Chalkdust, a “magazine for the mathematically curious” created by students in the Department of Mathematics at UCL, is publishing the Chalkdust Advent Calendar, with a mathematical curio posted every day by a team of contributors.

Matthew Scroggs, part of the Chalkdust team, has his own mscroggs.co.uk Advent Calendar with a prize competition. Here’s what his site says about that:

Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers is one clue to a murder mystery logic puzzle, revealing the murderer, motive, location, and weapon. Ten randomly selected people who solve all the puzzles and submit their answers to the murder mystery using the form behind the door on the 25th will win prizes!

With The Indisputable Santa Mathematical Advent Calendar, Hannah Fry and Thomas Oléron Evans promise “Christmathsy bits and pieces, one a day, advent calendar style. Assuming we don’t run out of ideas, that is…” Given that their new book, The Indisputable Existence of Santa Claus, offers “a dazzling, magical mathematical tour of the festive season with the most elegant mathematical solutions to your Christmas conundrums”, you’d hope they wouldn’t run out of ideas!

Plus Magazine offers The 2016 Plus Advent Calendar, promising that “each door of this year’s advent calendar conceals a favourite item from our Maths in a minute series, explaining important mathematical concepts in just a few words.”

The Nrich Primary Advent Calendar, with its attractive three houses, offers “twenty-four activities, one for each day in the run-up to Christmas”, saying: “This year, the tasks focus on encouraging mathematical habits of mind: being curious, being thoughtful, being collaborative and being determined”.

The Nrich Secondary Advert Calendar, with its pleasingly non-rectangular layout, offers behind each door “one of our favourite mathematical questions – a mixture of short and longer tasks”, noting that “this year, many of the tasks have been chosen to encourage mathematical creativity”.

]]>Q for my maths tweeps – recommendations wanted for Maths Journals suitable for a bright and engaged Sixth Form student. Suggestions?

— Colin Wright (@ColinTheMathmo) November 24, 2016

It led to a flurry of interesting replies, and here’s some of them.

Run by a group of students based at UCL, Chalkdust comes out four times a year and has editorials, fun features, interesting articles, cartoons and a prize crossword. It’s also available in a print version if you ask nicely and pay for postage.

Based at the University of Cambridge’s Millennium Maths Project, Plus has been a free online maths magazine for a good while – almost 20 years, with articles dating back as far as 1997. Plus has articles on diverse related topics, news, reviews, interviews and puzzles.

MAA’s Mathematical Monthly and MAA’s Math Horizons

The Mathematical Association of America has two regular journals, both requiring MAA membership to read online, but there are discounted student membership rates.

The Mathematical Association is a UK maths teachers’ organisation, and its magazine SYMmetry Plus, part of its Society of Young Mathematicians, is aimed at 10-18 year olds, with issues coming out three times a year. SYM members get a free copy, and it’s also available by subscription, costing around £20 for three issues.

Run by publishers Springer, the Mathematical Intelligencer is a proper journal, and while some articles require a subscription they have a subset of them available as open access.

The Royal Statistical Society runs this monthly stats-focused mag with a subscription or RSS membership, and online articles appearing a year after publication.

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The trademark was registered in 1999, but since the original design of the cube was never patented, it’s long been on shaky ground. The court has ruled that the shape of the cube alone is not enough to protect it from copying, and that a patent would be needed to do so. The implications are that licensed manufacturers of the game could now face more competition from cheaper overseas sellers.

Rubik’s Cube puzzled after losing EU trademark battle, at The Guardian

Rubik’s Cube shape not a trademark, rules top EU court, at BBC News

No fewer than **three** people involved in the production of this new series from PBS Digital have emailed us to tell us about it. In the face of that withering PR onslaught, who am I to ignore it?

*Infinite Series* is a new series of short videos about maths, presented by Cornell PhD student Kelsey Houston-Edwards. The first episode is all about recent advances in sphere-packing (covered here first a couple of years ago, and then didn’t cover this March)

She does wave her hands around a lot, doesn’t she? That was the genius of Numberphile – having some paper in front of the Clever Person gives them something to do with their hands other than shake them in the air like they just don’t care.

