On Wednesday 4th January, an error was discovered in the proof. Harald Helfgott (of the University of Göttingen in Germany and France’s National Center for Scientific Research), who studied the paper for several months, discovered that the algorithm was not *quasipolynomial* ($\displaystyle{ 2^{\mathrm{O}((\log n)^{c})} }$ for some fixed $c>0$) as claimed, but merely *subexponenential*: growing faster than a polynomial but still significantly slower than exponential growth).

Adorably, Babai posted this message on his website:

I apologize to those who were drawn to my lectures on this subject solely because of the quasipolynomial claim, prematurely magnified on the internet in spite of my disclaimers. I believe those looking for an interesting combination of group theory, combinatorics, and algorithms need not feel disappointed.

But maths is all about the drama, so on Monday 9th January Babai announced a fix for the error, and it’s now back on the quasipolynomial table. This has now been confirmed (as of 14th Jan) by Harald Helfgott himself at the Bourbaki seminar in Paris. Amusingly, Helfgott had only been studying the paper in such detail in order to give the seminar, and it was this close scrutiny which allowed him to discover the mistake.

Announcement on Babai’s website

Fixing the UPCC Case of Split-or-Johnson – Babai’s paper detailing the fix (PDF)

Graph Isomorphism Vanquished — Again, at Quanta Magazine

Bourbaki Seminar – Harald Helfgott, on YouTube

]]>I’m so far from understanding the mind of a mathematical genius that it’s simply inconceivable that you could tell a person an apparently random number and he could intuit or deduce the kind of fact that he deduced about that taxi license number. I mean, I can’t run a four-minute mile, but I once ran a five-minute mile, and I can extrapolate from my own experience, in a way understand how someone might just be a lot better than me at something that, in an inferior way, I can also do. But Ramanujan isn’t like that. It’s as though this man were a different species, not just a superior example of the same species. Can you learn to do this kind of thing? Could I, if I had applied myself? Or is it that goddess again, is it really just genius?

Answers on a postcard!

]]>The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>- Tricia Dodd, Chief Methodology Officer, UK Statistics Authority, appointed MBE “for services to Statistics and Research”.
- Dave Watson, director of IBM Research in the UK, who apparently has a focus on big data, appointed CBE “for services to Science and Engineering Research”.
- Maggie Philbin, appointed OBE “for services to Promoting careers in STEM and Creative Industries”.
- Anne-Marie Imafidon, co-founder and CEO of Stemettes, appointed MBE “for services to Young Women within STEM Sectors”.

I think every time I have done this (for New Year and Birthday Honours since 2013), there has been at least one person on the list, and usually several, specifically included for services to mathematics or mathematics education. This time, this is not the case, though there is one mention of statistics.

Are there any others I’ve missed? Please add any of interest in the comments below. A full list may be obtained from the UK Government website.

]]>The Math Teachers at Play (MTaP) blog carnival is a monthly collection of tips, tidbits, games, and activities for students and teachers of preschool through pre-college mathematics. We welcome entries from parents, students, teachers, homeschoolers, and just plain folks. If you like to learn new things and play around with ideas, you are sure to find something of interest.

I’ll be hosting the January 2017 edition of MTaP here at Travels in a Mathematical World. Of course, a blog carnival is only as good as its submissions, so if you join me in aspiring to the claim “you are sure to find something of interest” then please keep your eyes open for interesting blog posts and submit them to MTaP. Please submit posts you’ve enjoyed by others or yourself. Posts you wrote that are appropriate to the theme are strongly encouraged. Submit through the MTaP submission form, leave a comment here or tweet me. Thank you!

Submissions are open now, and anything received by Friday 20th January 2017 will be considered for the edition hosted here.

]]>The Joint Policy Board for Mathematics (JPBM), which consists of the American Mathematical Society (AMS), the American Statistical Association (ASA), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM) annually award their communications prizes to people they think are doing good work in mathematical outreach. This year, the prizes were awarded to two people.

**Arthur Benjamin** was awarded the Public Outreach prize, for a range of activities including his books aimed at general audiences, his TED talks, his popular video courses from *The Great Courses*, and his “mathemagics” performances. The prize citation says that his work demonstrates “his ability and commitment to share the joy of mathematics, and excites and engages audiences at all levels”.

The second award goes to **Siobhan Roberts**, a journalist and biographer based in Toronto. She’s written several mathematical books, the most recent of which was *Genius at Play—The Curious Mind of John Horton Conway.*

The awards will be presented today (5th January), at the Joint Mathematics Meetings in Atlanta.

*Arthur Benjamin to receive 2017 JPBM Communications Award for Public Outreach, at Eurekalert*

*Siobhan Roberts to Receive 2017 JPBM Communications Award for Expository and Popular Books, on the AMS website*

Breakthrough Prizes 2016

In early December, America’s rich and famous came together to celebrate achievements in science and mathematics and give away millions of dollars (no really, this is a thing that happens) – this year’s winners included mathematician **Jean Bourgain**, whose work on decoupling oscillators is described as a more complicated version of the Pythagorean theorem, but applied to waves.

