Paint every point of the plane with either blue or red color. Show that there are 2 points on the plane (same color) one inch apart.

Question

I need help with this question, because I do not understand some points.

Pigeonhole Question: Paint every point of the plane with either blue or red color. Show that there are 2 points on the plane (same color) one inch apart.

Hint I have received: Imagine an equilateral triangle with sides of 1 inches.

Problem with the hint: Why an equilateral triangle? Where does the triangle come in?

My Proof Attempted With the Hint:

If there is an equilateral triangle with sides of 1 inch apart, and two of the vertices were different colors. The third vertical point must be one of the two colors, therefore proving that there are two points, same color, one inch apart.

Answer 1

Your question is essentially how to derive the solution (including the hint). In general there is no deterministic way (and there cannot be), but often we try to prove stronger claims if we believe the result is true and weaker than necessary.

The original statement is:

Given any 2-colouring of the points in the plane, there are two points of the same colour with distance 1.

The stronger statement is:

There is a finite set $S$ of points in the plane, such that given any 2-colouring of $S$, there are two points of the same colour with distance 1.

Why do we think this is true, in this problem?

To solve any problem one has to first experiment. If one doesn't even try special cases, one can forget about solving any significant problem at all. In this problem, we could start with simple 2-colourings of the plane (say the left half red and the right half blue), but we also should try colouring one point at a time. This latter approach leads to the solution directly:

Start with some arbitrary point $P$. Since we haven't coloured anything yet, we can colour this one red (blue's the same).

Now a lot of points can't be coloured red anymore, in fact a whole circle of them of radius 1 centred at $P$ must be coloured blue.

Wait a minute now there are obviously two points on that circle coloured blue!

The solution is complete, but if you look closer you see that it didn't really look at all the points on the circle; it only needed to look at 2 of them that form an equilateral triangle with $P$.

So the moral of the story is that the hint was crafted by someone who knew the solution and didn't really know how to motivate finding the solution by oneself...