, take any point on the circle and draw another circle of radius 1 inch around it
All the points on this circle have to be blue

Answer 7

You'll find that there will be two points which will be blue according to ^ and red according to ^^
A point can't have two colours at once so the assumption is false

Answer 8

Another way to solve this: start by drawing an equilateral triangle with a side of 1 inch anywhere on the plane. Since the plane is coloured only red and blue, two things can happen:
i) all the vertices of the triangle have the same colour (i.e., they are either all blue or all red);
ii) two of the vertices have the same colour and the third one has another colour (i.e., two of them are blue and the third one is red or two of them are red and the third one is blue).
In both situations i) and ii) we have found two points of the same colour exactly 1 inch apart, just like we wanted to prove.

Answer 9

I like both answers but I particularly like the above answer. Thank you.