Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim

Er no, there would never be an infinite gap. There would be arbitrarily large gaps, but each would be finite, and so would take only finite time to get from the starting point to the finishing point. Not a problem, really. Just cryogenically freeze the people for whom it will take, say, 50 years or more for them to reach their room.

But again, as Pesto mentioned, that sort of difficulty is not really the crux of the problem. What we're really doing is demonstrating an interesting feature of infinite sets: you can take a proper subset of them -- that is, you can remove elements -- and the size of the set will not have changed. You do thing by putting the elements in a one-to-one correspondence. In the case of the hotel, for example, the one-to-one correspondence is N<-->2N. The fact that you can do that shows that there's the same number of even integers as there are positive integers, which is kinda counterintuitive.

Another interesting result, along the same lines, is that there are the same number of real numbers between 0 and 1 as there are real numbers in total. I'll let you try to put these numbers in a one-to-one correspondence with all real numbers. The next one to try after that is that there are the same number of rational numbers (numbers of the form a/b, where a and b are integers and b is not 0) as there are postive integers. Next, and hardest, is that there are more real numbers than there are positive integers: you do that by showing that if you put positive integers and real numbers in a one-to-one correspondence, you'll have some real numbers left over, always. The reason that's cool is it shows that there is more than one infinity, and that some infinities are larger than others. Mind-blowing.

-Erasmus