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Do you use mixed fractions?

I’m at the MATRIX conference in Leeds, where I’ve just been talking to Adam Atkinson. He told me that he’s trying to compile a definitive list of countries that don’t use mixed fractions.

Here’s a mixed fraction: \[ 2 \frac{2}{3} \]
And here’s a non-mixed fraction: \[ \frac{8}{3} \]
Actually, here’s an interesting fact about that number: \[ 2 \sqrt{ \frac{2}{3} } = \sqrt{ 2 \frac{2}{3} } \]
This only makes sense if you believe in mixed fractions (and unicode character U+2062, “invisible times”)

This is going to be one of those wipe-your-bum-standing-up situations: it’s entirely possible that you can be on either side of this divide and not know the other exists. Apparently, in some countries mixed fractions just don’t exist: an integer written next to a fraction is incorrect.

So, to help Adam on his way, I thought I’d start another in our long-running series of Aperiodical Surveys. Please tell us where you live, and if mixed fractions are OK in your book.

Integer Sequence Reviews: A075771, A032799, A002717

It’s been almost two years since I last sat down with my friend David Cushing and did what God put us on this Earth to do: review integer sequences.

This week I lured David into my office with promises of tasty food and showed him some sequences I’d found. Thanks to (and also in spite of) my Windows 10 laptop, the whole thing was recorded for your enjoyment. Here it is:

I can only apologise for the terrible quality of the video – I was only planning on using it as a reminder when I did a write-up, but once we’d finished I decided to just upload it to YouTube and be done with it.

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.

A new home for Interesting Esoterica

interesting-esoterica-homepage

Since 2010, I’ve been maintaining a list of “interesting esoterica” – papers, books, essays and poems that I find interesting entirely on their own merits. It’s mainly bits of esoteric maths – hence the name – but I’ve also included quite a few things just because they have amusing titles. The main idea is that when I’m talking to someone and want to show them a cool thing that I’ve half-remembered, I can look up the exact reference: I’ve shared the paper “Orange peels and Fresnel integrals” more times than I can count (probably the same as the number of times I’ve eaten an orange).

Maths Object: Nobbly Wobbly

My maths object this time is one of my dog’s favourite toys: the Nobbly Wobbly.

In the video, I said it was invented by a mathematician, but Dick Esterle’s bio normally goes “artist, architect, inventor”. I’ll leave it up to you to decide if Everyone’s a Mathematician.

It’s a particularly pleasing rubbery ball thing made of six interwoven loops in different colours, invented by Dick Esterle.

On Google+, various people told me the unexpected fact that the outer automorphism group of $S_6$ is hiding inside this dog toy.

I’ve also found this Celebration of Mind livestream starring Dick Esterle from 2013 talking about all sorts of mathematically-shaped toys, including the Nobbly Wobbly.

Are you sure 51 isn’t prime? – Analysing the results of the “Is this prime?” game

Two months ago, I bought isthisprime.com and not only set up the internet’s fanciest primality-checking service, but also invented a rather addictive game.

It quite quickly went viral, or as relatively viral as a maths game can get, with people tweeting their high scores and posting the link to reddit and Hacker News. I realised fairly soon that I should put in some stats tracking, to see if there were any interesting patterns in the data (and also to inflate my ego as the “games played” counter went up). I missed the first big spike in traffic, but on the 9th of March I wrote a script which saved a record of each game to a database.

isthisprime game dates

The mad rush settled down quite quickly but there were still occasional spikes as different sites or people with lots of twitter followers found the game. Now, after two months, I’ve got data for just under 350,000 games. That’s a decent amount of information!

Approximate a ratio by folding a piece of paper

towel-ratio

Warning: you could make a very strong argument I’ve thought far too much about something inconsequential. If that makes your stomach turn, look away now.

This morning in the shower, I had an idle thought about my towel. It was, as always, folded neatly on the toilet seat. A problem that’s been bugging me for a few days is how to pick up the towel by a section of the long edge, so when it unfolds it’s the right way round.

* quiet in the back

The problem is that the short edge and the long edge look the same, and once I’ve folded the towel over a couple of times and had a shower only a madman* would remember which is which. But my towel isn’t square, so it occurred to me that either the longer or the shorter edge, after folding, could be the edge I want. Since I never make a diagonal fold, the long edge is only ever folded on top of the long edge, and likewise for the short edge. I fold the towel until it fits comfortably on top of the toilet seat, and by the time I’ve finished my shower I can’t be relied upon to remember which sequence of folds I did.

Which got me thinking about the ratio between the width and height of my towel: if I know this ratio then, by looking at the towel and counting the number of folds, I can work out which folds I’ve done, and hence which of the sides will unfold to be the long edge.