Pigeonhole & Extremal

If $n+1$ objects go into $n$ boxes, some box holds at least two; more generally some box holds at least $\lceil m/n\rceil$ of $m$ objects.

In plain terms. More items than boxes forces a box to double up. The trick is choosing boxes so that "two in a box" is exactly what you want.

Example. Among any $13$ people, two share a birth month: $13$ people, $12$ months, so some month repeats.

Consider the largest, smallest, or closest object; its extremal property forces structure.

In plain terms. Look at the most extreme case in the setup — the biggest, smallest, or nearest — and its special status usually breaks the problem open.

Example. Among finitely many points, the two closest ones can't have a third point strictly between them.