A conversation about combinatorics, the mathematics of counting, inspired by a robot caterpillar. Presented by Katie Steckles and Peter Rowlett.

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A conversation about combinatorics, the mathematics of counting, inspired by a robot caterpillar. Presented by Katie Steckles and Peter Rowlett.

Podcast: Play in new window | Download

Subscribe: Android | Google Podcasts | RSS

At the MathsJam annual gathering, one of the many activities attendees can participate in is a competition competition – entrants each come up with a competition and submit it into a larger competition, other attendees enter each of the competitions within the competition competition, and the organisers get the chance to make long and confusing (but strictly correct) announcements that contain the word competition a lot of times.

This year, we decided, after a spectacular last-minute MathsJam bake-off entry failure on the behalf of Katie, to enter a joint competition into the competition competition. Inspired by the ‘lowest unique answer’ style of competition, which has previously featured in various MathsJam Competition Competitions (and our recent lecture on game theory) we came up with an idea – what about a competition seeking a unique entry in a non-ordered set?

Pringles ran a Super Bowl advert. In case you’re looking for ways to give Pringles more money, apparently you can buy several tubes of Pringles and mix the flavours. (Pringles are a type of food. Super Bowl is a kind of sport. None of that matters, what matters is…) The advert shows a man stacking three Pringles together and claims there are 318,000 possibilities.

If you are like me, you have played the game *SET* and have probably been perplexed at how quickly some people can play the game! Even as the game is quite easy to explain, it takes some time to build various strategies and pattern recognition to play the game effectively. If you have never heard of *SET*, don’t fret because we will soon review its layout. For my final masters project at Texas A&M University, we had the autonomy to research any higher-level mathematical topic and I felt *SET* would be a great venue to tap into some deeper mathematics. Little did I know how truly complex and elegant *SET* really is with connections to combinatorial geometry, finite affine geometry, and vector spaces over finite fields, some of these problems still open in research-level mathematics. All of these topics (and more) are included in a great resource I highly recommend for some summer reading. Check out *The Joy of Set* by McMahon, et al. to dig deeper into what is presented below.

A long-standing mathematical problem has had a recent breakthrough – scientist Aubrey de Grey has proved that the chromatic number of the plane is at least 5.

I rediscovered this nice paper by Kenneth P. Bogart in my Interesting Esoterica collection, and decided to read through it. It turned out that, while the solution presented is very neat, there’s quite a bit of hard work to do to along the way. I’m not particularly experienced with combinatorics, so the little facts that the paper skips over took me quite a while to verify.

Once I was happy with the proof, I decided to record a video explaining how it works. Here it is:

*I probably made mistakes. If you spot one, please write a polite correction in the comments.*

*Sam’s dad is in a mathematical conundrum – so she’s asked Katie, one of our editors, if maths can save the day.*

My dad is going away on a golfing holiday with seven of his friends and, since I know a little bit about mathematics, he’s asked me to help him work out the best way to arrange the teams for the week. I’ve tried to work out a solution, but can’t seem to find one that fits.

They’ll be playing 5 games during the week, on 5 different days, and they’d like to split the group of 8 people into two teams of four each day. The problem is, they’d each like to play with each of their friends roughly the same amount – so each golfer should be on the same team as each other golfer at least twice, but no more than three times.

Can you help me figure it out?

Sam Coates, Manchester