Mr. Hilbert is busy renovating his hotel. The hotel rooms are going to be made larger, which means that there will be excess beds, furniture etc. that will be sold. For every ten items of one kind one will be sold. His workers went about emptying the rooms one by one, for every ten items of one kind, one was shipped away. When he visited his hotel to see how things are progressing, he was shocked to see that all items had been sold.

His workers, however, assured him that they stuck to his system of selling only one out of ten items taken from the hotel. All items taken from the hotel were moved to a storage site, when ten new items of one kind were moved there, one item of the same kind from the storage site was sold. The question then is how all items can end up having been sold if only one item out of ten taken from the hotel is going to be sold.

  • 2
    Does the hotel have a circular layout by any chance? – BmyGuest Oct 22 '16 at 21:27
  • @BmyGuest It's not circular but there is something peculiar about the hotel build by Hilbert. This is a well known fact that you can look up without risk of finding a direct answer to this question. – Count Iblis Oct 22 '16 at 21:37
  • When did he visit? "When he visited his hotel to see how things are progressing" might suggest that the the renovation was still ongoing. In that case, some items should still be available. If he only visited after the renovation was finished... well, the renovation would have taken quite some time, right? Probably room 1 would have to be renovated again by the time the hotel was finished. – bers Oct 23 '16 at 13:55
up vote 10 down vote accepted

It's fairly simple. At the storage room you count the arriving beds from this well-known infinite hotel, one by one, and you put a sticker on them with its arrival number:

1, 2, 3, ...,

Once the arrival number of a bed is a multiple of ten, you then simply discard the bed with the smallest arrival number in the storageroom. So when the 10th bed arrives, you discard the first. When the 20th bed arrives, you discard the second, etc. You have followed Hilbert's rule.

The claim is that there are no beds left in the storageroom in the end. Indeed, bed number k was removed once bed number 10k arrived.

This hotel could be

empty (without any room) !
So 0 item is taken from the hotel and 0 is sold, which implies that everything taken out has been sold.

I think that the hotel....

.. is the infinity hotel.
A hotel with an infinite number of rooms.
Since 10% out of infinity is still infinity, it means that and infinity of furniture sets were sold. Since there was an infinity of furniture sets at the beginning it is possible that all of them to be sold.
infinity - infinity can be anything, including zero.

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