Definition
Pigeonhole principle
If more objects are assigned to fewer categories, at least one category receives two or more objects.
What it means
Think of putting items into labelled boxes. If every box were allowed only one item, the boxes would hold at most as many items as there are boxes. One extra item makes sharing unavoidable. The clever step in a problem is choosing the right items and boxes.
Remember it
More guests than chairs means at least one chair must be shared.
Example
Six homework notes are filed into five subject folders. Even without seeing the filing, at least one folder must contain two or more notes. If every folder held at most one note, the five folders could hold at most five notes, not six.
Watch for
When the number of objects is not greater than the number of categories, sharing may happen, but the principle does not guarantee it.
Common mix-up
The principle proves that a shared category exists; it usually does not tell you which category is shared.
Exact meaning
The pigeonhole principle says that if more objects are assigned to fewer categories, at least one category must receive at least two objects. It guarantees that some category is shared, although it does not identify which one.