**Subscribe on YouTube: **PBS Infinite Series. None of the many emails we received said how often new episodes will appear.

I took the National Numeracy Challenge with my friend David Cushing back in 2014. I remain undecided about how useful it is.

The National Numeracy continues to fight the good fight, promoting numeracy across all age groups and sections of society.

They’ve got a new game out called *Star Dash Studios*, and it’s surprisingly good!

You play a runner on a film set. The majority of the game is, fittingly, running – it’s one of those things where you have to avoid obstacles and pick up coins for as long as possible. Every now and then, you’ll bump into a member of the crew, who asks you to do a job which – *and here’s the very cleverly hidden educational content* – involves some mental maths.

My favourite task is the make-up artist, who asks you to work out when they should start glamming up an actor so they’re ready in time for shooting. It’s literally a single subtraction, but I don’t think I’ll ever tire of seeing a nice actress turn into a groady Orc.

Other jobs involve balancing the camera crane, paying extras, or cutting bits of wood for the carpenter. Unless I’m missing something, the tasks that involve approximate measurement are particularly weak: you might be asked to halve a 3m piece of wood, but the measuring stick is only marked every 1m, meaning you have to guess where 1.5m is. I failed at that quite a few times, and I don’t know why.

The maths is really simple, and scattered fairly sparsely between mindless running sections. But it’s all *useful*, the sort of maths that I wouldn’t think twice about doing, but that a large section of the population wouldn’t be confident with. I don’t know who to recommend *Star Dash Studios* to – the school-age kids I know are all expected to do much more complicated stuff, and I’m not sure how an adult would react to me suggesting they fix their terrible numeracy skills with a game.

It’s really good though. Well done!

**Play: ***Star Dash Studios is available on iOS and Android from National Numeracy.*

*Math Snacks* is a collection of animations and minigames from New Mexico State University’s Learning Games Lab.

They’re much more conventional ‘edutainment’ – games that are really pretty terrible when considered as games, each very clearly designed to teach you a certain maths concept. They’ve done a tonne of research about it, which I haven’t read. If you’re a teacher, maybe it’ll look good to you.

An email in our collective inbox contains the interest-piquing phrase “we feel Aperiodical is one of the coolest maths related website”. It’s from Logic Roots, a company who make educational board games. Titles include *Say Cheese* and *Froggy Fractions* (not *that* Frog Fractions).

If that email wasn’t mail-merged and was in fact heartfelt, thank you Aditi, but sorry – they look much of a muchness to me!

]]>Three married couples want to cross a river in a boat that is capable of holding only two people at a time, with the constraint that no woman can be in the presence of another man unless her (jealous) husband is also present. How should they cross the river with the least amount of rowing?

I’m planning to use this again next week. It’s a nice puzzle, good for exercises in problem-solving, particularly for Pólya’s “introduce suitable notation”. I wondered if there could be a better way to formulate the puzzle – one that isn’t so poorly stated in terms of gender equality and sexuality.

There’s a related, but not identical, problem – but this doesn’t help as it has its own, different issues. Here’s the version of the missionaries and cannibals problem given by Wikipedia:

Three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries). The boat cannot cross the river by itself with no people on board.

Wikipedia says the jealous husbands problem is older, dating back in some forms in Europe to the 800s, with the ‘husbands and wives’ formation coming between the 13th and 15th centuries.

Anyway, absent of a clever revelation I asked Twitter. There are minor spoilers below, so you might want to have a go at the puzzle first if you haven’t seen it before.

First, Christian Lawson-Perfect suggested simply to replace each wife with a heavy, inanimate object that belongs to one person and is coveted by the others. The object must be heavy, or at least bulky, in order that the boat can only hold one person and one object on each journey. I pointed out that whatever those coveting the object want to do with it must be done during a boat ride. In the classic formulation, I suppose each husband fears his wife would be charmed during time alone with another man. Christian suggested unlocked suitcases and Colin Beveridge suggested that these could contain top-secret information. Matthew Arbo pointed out what I had missed: at some point in the solution, we’d require one of these suitcases to row the boat.

