Mathematics
Finite Automata and the Languages They Recognize
Quick fact
A finite automaton with n states can recognize a language that contains a string of length n only if there is a cycle, meaning that recognizing unbounded patterns requires some form of repetition.
Why this is interesting
Have you ever wondered how a vending machine knows you've inserted the right amount of money? It doesn't count every coin—it just remembers a few states. That's the essence of a finite automaton.