Season 11 Episode 12
James has a problem to do with road networks on this episode of the Oxford Online Maths Club. This last episode of the term also includes an OOMC REWIND looking back at the other episodes and further reading notes.
Further Reading
Starter problem
This problem reminded me of the Collatz conjecture.
Here's a video you could watch; The Simplest Math Problem No One Can Solve - Collatz Conjecture | Veritasium | YouTube
Or if you prefer to read an article, how about this explainer on Quanta? The Simple Math Problem We Still Can’t Solve | Quanta magazine
My personal hope is that the Collatz conjecture turns out to be undecidable, so that all these headlines that say things like “no one can solve” turn out to be mathematically true. There are generalisations of the Collatz conjecture that are known to be undecidable, so this is not a crazy thing to hope for!
Soap film minimisation
The problem that I discussed with the four towns is sometimes called the Steiner Tree Problem. One of the places that I've seen this is in one of David Acheson’s popular mathematics books. Here's a video of David Acheson demonstrating with a real live soap film!
The general problem (not a square) is discussed in these very accessible notes by David Wakeham here.
For regular polygons with more sides, see the diagrams on Wikipedia.
Perhaps this is disappointing, but with six or more sides on a regular polygon, the best road network just goes around all-but-one of the outside edges!
Rewind
You can find out more about all the previous episodes on the Maths Club homepage.
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.