Suppose every point in a plane is colored red or colored blue. Show that there exists some line segment in the plane such that its end points and its midpoint are all the same color.

## Solution

Pick two points $x$ and $y$ that are, say, red. We propose to work with the line $xy$ (see the figure). On this line, locate points $M, N,$ and $P$ such that $Mx = xy = yP$ and $xN = Ny$.

Now, by cases:

- If $N$ is red, we’re done: $xNy$.
- If $N$ is blue, then look at $M$.
- If $M$ is red, we’re done: $Mxy$.
- If $M$ is blue, then look at $P$.
- If $P$ is red, then we’re done: $xyP$.
- If $P$ is blue, then we’re still done: $MNP$.

Lovely problem, eh? (Only if you like proof by cases!). Note: there are several more problems about this red and blue plane, all with the 3060 Stella number.