The base of the development is a tool to “semantically enrich” mathematical expressions, inferring information about the meanings of individual elements. This is not an easy job, and has required the development of thousands of pattern-matching rules to identify different notational conventions.

This semantic enrichment is passed directly onto a tool which works with a screen reader to read mathematical expressions aloud; the exploration tool lets you work through an expression piece-by-piece, using the keyboard to navigate.

The auto-collapse extension also makes MathJax behave better in responsive designs where the available width for rendered expressions can change, by intelligently picking break points and collapsing sub-expressions so they fit on smaller screens.

We’ve enabled the accessibility extensions on our site – right-click on the expression below and play with the “collapsible expressions” and “explorer” menus.

\[ c_p\rho \int_{x-\Delta x}^{x+\Delta x} [u(\xi,t+\Delta t)-u(\xi,t-\Delta t)]\, d\xi = c_p\rho\int_{t-\Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x} \frac{\partial u}{\partial\tau}\,d\xi \, d\tau \]

In the future, the accessibility extensions will be included in the standard MathJax configuration, but for now you need to manually load them.

To load the accessibility extensions on your own site, you just need to add a line to your MathJax config script. Here’s a basic configuration:

<script type="text/x-mathjax-config"> MathJax.Ajax.config.path["Contrib"] = "https://cdn.mathjax.org/mathjax/contrib"; MathJax.Hub.Config({ extensions: ["[Contrib]/a11y/accessibility-menu.js"], tex2jax: { inlineMath: [ ['$','$'], ["\\(","\\)"] ], processEscapes: true } }); </script> <script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML"></script>

When you’re reading a site that has loaded MathJax without the accessibility extensions, you can use this bookmarklet (drag the link to your bookmarks) to add in the extensions.

**More information**, including detailed instructions on how to use the extensions and demos of the new features, is at the MathJax blog.

Puzzlebomb is a monthly puzzle compendium. Issue 54 of Puzzlebomb, for June 2016, can be found here:

Puzzlebomb – Issue 54 – June 2016

The solutions to Issue 54 will be posted at the same time as Issue 55.

Previous issues of Puzzlebomb, and their solutions, can be found at Puzzlebomb.co.uk.

]]>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.

]]>- Prof. Alice Rogers, Emeritus Professor of Mathematics, King’s College, London, appointed OBE for services to Mathematics Education and Higher Education.
- John Sidwell, volunteer, HMP Hewell appointed MBE for services to Prisoners through One to One Maths.
- Danielle George, vice-dean for teaching and learning, Faculty of Engineering and Physical Sciences, University of Manchester, appointed MBE for services to engineering through public engagement.
- Anthony Finkelstein, professor of software systems engineering, University College London and the Alan Turing Institute, for services to computer science and engineering.
- Economist Angus Deaton, professor, Princeton University, Nobel laureate, for services to research in economics and international affairs.
- Prof. Alan Thorpe, lately Director-General of the European Centre for Medium Range Weather Forecasts, appointed OBE for services to environmental science and research (thanks to Philip Browne on Twitter).
- Prof. Nalini Joshi was made an Officer of the Order of Australia (AO); the citation is more involved than the UK ones and reads “for distinguished service to mathematical science and tertiary education as an academic, author and researcher, to professional societies, and as a role model and mentor of young mathematicians” (added in an update 16/06/16).

It’s also worth mentioning the new batch of Regius professorships, 12 posts created at universities around the UK to celebrate the Queen’s 90th birthday: Oxford University has been given a professorship in maths, but no appointment has been made yet.

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

]]>“Principia Mathematica”, published in three volumes in 1910, 1912 and 1913, was a major work by mathematician and philosopher Bertrand Russell, with help from Alfred North Whitehead. The book contains a proof, starting from very basic axioms, that 1+1=2 – which takes over **360 pages**! It might seem excessive, but they work from only the most basic assumptions, and have to define firstly what they mean by ‘1’, ‘2’, ‘plus’, and ‘equals’. It’s all done in formal logic, and must surely be one of the longest proofs relative to the length and complexity of the statement it’s proving.

**PROOF SIZE: 0.1148 tennis courts**

- Principia Mathematica, on Wikipedia
- Russell and Whitehead, on The Story of Mathematics
- Restatement of the proof in more modern notation, at MetaMath Proof Explorer
- The proof starts on page 362 of Principia Mathematica Vol. 1

Famously one of the first proofs that required so many fiddly cases to check that its authors resorted to a computer, the Four Colour Theorem simply states that any diagram (or equivalently, any graph) drawn on a flat piece of paper can be coloured using at most four colours, so that any two adjacent parts are different colours. The original problem was stated in 1852, and while purported proofs were published in the 1880s, they were later found to be incorrect.

The first real proof, given by Appel and Haken in 1977, roughly consisted of finding a set of 1,936 minimal reducible structures that any diagram/graph that might not be four-colourable could be made up from, then checking each one by computer. Checking the maps one by one took over **1000 computer hours**, and accompanied a hand-checked component of the proof on **400 pages of microfiche**.