Christian suggested replacing the wives with people who know TV spoilers. It’s a nice thought, but I think this would be very complicated to state because of the pairing of characters in the puzzle. We’d need each person who knew spoilers to know different spoilers and be paired with one of those who don’t know spoilers known by the others.

Ian Preston suggested a formulation that I wrote up like this:

Three children, each accompanied by one of their parents, each want to cross a river in a boat that is capable of holding only two people at a time. Children behave very well with each other and with their parent, but misbehave in the presence of other adults when their parent is not present. Everyone must therefore cross the river with the constraint that no child can be in the presence of an adult who is not their parent unless their parent is also present. How should they cross the river with the least amount of rowing?

This is longer than the classic statement and more convoluted. The requirement that children behave together is necessary so that we don’t think they need to stay with the parent at all times, but it’s a big hint that at some point some children are going to be left alone. Even so, there is a further problem. James Grime was confused about whether the children could row the boat, suggesting I replace children and their parents with dogs and their owners. Since at some point we require children to row the boat, perhaps I should say they can do this in the statement – yet another hint.

James Grime also suggested prison wardens and prisoners on a boat to Alcatraz. This is a creative idea, but at some point in the solution we have all the guards at Alcatraz and the prisoners, with the boat, on the shore at San Francisco. Plus, I think this is closer to the missionaries and cannibals than the jealous husbands because of the lack of pairing.

Alison Kiddle suggested a formulation in which we have three mods and three rockers, with each mod having a rocker sibling. People tolerate their own clique or their own sibling, and in a mixed group they won’t kick off if their sibling is present. I think this is a good statement of the problem and I like it quite a lot, though the cultural reference might need updating and its a bit more complicated to explain what will happen if the two groups are allowed to mix.

out of the norm said he’d heard it with Harry, Ron and Hermione with three ogres, or three nuns and three ogres, since overpowering is equivalent to jealousy. Karen Hancock suggested the allergies puzzle at the bottom of this list of interesting river-crossing problems. Nice statements, but I don’t think either is equivalent to the jealous husbands.

Then we came to the suggestion I think I am happiest with. James Sumner made a suggestion that I’ve written up as the following:

Three actors and their three agents want to cross a river in a boat that is capable of holding only two people at a time, with the constraint that no actor can be in the presence of another agent unless their own agent is also present, because each agent is worried their rivals will poach their client. How should they cross the river with the least amount of rowing?

This maintains the jealousy, so is hopefully easy to understand and should minimise the need for additional explanation. As James pointed out, we might wonder why on earth they’re crossing a river in a boat made for two, but I think that’s a minor quibble.

]]>Christian’s put together this fun applet for exploring the Zeta function – you can move your pointer around to reveal the value of $\zeta$ at each point in the complex plane.

The hue (colour) revealed is the argument of the value, and the lightness (bright to dark) represents the magnitude. There’s a blog post over at Gandhi Viswanathan’s Blog explaining how it works.

The resulting plot has contour lines showing how the function behaves.

]]>100 years ago (on 14^{th} November) was born a Frenchman called Roger Apéry. He died in 1994, is buried in Paris, and upon his tombstone is the cryptic inscription:

\[ 1 + \frac{1}{8} + \frac{1}{27} +\frac{1}{64} + \cdots \neq \frac{p}{q} \]

The centenary of Roger Apéry’s birth is an appropriate time to unpack something of this mathematical story.

Let’s start with something relatively simple:

What happens if you add up all the unit fractions?

\[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots \]

Answer? Not a lot.

Or, rather more accurately – a lot. This is the tolerably well-known harmonic series, and it is relatively easily demonstrated that it *diverges*, or as you might like to think of it, has an infinitely large answer.

