This year pickings are slim – quite a few mathematical Fringe regulars are having the year off – but there are a couple of shows which have maths as a definite theme:

Mathematician, maths teacher and UK FameLab champion Kyle Evans is taking his maths/folk/comedy music show to Edinburgh, and will be performing his trademark mathematical songs.

Self-proclaimed ‘Mathemagician’ Jason Davidson performs another of his family maths magic shows, suitable for all ages.

The Edinburgh Skeptics have put on a whole month of talks on different topics, although sadly only one of them has a maths theme (and that’s because it’s literally me, doing my Mathematical Life Hacks show on Thursday 24th August).

Thu 24th 7:50 @B_Labyrinth it’s @stecks who’ll make all our lives better with her Mathematical Life Hacks pic.twitter.com/dchtxJXBYO

— Edinburgh Skeptics (@edskeptics) July 7, 2017

I’ve also found a collection of things which aren’t mathematical, but definitely involve science in some way and you may enjoy them. Here’s their programme blurbs and ticket links – enjoy!

- Katy Brand: I Could’ve Been An Astronaut (comedy)

Katy Brand explores her amateur and somewhat improbable love of astronomy and her crapness at maths in this follow up to last year’s hit show I Was a Teenage Christian. - Kevin Quantum: Anti-Gravity (magic)

Impossible illusions and levitations brought to you by the magician-scientist hybrid tutored by Penn & Teller. Having spent half of his adult life studying physics and half studying magic, Magic Circle member Kevin explores the exotic space where science and magic meet. - The Amazing Bubble Man (magic)

Louis explores the breathtaking dynamics of bubbles, combining comedy and artistry with audience participation and enough spellbinding bubble tricks to keep everyone mesmerised. - Baba Brinkman’s Rap Guide to Consciousness (rap, comedy)

Fringe First winner and “peer-reviewed rapper” Baba Brinkman (Rap Guide to Evolution, Rap Guide to Religion) explores the scientific study of consciousness in his latest hip-hop comedy. Baba’s brain consists of roughly 90 billion neurons with trillions of connections, and none of them has any clue that he exists. And yet those cells come together to produce a steady stream of ill rhymes, laughs, and mind-blowing scientific findings. Come and find out how. - Alice and the Black Hole Blues (theatre)

This world premiere devised theater piece imagines that Lewis Carroll’s Alice in Wonderland falls through a black hole and meets five visionaries who challenge societal assumptions about women and science. Hypatia, Marie Curie, Vera Rubin, Rosalind Franklin and Lise Meitner prove that even through you cannot see it, it does exist. - Edison (theatre)

The life, career, and brutally stolen dreams of mastermind Nikola Tesla culminate in this genre-bent play, sponsored by one of the biggest a-holes in history, Thomas Alva Edison. Featuring Tesla: troubled genius and brilliant inventor, and Edison: cunning businessman and crook, Edison brings to light this true story based on the life of the original electrical wizard. - Ensonglopedia of Science (music, comedy)

A song about science for every letter of the alphabet. Expect atoms, big bangs, cells, DNA… Expect the unexpected. And expect it to rhyme. (**Katie says**: full disclosure, my mum once gave me a flyer for this guy because he was doing a show in Heywood and she thought I might like it. I never went.) - Doctor Google Will See You Now (talk)

Your supermarket knows when you’re pregnant; Google knows what medical conditions you have; Facebook could help your doctor diagnose you. What if Google sent a record of your search terms to your doctor to notify them of likely health conditions? What if Facebook sent an alert to midwives when pregnant ladies posted photos of them drinking alcohol or smoking? Join**Mhairi Aitken**(University of Edinburgh) as she discusses how this information could be used and have your own say on what should remain science fiction and what should become reality. - Dr Data: The Answer to Cancer (talk)

Data is everywhere. From diagnosis to treatment to recovery, a patient’s cancer journey leaves a data trail. What if the answers we seek about cancer are hidden in there? It could help us refine diagnosis, select treatments and improve the patient experience. - Measuring Humanity (talk)

Someone smart once said ‘if you can’t measure it, it doesn’t exist’, so**Marisa de Andrade**(The University of Edinburgh) is out to measure humanity! Join her as she pushes academic boundaries on her mission to measure health and inequalities through creativity and connectivity. Is there such a thing as hard-to-reach communities? Is there only evidence where there’s money to be made? How do you measure soft touchy-feeling things like compassion and relationships – and should researchers be doing this anyway? Can you prove that acting helps you stop smoking, or that 3D virtual reality landscapes beat depression? - Physics vs Psychology: Which is the Hard Science? (talk)

