A hotel with infinitely many rooms can still take new guests by moving each occupant to the next room.

The longer version

Every room is occupied, yet a new arrival can be placed by asking the guest in room one to move to room two, the guest in room two to move to room three, and so on down the corridor. Room one falls empty. No one is evicted, because the list has no last occupant to be pushed out.

An entire busload can be accommodated too, by sending everyone to the room numbered twice their current one, which vacates all the odd numbers at once. A finite hotel has no such trick, since shifting occupants always strands the final guest. The thought experiment is really a demonstration that a set can match a proper part of itself.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.