Now do the same thing with the unit fractions with *squares* in the denominator:

\[ \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \cdots \]

It may be something of a surprise, when first encountered, that this time the sum *converges* – and has a well-defined answer, which is in fact: $\frac{\pi^2}{6}$. Proving this is known as the *Basel Problem*, and was first demonstrated by Leonhard Euler in the 18^{th} century. It also turns out to be the reciprocal of the answer to the question:

*“What is the probability that two positive integers selected at random are relatively prime?”*

Moving on one more step, what happens if you use the *cubes* instead of the *squares.*

\[ \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \frac{1}{6^3} + \cdots \]

This time nothing is very obvious. It is tempting to believe immediately that this too will converge, since the squares version did (and in fact, this is correct) but it is certainly not clear *what* it might converge to. Mathematicians made some progress (including Euler again, who calculated the first 16 decimal digits of the sum), arriving at the conclusion that

\[ \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \frac{1}{6^3} + \cdots \simeq 1.2020569\ldots \]

But what is this number? Will the decimals repeat or end (i.e. – is it a rational number)?

Nobody knew – until a French mathematician named **Roger Apéry** made an announcement in 1978, followed by some clarifications in 1979 (for a surprised mathematical community). He had proved that this number is indeed *irrational*. As a result, it is now known as **Apéry’s constant**.

What is more, this sort of result has proved extremely difficult to produce. The examples we have been looking at are all particular examples of the Riemann Zeta Function:

\[ \zeta(s) = \sum_{n=0}^{\infty} \frac{1}{n^s} \]

The harmonic series is $\zeta(1)$, the sum of the reciprocals of the squares is $\zeta(2)$, and Apéry’s sum is $\zeta(3)$. To put these in context, it is currently not known specifically whether any other particular $\zeta(n)$, for $n$ odd, is irrational. The best we’ve got is from Wadim Zudilin, in 2001, who showed that at least one of $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ must be irrational, and Tanguy Rivoal, in 2000, who showed that infinitely many of the $\zeta(2k+1)$ must be irrational.

It is also not known whether Apéry’s constant is transcendental.

But it *is *the reciprocal of the answer to the question:

*“What is the probability that any three positive integers selected at random are relatively prime?”*

Hooray!

Some Classical Maths Blog – a review of Apery’s proof and some pdfs of the original

Rivoal, Tanguy (2000), “La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs”

Zudilin, Wadim (2001), “One of the numbers* **ζ(5)*,* **ζ(7)*,* **ζ(9)*,* **ζ(11)** *is irrational”

Today is the 100th anniversary of Roger Apéry’s birth, and we’re The Aperiodical, so we just had to make a big deal of it.

So, for all of today, we’re The Apéryodical. Throughout today we’ve got a few posts about Apéry and the thing he’s most closely associated with: the Riemann zeta function.

For now, here’s a really big ζ. You’ll need it later.

ζ

]]>The **Alan Turing** centenary shows no signs of abating.

First of all, there’s a marvellous new art installation under Paddington Bridge in London, in memory of Turing. There’s also a theatre piece called Breaking the Code, showing at Manchester’s Royal Exchange Theatre until 19th November.

Secondly, work continues to introduce legislation in the UK pardoning all gay men who were convicted of crimes related to homosexuality, in the same way Alan was a few years ago. Ministers said they were ‘committed’ to getting the law passed, but in an emotional session the bill was “talked out” by minister Sam Gyimah, meaning it wasn’t voted on.

The **London Mathematical Society** (LMS) has been honoured this autumn by receiving the first Royal Society Athena Prize to recognise its advancement of diversity in science, technology, engineering and mathematics (STEM) within the mathematical community. The prize was awarded in a ceremony at the Royal Society’s annual diversity conference on 31 October.

Mathematician, author and friend of the site **Rob Eastaway** has received the 2016 Christopher Zeeman medal, awarded to recognise and acknowledge the contributions of mathematicians involved in promoting mathematics to the public and engaging with the public in mathematics in the UK.

There will be an award lecture taking place on 22 March 2017, and details will be announced in Mathematics Today and the LMS Newsletter.

IMA website article on the award

Rob Eastaway’s citation (PDF)