>[!abstract] >The pigeonhole principle in mathematics states that if there are $n$ elements and $m$ categories, and $n \gt m$, then at least one category must contain more than one element. > >The principle sounds trivial but has deep mathematical applications, including in Dirichlet’s approximation theorem, Ramsey theory, etc. >[!example] Examples >- In a group of 366 or more people, at least two share a [[Birthday paradox|birthday]] (i.e., the probability $p$ of a collision is strictly equal to $1$). >- In a drawer containing socks of three colors, picking any four socks blindly guarantees that at least two are of the same color. >- The human head has no more than 150,000 hair. In any city with a population greater than 150,000, at least two people have the same count of hair. >- If 10,000 emails are received daily by 100 employees, one of them receives at least 100 emails. >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —