Mathematics
The Pigeonhole Principle and Its Surprising Applications in Combinatorics
Quick fact
The pigeonhole principle, though obvious, is the engine behind deep results like the Erdős–Szekeres theorem: any sequence of more than (r-1)(s-1) distinct real numbers contains an increasing subsequence of length r or a decreasing subsequence of length s.
Why this is interesting
Have you ever wondered why, in a room of just 23 people, there's a better than even chance that two share a birthday? The answer hides in a surprisingly simple idea that seems too trivial to be powerful.