Explore how an infinite hotel can always make room for more guests.
Last updated: June 2026 | By Patchworkr Team
Green = vacant, Cyan = occupied (Guest #)
The hotel demonstrates that countably infinite sets can be rearranged to make room for more guests.
Move guests to room n + 1 for one new guest, or to room 2n for infinitely many new guests.
One new bus arrives with infinitely many guests.
1. Move guests from room n to room 2n
2. Free all odd-numbered rooms
Final answer: all guests accommodated
Is this a real hotel?
No. It is a thought experiment about infinity.
Why does room 1 free up?
Because everyone moves from room n to room n + 1.
How do infinite guests fit?
By moving current guests to even-numbered rooms.
What does it prove?
It shows that countably infinite sets can be rearranged without changing size.
Related Tools
Explore Collatz sequences.
Infinity paradox.
Taylor series error bound.
Find magic numbers.
Generate magic squares.
Möbius strip properties.