# Placing Pandas in a Triangle Pen

I am working on a bit of a silly problem in my introductory discrete mathematics course. I have five pandas that I need to place in a pen, and I have a pen that is the shape of an equilateral triangle with sides of length 2. I want to prove that If I were to place these five pandas in the pen, there must be at least two pandas that are of less than or equal to unit distance from each other. I definitely think this is a pigeonhole principle problem, although I am not sure what should be the "boxes'' in this situation. Any assistance would be greatly appreciated.

• Think about triforce. – user296602 Jan 15 '16 at 1:05
• Yes, pigeonhole principle is your friend here. For a more difficult challenge, determine the maximum of minimum distances between pandas (across all possible placements). – hardmath Jan 15 '16 at 3:04

In this diagram, the large triangle has sides of $2$ while the four smaller ones have sides of $1$. So place five pandas into those four small triangles. The maximum distance between any two points in any small triangle is clearly just $1$. Use the pigeonhole principle and finish from here.
Hint: you can cut the pen into four triangles with side $1$