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The chromatic number of the plane is at least 5

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.

CLP reads “Non-sexist solution to the ménage problem”

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.

From the Mailbag: Golfing Combinatorics

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

From the Sartorial Arts Journal, New York, 1901Dear The Aperiodical,

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

Information and Inference: new journal with free content for two years

The Institute of Mathematics and its Applications has launched a new journal, Information and Inference: a Journal of the IMA. This aims to

publish high quality mathematically-oriented articles, furthering the understanding of the theory, methods of analysis, and algorithms for information and data.
Articles should be written in a way accessible to researchers in the associated topics in pure and applied mathematics, statistics, computer sciences, and electrical engineering. Articles are published in, but not limited to: information theory, statistical inference, network analysis, numerical analysis, learning theory, applied and computational harmonic analysis, probability, combinatorics, signal processing, and high-dimensional geometry.

According to the website, “all content will be free to access for the first two years of publication of the journal”. You can sign up for free email table of contents alerts.

The first paper, ‘The masked sample covariance estimator: an analysis using matrix concentration inequalities‘, has been made available for advanced online access.

More information: Oxford Journals: Information and Inference: a Journal of the IMA.

Like everybody else, you too can be unique. Just keep shuffling

The first take-home lesson of this note is that you too can be unique. You’ll have to keep shuffling to get there, but it is an attainable goal.

Several years ago it dawned on me that the number of possible ways to order or permute the cards in a standard deck of size $52$ was inconceivably large. Of course it was — and still is — $52!$. That’s easy enough to scribble down (or even surpass spectacularly) without understanding just how far we are from familiar territory.

A 17×17 4-colouring with no monochromatic rectangles

Described on The Math Less Traveled