A shorter proof, involving a mere 633 reducible structures, was produced in 1996, and that proof was formalised in 2005 using the COQ computer proof assistant, which means it’s less reliant on cases checked by computer.

**PROOF SIZE: 1.8 times the area of an olympic swimming pool plus 370.4 watches of the film Avatar**

- Four Colour Theorem, on Wikipedia
- The Four Colour Theorem, at NRICH
- Formal Proof – The Four-Color Theorem, an article explaining the shorter COQ proof by Georges Gonthier in the Notices of the AMS (PDF)

‘The Enormous Theorem’ is always given as an alternative name for the classification of finite simple groups, but of course nobody actually ever calls it this in their work.

The classification of the finite simple groups — otherwise known as ‘CFSG’ or simply **The Enormous Theorem** — is a bit different to the other massive proofs listed here. It’s not a single work from one person or team, but rather a joint effort among dozens of mathematicians through the (nineteen-) eighties and nineties, a patchwork of results published in separate articles which together constitute the overall theorem.

A *group* is roughly speaking the set of interlocking symmetries of some concrete or abstract object, and the simple ones are those that are not ‘made by’ mashing two smaller symmetry groups together. The CFSG states that any finite simple group must be either a member of one of a set of precisely-defined infinite families of groups, or one of 26 specific outliers: the sporadic simple groups.

Wikipedia reckons that the aggregated length of the papers contributing to the proof is around **10,000 pages**. Work on a consolidated ‘second-generation’ proof, led by Daniel Gorenstein, is ongoing and is expected to result in an eleven-volume proof a mere 5,000 pages long. In fairness to the competition, it should be pointed out that these page counts are for the full articles/books, including all exposition as well as the bare proofs.

**PROOF SIZE: 1.4 basketball courts**

- Classification of finite simple groups, on Wikipedia
- An enormous theorem: the classification of finite simple groups, by Richard Elwes at Plus Magazine
- Rewriting the enormous theorem, by Rachel Thomas at Plus Magazine
- The Status of the Classification of the Finite Simple Groups, by Michael Aschbacher (PDF)

Kepler Conjecture/FLYSPECK

The Kepler Conjecture, originally posited in the 17th century by Johannes Kepler, relates to the density of spheres packed in 3D space. Kepler conjectured that the ‘cubic close packing’ (the one where you put a hexagonal grid of spheres on a flat surface and then stack another one on top, but offset so the balls are over the gaps) is the most efficient way to pack spheres in 3D space – the one with the least empty space left in between.

While it was long suspected to be true, nobody managed to formally prove it until Thomas Hales in 1998. Hales’ proof involved around **300 pages of notes** and **3 gigabytes of computer programs, data and results**. It was a ‘proof by exhaustion’ which involved checking many individual cases. Referees on the proof said they were ‘99% certain’ the proof was correct, and hence that this was the most efficient 3D packing.

But that wasn’t enough for Thomas Hales – as we covered here when it was completed, the FLYSPECK project (named as it is a Formal Proof of the Kepler conjecture, and ‘flyspeck’ is a word that contains all of those letters in that order, but it’s not as good as the word ‘flapjack’) was a further project undertaken to formalise and check the previous proof. It took from its start in 2003 until September 2014 to complete, and used proof assistants Isabelle and HOL Light.

**PROOF SIZE: 1.43 downloads of the film Titanic in HD, plus about 0.94 Harry Potter and the Prisoner of Azkabans**

- The Flyspeck project is complete: we know how to stack balls!
- Git Repo for FLYSPECK
- The Kepler Conjecture, on Wikipedia

Another recent bit of maths that’s made headlines by having a massive proof was the Erdős Discrepancy problem – back in February 2014, a proof using an SAT solver by Boris Konev and Alexei Lisitsa of the University of Liverpool hit the headlines because it was ‘the size of Wikipedia’ (around 13 gigabytes).

The problem asks whether it’s possible to come up with an infinite sequence of +1s and -1s and a ‘target’ number in such a way that you can never get past the target by adding together regularly-spaced terms from the sequence. (Going lower than minus-the-target-number also counts, in case you thought you had trumped the proof with your clever sequence of just -1s.) James Grime has explained the problem in a video with snakes and a cliff. The proof showed that in fact you can’t always make the required sequence.

Luckily, Terence Tao came to the rescue in September 2015, with a smaller hand-crafted proof developed in collaboration with the Polymath project.

**PROOF SIZE: around 3,250 holiday snaps taken on a 10 megapixel camera**

- A SAT Attack on the Erdős Discrepancy Conjecture by Boris Konev and Alexei Lisitsa
- Erdős’s discrepancy problem now less of a problem
- New Wikipedia sized proof explained with a puzzle – James Grime on YouTube
- Terence Tao has solved the Erdős discrepancy problem!

