h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

With a little forethought, Hilbert could have avoide the problems given on this page, observe: Assume we start with the hotel empty. Someone comes along for a room, he gets no 1. Next person no 2, then 4, 8, 16, doubling each time, so the powers of two are used: 2^n (n=0,1,2,...). Now, when an infinite number have booked in, we put the next person in 3, then 5, 9 , 17 i.e. 2^n +1 (for n=1,2,...). When those have been filled up, we start with 6, 10, 18 (2^n +2: n=2,3,...). Next 11, 19, 35 (2^n +3: n=3,4,...), then 20, 36, 68 (2^n +4: n=4,5,...)

In this way, we can even cope with the infinite number of buses, each containing an infinite number of people and even this will not fill the hotel up as we have missed, for example 7, 12, 13, 14, 15 and an infinite number more.

An even easier method is to first use room 1, then fill up all rooms 2^n (n>0), then 3^n, 5^n, 7^n, 11^n and so on for all prime numbers. Easier to define, simply.
However, the whole idea of the (non-)problem was to demonstrate how you could fit in more people even if ALL rooms were occupied.

That's a good point.

I was aware of that, just being contrary.