In addition to the big \$3 million prizes, there were six \$100,000 New Horizons prizes – half in physics and half in mathematics. The awards were given out in an Oscar-style ceremony held at NASA’s Ames Research Centre, with Hollywood celebrity guests including Alicia Keys, Morgan Freeman and Jeremy Irons.

*News story at New York Times
Breakthrough Foundation website*

David H. Bailey, Andrew Mattingly, Glenn Wightwick and the late Jon Borwein are to be awarded the AMS’s 2017 Levi L. Conant Prize for their article *The Computation of Previously Inaccessible Digits of π2 and Catalan’s Constant*, which appeared in the Notices of the AMS in 2013. The Conant Prize is awarded annually to the best expository paper in either the Bulletin or the Notices of the AMS in the preceding five years.

*Previous winners of the Conant Prize*

*via David Bailey’s Math Scholar blog*

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Tomorrows pin flags at the #AAMSO will have a slight twist! See if you can work them out #JustForFun @MathsScot pic.twitter.com/6AlMGZ503v

— AAM Scottish Open (@AAMScottishOpen) July 5, 2016

**Katie: **This year, instead of the standard 1 to 18 written on the flags, the Scottish Open Golf Tournament used simple mathematical formulae – such as $12-7$ or $2^3 + 2^3$ to label their holes. It was a fun twist, and the tweet mentions @MathsScot – the Scottish government’s Making Maths Count campaign, which I suppose means they may have been involved.

**CLP:** A while ago we received an email from the marketing director of an online journal titled “Inference: International Review of Science”. All of our fake-journal alarm bells immediately started ringing.

Surprisingly, *Inference* is not a journal with dubious standards of peer review set up to diddle authors out of publishing fees – it’s a marvellous scholarly review of the sciences, from a humanist perspective, like an even more high-brow *Nautilus* magazine.

The email suggested we might like an article by Alexander Kharazishvili about Cantor’s diagonalisation method. While it looks at first glance like the same potted history of formal logic that you might have seen a hundred times before, they’re not afraid of a bit of notation, and in fact it goes into a lot of depth – it’s a real review of the subject. Kharazishvili has written another article for Inference, about divergent sequences, which I prefer even though it contains the phrase “the proof is easy”.

They’ve got some big names to write for them – to name one, Gregory Chaitin has written an article about experimental/empirical mathematics.

*Inference: International Review of Science*

**Katie: **Mathematicians from the University of Oxford Mathematical Institute have teamed up with the Ashmolean museum to produce a series of films demonstrating the mathematics behind the museum’s historical artefacts. From Babylonian tablets to symmetrical tile patterns, there are three 6-8 minute films, plus one that introduces the whole series.

*Random Walks – the Mathematics of the Ashmolean, at the Univeristy of Oxford website*

**CLP:** Fermat’s Library is like a big online journal club – they post a paper that somebody has suggested, and then readers are free to annotate and comment on it.

The community seems to comprise a large contingent of Silicon Valley types, so the papers posted tend towards the kind of thing they enjoy, but there’s some maths.

The interface is beautiful, and very well designed – comments are denoted by little dots positioned just off the margins of the paper, and open up into a sidebar that can display images, equations, or even videos. There’s also a traditional comments thread at the bottom for discussion of the paper as a whole.

I’ve had this link saved in my inbox since January of this year because I really wanted to do a proper treatment of it, but never got round to it. Sadly, it seems like not too many people use the site – most of the papers I’ve looked at only have annotations from one person, although they’re almost always of a high quality.

Thanks to Aperiodifan Nathan Day for sending this in. He said, “as an A-level student who wouldn’t otherwise understand most of the papers, I have found it really interesting and useful to have these papers explained and suggested to me each week.”

**Katie: **Back at the start of December, we found this nice story about one of the train lines in Singapore’s underground system on which the trains weren’t behaving themselves, and how they used data analysis to figure out what was going on.

*How the Circle Line rogue train was caught with data, at the data.gov.sg blog*

*Fermat’s Library*

**Katie:** Buzzfeed have collected a set of mathematically interesting GIF animations (just under half of which are from @solvemymaths’ Twitter feed), including images from geometry, fractals, trig and data vis. The title of the article and the post URL seem to disagree on whether they’re deeply soothing or will make you understand maths, but I’m enjoying them either way.

*17 Mathematical GIFs That Are Deeply Soothing, at Buzzfeed*

**CLP:** Étienne Ghys has uploaded an entire book to the arXiv! It’s called *A Singular Mathematical Promenade* and tells a winding tale that starts with one of his colleagues writing down a New Maths Fact on the back of a Paris metro ticket.

Over the course of about 300 pages, Ghys takes us on a tour of a smorgasbord of mathematical concepts, leading up to “the precise description of the chord diagrams that occur in the neighborhood of a singular point of a planar real analytic curve”.

It’s like a mathematical road trip, and it’s beautifully presented. Ghys must be a TeX wizard!