The year is… not important; this is fiction. But Susan, the last secondary school student on Earth, is deciding what to study at university. Physicist**Helen Cammack**and psychologist**Kate Cross**(University of St Andrews) are here to do battle for Susan’s soul. Should she pursue physics, the study of the universe? Or psychology, the science of the mind? Is there a hard option and an easy option? Susan needs you to come along, ask hard questions of our two scientists and help her choose her path… - You Don’t Matter (talk)

You’re completely insignificant; just one out of 7 billion people on planet Earth, orbiting just one star out of 100 billion stars in our Milky Way galaxy, just one galaxy out of 100 billion galaxies in our observable universe. And our universe is likely just one universe in an almost infinite number of universes where everything can and will happen. No longer should you waste time worrying about life’s indecisions, because in one universe, somewhere out there, you’ll have made the right choice.**Catherine Heymans**and**Joe Zuntz**(University of Edinburgh) will be your guides. - The Principle of Uncertainty (theatre)

Dr Laura Bailey knows how to explain to everyone counter-intuitive double slits experiments or the paradox of a cat in a box which is dead and alive at the same time. She believes the equations that tell her unequivocally of the existence of a myriad of different universes, but when faced with a crucial event, science falls short of showing her a way out. - 1 Woman, a High-Flyer and a Flat Bottom: Samantha Baines (comedy)

Award-winning comedian**Samantha Baines**(The Crown, Sunny D, BBC Radio 4) returns to Edinburgh after a smash-hit, sell-out run in 2016. This year she’s exploring the lost women of science. Expect facts, puns and an ear trumpet attached to a whiskey bottle. - Double Dome Nights: Dark Side of the Moon – the Full Dome Experience

Double Dome Nights is showing a combination of two films from Dynamic Earth’s spectacular bank of out of this world 360-degree immersive experiences. Inspired by the music of Pink Floyd, this dome spectacular features the 1973 ‘Dark Side of the Moon’ album in explosive surround sound. - Scottish Superwomen of Science – Minerva Scientifica (music)

2016 Fringe award-winner Frances M Lynch returns, celebrating breakthroughs by Scottish women scientists in genetics, geology, computing, engineering, astronomy, marine biology, mobile phones, etc. Musical drama featuring new sounds fashioned from traditional and classical songs by Scottish women composers, some co-created with women scientists for Minerva Scientifica. - Wanna Dance With Somebody! Or, A Guide To Managing Social Anxiety Using Theoretical Physics (theatre)

Josh is good at dancing, but not at people. Is the hokey cokey really what it’s all about? Josh doesn’t know, but he’s damn well going to find out. A new show about social anxiety, about coping mechanisms, about big ideas and how they impact on small things. About taking chances. And about dancing. Mixing storytelling and physical comedy, Running Dog Theatre create a show that is part physics lecture, part dance lesson and part school disco.

There are two main routes to becoming an astronaut – you’ll either start by being a test pilot, in which case you’ll train to fly experimental aeroplanes; in that case, before you start, you’ll need a degree in a science-related subject (maths, physics, biology or engineering) plus three years experience building on that (usually in research); so, the equivalent of a PhD. You’ll also need to be between 64 and 76 inches tall (I am in fact 64.9 inches, so it’s good to know I’d have had the option). The other way to become an astronaut is as a payload specialist, which means you’re a scientist going up to run or oversee a scientific experiment – again, you’ll need to be a research scientist and hence a pretty capable mathematician.

It’s obvious that there’s plenty of maths involved in space exploration – from calculating the physics of trajectories and launch escape velocities, to fuel quantities, and then once in space, maintaining all the systems needed to move things, sustain life support and generally keep things going. You might argue that most of the mathematics here can be done by ground crews, and to a large extent that’s true – but in space, and especially in emergency situations, the astronauts themselves sometimes have to pitch in and crack on with the sums.

One famous example of this was in 1997, when a team aboard the Russian space station Mir (pictured above – ‘*mir*’ in Russian means ‘*peace*‘, or with a capital M ‘*world*‘, but it’s also used to mean a community or village, aww) found themselves in exactly such an emergency situation – while testing a new manual docking system for one of the incoming Progress modules, used to deliver goods and equipment to the station, there was an accident and the Progress module collided with part of the station, causing an air leak and damage to the Mir solar panels.

After scrambling to seal off the decompressed section of the station, and restoring a balance of air pressure, the crew found themselves in a bit of a spin – quite literally. The station had been knocked out of its usual stable orientation by the collision, and the gyrodynes, or momentum storage devices, usually used to keep the station correctly oriented in orbit, were unable to keep it pointing in the right direction. This was a bit of a problem – the astronauts themselves didn’t mind too much, since they’re in zero gravity and don’t know which way is up anyway – but the station’s solar panel array, used to power all the systems on board, was no longer pointing at the sun.

