Hilbert's Hotel Paradox Calculator

Step through the classic Hilbert's hotel paradox: how a fully booked infinite hotel can always accommodate one more guest, a busload, or infinitely many busloads of new guests.

Frequently Asked Questions

What is Hilbert's hotel?

A thought experiment about a hotel with infinitely many rooms, all occupied. Even so, more guests can always be accommodated via clever reassignment.

How can a full hotel take new guests?

Move every current guest from room n to room n plus one. Room one is freed up. The hotel was full only in the sense that every room had a guest; with infinite rooms, you can always shift.

What about infinitely many buses?

Use a pairing function (such as bus i, seat j to room two to the i times three to the j) to assign every newcomer a unique room. Infinitely many guests still fit.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.