How this instrument works
Hilbert's Hotel is a thought experiment, devised by mathematician David Hilbert, about the strange behavior of infinite collections: imagine a hotel with infinitely many rooms, numbered 1, 2, 3, and so on without end, every single one occupied. It seems like there's no room for anyone new — and yet the hotel can still accommodate more guests, simply by asking every current guest to shift over.
If k new guests arrive, the guest currently in room n just moves to room n+k — every existing guest shifts over by the same fixed amount, freeing up rooms 1 through k for the newcomers, with nobody left without a room. This only works BECAUSE the hotel has infinitely many rooms to begin with; a finite hotel obviously has no such trick available.
The paradox is a vivid, concrete illustration of a genuinely surprising fact about infinite sets: a 'full' infinite collection can still absorb more elements without anything overflowing, unlike a full finite collection, which simply has no room left at all.
- Enter the guest's current room number into the Current room number field.
- Enter how many new guests are arriving into the Number of new guests arriving field.
- Read New room number: where that guest moves to, freeing up the lower-numbered rooms.
- Try a large number of new guests to see the shift still works no matter how many arrive at once.
Worked example — room 5, three new guests
A guest currently in room 5, with 3 new guests arriving, shifts to room 5+3=8 — and since every OTHER guest shifts by the identical amount, rooms 1, 2, and 3 open up for the newcomers, with no guest left stranded and no room double-booked.
A guest in room 100, with 50 new guests arriving all at once, shifts to room 100+50=150, freeing up rooms 1 through 50 for them all simultaneously. However large the group of new arrivals, the identical shift-by-k trick keeps working, since the hotel has infinitely many rooms still further out to shift into.
Questions
What is Hilbert's Hotel?
A thought experiment about a hotel with infinitely many rooms, all occupied, that can still take in more guests by shifting every current guest over by a fixed amount — a vivid illustration of how infinite collections behave differently from finite ones.
How does a fully booked hotel make room for new guests?
Every existing guest moves from their current room n to room n+k, where k is the number of new guests arriving — this frees up exactly k rooms (numbered 1 through k) at the low end, with no guest left without a room.
Why doesn't this work for a normal, finite hotel?
A finite hotel has a highest-numbered room — shifting every guest up by k would push the last few guests past that highest room number entirely, with nowhere for them to go, which is exactly the difference infinity makes here.
Who came up with this paradox?
It's attributed to the mathematician David Hilbert, who used it in a lecture to illustrate counterintuitive properties of infinite sets in a way non-mathematicians could follow directly.
Does the number of new guests matter to whether the trick works?
No — the identical shift-by-k logic works whether one new guest arrives or infinitely many do (in the infinitely-many case, guests shift to double their room number instead), since there are always further rooms out along the infinite sequence to shift into.