Claiming to be the ‘largest proof ever’, the Boolean Pythagorean Triples theorem relates to the question of whether it’s possible to split all numbers into two groups, neither of which contains a complete Pythagorean triple. For example, 3, 4 and 5 form a triple, and to find a valid split they would have to not all be in the same half – but then 5 couldn’t also be in the same half as 12 and 13, and so on.

It’s much better explained by Evelyn Lamb in her post in Nature, but a team of researchers have shown that not only is it not possible to do this, it’s not even possible to split the numbers 1 to 8000 in this way without getting stuck. It might not sound like ground-breaking mathematical knowledge we need right now, but it ties in to Ramsey Theory and other combinatorial questions. Proving it took 2 days for a computer running 800 parallel processors, and generated **200 terabytes of data**.

**PROOF SIZE: Amount of data generated by CERN every 2.92 days**

- Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer by Marijn J. H. Heule, Oliver Kullmann and Victor W. Marek
- Two-hundred Terabyte maths proof is largest ever, by Evelyn Lamb at Nature
- Boolean Pythagorean Triples theorem on Wikipedia

More information at Wired.

*via math-fun.*

I thought it might be interesting (to me, at least) to list the types of assessment I’ve been involved in marking in the 2015/16 academic year.

These are not all of my invention (i.e. some are things I made up in teaching I ran, others are pieces I delivered as part of some else’s design). In no particular order (numbers are approximate):

- 120 short individual tests (four tests times thirty students) — a series of short, unconnected questions;
- 16 multiple-choice tests;
- 32 group activities (four activities times eight groups) — students had to solve a slightly open-ended question as a group and I marked them on the written description of their solution and how well they had communicated and worked as a group during the task;
- 266 short individual courseworks — well, one was not particularly short, but they were all a series of short, unconnected questions;
- 30 in-depth individual courseworks — this had a series of connected and increasingly open-ended questions to investigate a topic;
- 6 group essays — students worked in groups to research history of maths topics and wrote their findings as a short (500 word) essay plus a brief (100 words) account of their estimation of the reliability of the sources they used; they did this formatively weekly for half a term before handing one in summatively;
- 25 individual history of maths essays — topic of student’s choice (with agreement);
- 15 group presentations accompanied by two-page handouts — this was to describe the findings of an open-ended group investigation;
- 25 group project plans and minutes of 75 group meetings — for the above investigation;
- 99 self- and peer- reflections on contribution to group work — for the same;
- 36 reflective personal statements discussing career plans, skills relevant to those and ethical issues;
- 10 individual presentations — interim reports on final year projects;
- 6 dissertations — final reports of year-long final year projects, each with a corresponding viva;
- 4 group presentations — to report on findings of a semester-long, open-ended group investigation;
- 16 group posters — to report on the above investigations;
- 1 group report — report of the same;
- one quarter of the questions on 200 group-marked exam scripts (two exams).

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.*

Apparently those symbols winding their way around the garden are “plant growth algorithms”, whatever those are.

There’s also a golden-ratio-thingy water feature, of course.

You can thank Winton Capital, sponsors of all sorts of worthy maths projects, for this bit of mathsy art.

]]>**Theorem: **every 5-connected non-planar graph contains a subdivision of $K_5$.

The above statement, conjectured independently by Alexander Kelmans and Paul Seymour in the 70s, is very easy to say. And the video below, starring Dawei He, Yan Wang, and Xingxing Yu, makes it look very easy to prove:

It’s like they got Wes Anderson to film an academic PR video. In the normally uninspiring world of maths press releases, it’s quite refreshing. And the written press release is pretty snappy, too. Let’s not make this a *thing*, though.

However, as “one of those maths whizzes out there”, I wanted to know a bit more about the work than a two-minute video can impart, so I’ve looked up the working-out. There’s a pair of papers building up the proof: “The Kelmans-Seymour conjecture I: special separations”, and “The Kelmans-Seymour conjecture II: 2-vertices in $K_4^{-}$”. They’re decidedly *not* as aesthetically pleasing as the video: here’s an excerpt from paper 2:

Maybe a publisher will *add value* to the paper in the form of some line breaks.

Anyway, I thought I’d already been told this theorem as a fact, so congratulations to Dawei He, Yan Wang, and Xingxing Yu for finishing off such a lovely theorem. That’s assuming the proof works: so far, there are just the two preprints on the arXiv and a press release from Georgia Tech. I haven’t been able to find anything from other experts in the field to add credibility to the claim of a proof.

40-Year Math Mystery and Four Generations of Figuring – press release from Georgia Institute of Technology.

**Read the papers****: ** The Kelmans-Seymour conjecture I: special separations, and The Kelmans-Seymour conjecture II: 2-vertices in $K_4^{-}$. There isn’t much there for the tourist, though.

Finally, I can’t restrain myself from pointing out that paper 1 cites a paper by my maybe-relative, Hazel Perfect, “Applications of Menger’s graph theorem”. So cool!

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