Hilbert’s Hotel has made many appearances in the literature and the media over the years. See, for example, the Wikipedia article https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel, which mentions quite a few examples in the body, and a few more in the External Links at the end. I have always enjoyed reading fiction where the Hotel turned up! However, authors haven’t always taken the opportunity to look in detail at more sets of numbers, such as the rational numbers (which do fit into Hilbert’s Hotel) and the real numbers (which don’t fit). Also, the case of (countably) infinitely many buses each with (countably) infinitely many passengers is relatively easy to handle using prime powers or prime factorization, and the story really shouldn’t stop there. In any case, I thought that this topic would make for a fun maths talk, and back in 2006 I finally got round to producing a Beamer presentation, with the title Beyond Infinity? It was a lot more work than I thought getting the “transitions” to play nicely! But I presented the talk in 2006 and 2007 at various University of Nottingham enrichment and recruitment events. I also recorded audio (15 and a half minutes), but not video.

In 2008, I combined the slides and audio using Windows Movie Maker (I have never had so many crashes!) to produce a video of the talk. At the time, 15+ minutes was too long, so I split it into Part I (https://www.youtube.com/watch?v=OdhD-cx0OHQ) and Part II (https://www.youtube.com/watch?v=BKQjL-nChv8). The slightly more advanced material about the real numbers not fitting into the hotel is in Part II. Once YouTube allowed videos to be longer, the University of Nottingham published the full video (https://www.youtube.com/watch?v=Tj6DwD6c4ro) on the official University YouTube channel.

I wasn’t completely happy with this edition. I felt that I could have said more in places, and that it might be good to have some webcam footage of me in the corner. Also, I decided that “Hotel Uncountable” wasn’t very good, and that “Hotel Continuum” would be much better. So in 2010, when I was presenting the talk again, I took the opportunity to record a new screencast together with webcam footage. The resulting video (https://www.youtube.com/watch?v=3FDXnChPVm8) was longer, at 24 and a half minutes. I think it was an improvement, but it has had fewer views. Perhaps that is because of the length? Maybe I should split it in two?

In 2020 I produced a new version for the University of Nottingham’s Open Days (which were online that year). My original slides did not conform to the latest University branding, but Helen Preston kindly created a new set of slides in PowerPoint using a colour scheme based on different shades of blue. Rather than recording a live presentation, this time I pre-recorded the audio to go with the PowerPoint slides. Given that I had given the talk live several times and recorded it before, I wasn’t expecting it to need many takes. But somehow pre-recording is different, and it took a lot longer than I thought before I was satisfied. You can find the resulting edition via https://www.nottingham.ac.uk/open-days/catch-up?tag=Mathematics or directly at https://www.nottingham.ac.uk/open-days/video.aspx?id=bb776f5c-c66d-4b7a-bf2f-14b2be38f6fc, and it is also available on YouTube at https://www.youtube.com/watch?v=oq5MdaNnQ3E and on MediaSpace (which is available in China) at https://mediaspace.nottingham.ac.uk/media/t/1_8tmdqul4
This is the shortest of the three editions (the talk itself is only about 12 minutes long, though it is followed by a longer Q&A session).

I still have a preference for live presentations myself, and for including webcam footage in the corner, but I’d be interested in comments if anyone can stand to watch at least some of each video!

Several of my colleagues have also presented this talk, and we have also run it as a longer interactive session (30-50 minutes) with puzzles for the pre-university students to think about, though it is probably best to refer them to the full video for the details of the last section.