Why Too Few Categories Guarantee a Match
The key is that the set of possible hair counts is finite. A person’s count might be zero, one, or another whole number, but a head can hold only so many hairs. Once that upper limit is fixed, every person must fit into one of a limited number of count categories.
In general, imagine placing more objects than boxes into the boxes. If each box held at most one object, the boxes could contain no more objects than there are boxes. So at least one box must hold two or more. In the hair-count example, the people are the objects and the possible counts are the boxes. No one needs to inspect each person: the match follows from the limits alone.
The Principle Finds Collisions, Not Their Location
This kind of reasoning is useful whenever a system assigns many things to a smaller set of labels. For example, if a computer maps more distinct inputs to a finite set of output codes than there are codes, two inputs must receive the same code. That shared output is called a collision. The principle guarantees one exists, but it does not tell you which inputs collide or how to find them.
It also explains why the size of the categories matters. If there are one hundred possible labels, assigning one hundred items does not force a duplicate; each could get its own label. The hundred-and-first item does. The practical payoff is a certainty without a search: count the available categories, compare them with the items, and know when a repeat is unavoidable.
