Hilbert's Hotel Paradox Calculator

Your details

Choose which paradox scenario to explore.
How many new guests are arriving (k). Every current guest shifts up by this many rooms.
Pick any currently occupied room to see where that guest relocates.
Current guest's new room
8

Room n + k

First free room after shift1
New room (finite shift)8
Existing guest room (x2)-
New guest room (odd)-
Existing guest room (2^n)-
New bus guest room (p^s)-

3 new guests accommodated with a simple shift

  • Every existing guest moves up 3 rooms, freeing the first 3 rooms.
  • Adding a finite number to infinity leaves infinity unchanged: the hotel is still full.
  • This illustrates that aleph-null (the size of the natural numbers) plus any finite k still equals aleph-null.

Next stepTry Scenario 2 to see what happens when infinitely many guests arrive at once.

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