Season 5 Episode 10
In this episode, James tries to count all the cubes with faces painted red or green or blue, using two different methods.
Nets of 57 cubes
Here's a PDF that I've made with nets of the 10 black-and-white cubes then nets of the 57 red-green-blue cubes from the livestream. I made this because I wanted to have physical models of all 57 cubes. I've tried printing this on A4 paper and it doesn’t work; they're far too small and paper is too flimsy. If you're going to attempt to make these then
- Good luck. Seriously, 57 is a lot.
- Enlarge them somehow.
- Print them on card or something, perhaps.
- Take a photo when you’re done.
As far as I can tell, there are no photos of these cubes on the web, despite quite a few descriptions of why there are 57 of them. For example, Wikipedia just has one image of one of them, and we can’t even tell which one it is. And it's not even a photo!
57 cubes (you have been warned) (PDF, 3.64 KB)
Orbit Counting Lemma
Wikipedia has an article on this lemma which gives a couple more examples and a bit more of the history.
The theory behind this is much more general than the problem I showed you on the livestream. This is part of Group theory, which I think that some people get to study as part of A-level Further Maths or equivalent, and which anyone who does a Mathematics degree should get to study.
For an introduction, see Brilliant | Group Theory Introduction.
For an introduction with a bit more attitude, see these notes from Jake Wellens at MIT.
Jonah introduced some group theory on OOMC last year in Season 3 Episode 2
Counting and Fermat
Here's another bit of counting with rotations that is thematically similar to the problem we did on the livestream.
It’s possible to prove Fermat's Little Theorem (not Fermat's Last Theorem!) by counting necklaces. I saw this as a maths workshop as a teenager, and you can see it as a blog post here.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.