I haven’t had a chance to do any more than skim it so far, but I’m seriously considering printing it out and reading the whole thing – Étienne Ghys is very good at explaining very hard maths.

*A Singular Mathematical Promenade on the arXiv*

**Katie: **To round out the year nicely, mathmo-at-large Eugenia Cheng has yet again come up with the perfect formula for something, and this time as a nice festive twist it’s for the perfect mince pie. The formula involves calculating the ratio of pastry to filling; she’s used a nice bit of calculus to optimise the ratio, and worked out a way to improve your pies, regardless of the angle of slope/ratio of base/opening of your mince pie cases, by trimming one of the layers before assembly.

Sadly, the Telegraph seems to think that people would be more interested in watching a horrifically chopped down version of the video that doesn’t make any sense and loses any semblance of interesting. Luckily though, we’ve found the full version.

Bafflingly, this video seems to be a promo for the new maths gallery at the Science Museum. Don’t ask how, we don’t know.

*The mathematical formula for the perfect mince pie, at The Telegraph*

*How to make the perfect mince pie – using maths, at Barcroft TV*

Aperiodipal numero uno Samuel Hansen’s acclaimed podcast series *Relatively Prime* is back, on a new monthly schedule, with an episode about how PhD student Ibrahim Sharif designed a lottery to award licences to sell cannabis in the state of Washington.

When the stakes are so high (*geddit?! – Ed.*) you have to be really sure that your lottery is fair. That’s where a lot of fun maths comes in.

You can listen to *Lottery Daze* on relprime.com. Sam intends to fund this new incarnation of *Relatively Prime* through Patreon – you can pledge to pay Sam a certain amount (starting at a dollar) for every episode he releases, with perks for paying more such as a postcard from Sam or placing an ad in one of the episodes.

**Listen** to Lottery Daze on relprime.com

**Support** Relatively Prime on Patreon

We’re all (hopefully) aware that a pleasing property of numbers that are divisible by nine is that the sum of their digits is also divisible by nine.

It’s actually more well known that this works with multiples of three, and an even more pleasing fact is that the reason three and nine work is because nine is one less than the number base (10), and anything that’s a factor of this will also work – so, in base 13, this should work for multiples of 12, 6, 4, 3 and 2. Proving this is a bit of fun.

Once when I was thinking about this fact, an interesting secondary question occurred.

When thinking about the multiples of nine, I started to wonder: exactly which multiple of nine will you get when you add the digits together? Obviously, for all the small multiples of nine (less than, say, 81) they all add to $1 \times 9$, but once you reach 99 you have a sum of $18 = 2 \times 9$; and obviously bigger multiples of nine like 248426 have larger sums. So, how does this increase, and is there a pattern?

Since this struck me while I was out at dinner, and not equipped with my usual laptop/MS Excel combo, phone spreadsheets came to the rescue! I launched Google Sheets and created a quick table of values. Putting in a few early multiples of nine, highlighting down and using Auto-Fill created a list of multiples:

I then used the second column to work out the digit sum; my preferred method for this is using the `MID()`

function, which will return a sub-string of a sequence of letters or numbers. It takes three inputs – the thing you want a substring of (in this case, the cell to the left), the starting point, and the number of characters. I whipped up a quick formula which would find the first digit, and add it to the second, third and fourth digits (even though Google Sheets only gives you 1000 rows in a spreadsheet, so I wouldn’t ever need this many).

I also needed to add in the `VALUE()`

function, since `MID()`

returns text strings which it doesn’t always recognise as things you can add together, so I needed to convert them back to numbers before adding. This then gave me the list of digit sums for the multiples of 9. A quick final modification to the formula to divide this result by 9 gave me the information I was after – mostly, a string of 1s with a blip 2 at 99, then back to 1s again. How does this pattern progress?

You can see that the pattern alternates between 1s and 2s, increasing the number of 2s until it’s all 2s, then there’s the one 3 and it returns to the pattern, but this time starting slightly later on; then two 3s, and repeat, and the number of 3s increases until it’s all 3s, and then I assume we’d get a 4 (but my spreadsheet doesn’t have enough rows to actually see this).

This information looks nice in a table, but could probably be better represented in a graph, so I created a graph to display this data and make it more visible. It was around this point that the bill arrived, and my fellow diners started to get annoyed that I was just playing on my phone (and even though I offered to show them the graph, they weren’t interested). Here’s the graph:

You can see this number varying between 1 and 2, and the increasing and decreasing width of the segments until there’s a spike, which then increases in width too and the pattern continues. A more powerful package (and probably a bigger screen) could undoubtedly bring more clarity, but this was an interesting starting point and satisfied my curiosity – there IS a pattern, it IS quite pretty, and numbers are cool.

]]>Puzzlebomb – Issue 60 – December 2016 (printer-friendly version)

The solutions to Issue 59 will be posted around one month from now.

This will be the last regular monthly Puzzlebomb – in future, there will be occasional one-offs but regular editions are taking a break. If you have any ideas for puzzles, please send them in! Previous issues of Puzzlebomb, and their solutions, can be found at Puzzlebomb.co.uk.

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