Pigeonhole Principle Question: Given any 5 points inside a square of side length 2, there is always a pair whose distance apart is at most $\sqrt2$

Solution 1:

Credit to @Gerry_Myerson for inspiration.

We have a 2x2 square. We want to split this into 4 1x1 squares, simply by joining the centres of opposite sides. This creates a small grid.

Now imagine we want to insert our 5 points into these 4 1x1 squares. Here we apply the pigeonhole principle. Since we have more points than squares to place them, we know that some square must end up containing at least 2 points.

So now we know that given any 5 points inside a square of side length 2, we can find two of them that lie in some square of side length 1. What is the maximum distance apart that these can be? It's obvious (and prove-able, though I won't do it here) that the two points are maximum distance apart when they lie on opposite corners. By simple Pythagoras', we know that the length of the diagonal of a square of side length 1 is $\sqrt2$, so we have the required result.