The station lost power, and the backup batteries were soon exhausted. This meant the gyrodynes also powered down, leaving the station entirely without power, and rotating even more without the stabilising effect of the momentum storage devices.

This left the crew on board in a bit of a bind – without power, they could only maintain sporadic contact with the ground. Under normal circumstances, communication was only possible when the station’s orbit took it within range of one of the base stations on earth, but now they could only speak for a short time, and then only every few hours. The crew, which included two Russians and a British NASA astronaut called Michael Foale, needed to get on with some maths.

Foale realised they could use the station’s Soyuz module (Russian for ‘union’), which was the Russian equivalent of the space shuttle – it was used to fly up to the station, and remained docked to the back of Mir in case they needed to evacuate. While the station was powered down, Soyuz still had onboard thrusters and could in theory be used to manoeuvre the station so it was pointing at the sun again. While the ground crew tried to assist with calculations, they didn’t have enough information or telemetry to be able to send any useful suggestions.

Yes, they apparently carried a scale model of Mir on the space station Mir. Which hopefully itself had a tiny tiny model of Mir inside that, and so on.

Setting up a crude mock-up of the scenario – mounting a torch on the ceiling, shining down on a scale model of Mir held over a table representing the surface of the Earth, Foale and his colleagues modelled the motion of the spinning station. By holding his thumb up to the window, he could use the speed at which stars passed behind it to calculate roughly the speed and direction they were spinning in. They also had to work out the orientation of the Soyuz relative to the rest of the station, and what direction the thrusters would move them in when fired.

The problem was made more complicated by the fact that the actual moment of inertia of the station will depend on the distribution of mass within the different sections, which it’s impossible to know as things might have been moved around – they could only base their calculations on the positions of fixed hardware. On top of this, the point the Soyuz was mounted at meant they could only really use it to rotate in two of the three axes, as it wouldn’t have any control around the axis it was pointing in the direction of.

After a lot of scribbling and working out, and communicating with the ground crew when possible, Foale used Euler’s equations, some rotation matrices, and a few simultaneous equations to calculate that a 3-second burn on one side of the Soyuz would get them stabilised and stop them spinning. They tried it, and it worked! Maths in space success! Especially impressive to do maths while the room you’re in is spinning (as anyone who’s got a wheely chair in their office and has tried to do maths while rotating will know).

It did take a long time for them to recover fully from the incident – as well as recharging the station’s batteries, and getting all the systems back online (including the gyrodynes), they also had to fix the puncture – a hole only 3cm across – and repressurise the damaged section of the station, which took months of work. It included a difficult internal ‘space walk’ – unlike previous crews, whose space walks all took place on the outside of the station, they had to squeeze into the damaged part of the station wearing a full space suit in order to locate and patch the hole.

While waiting for the depressurised section to be fixed, Foale wondered if it’d be possible to do the mathematical calculations more rigorously – if anything similar happened again, it’d be useful maths to have. He wanted to use his favourite maths software, Mathematica, to model the spinning station. However, his laptop and Mathematica CD were both in the Spektr module – the part of the station that had been damaged. He also couldn’t get to any of his personal effects, clothes or toothpaste, but this was more important. So, he got on the phone. The space-phone.

Wolfram Research, who make Mathematica, were more than happy to oblige – he literally called Wolfram Tech Support from space, and asked them to send him another copy. They put a new hard drive containing all the right software onto the next Progress module being sent to the station, and within a couple of weeks he was back in business. (It’s a good job really, as when they did retrieve his belongings from Spektr, his laptop had not survived being exposed to the cold vacuum of space. If anyone’s wondering, it was an IBM ThinkPad.) Michael’s Mathematica notebook, detailing the calculations he did, and including 3D animated models of Mir, is available online for anyone who’s got Mathematica to play with.

Mir Spacecraft: Worst collision in the history of space flight – BBC Witness

Astronaut Places a Customer Service Call to Wolfram Research from Space Station Mir, on the Wolfram Blog

Michael Foale, on Wikipedia

]]>Welcome to the **145th Carnival of Mathematics**, hosted here at The Aperiodical.

If you’re not familiar with the Carnival of Mathematics, it’s a monthly blog post, hosted on some kind volunteer’s maths blog, rounding up their favourite mathematical blog posts (and submissions they’ve received through our form) from the past month, ish. If you think you’d like to host one on your blog, simply drop an email to katie@aperiodical.com and we can find an upcoming month you can do. On to the Carnival!

As is traditional, I’ll start with some facts about our Carnival number, 145.

145 is a pentagonal number, and a centred square number. This diagram of 145 things arranged in a pentagon was generated using Andrew’s ridiculous Polygonal Number Calculator, on Matt Parker’s Things to Make and Do in the Fourth Dimension website.

It’s also a factorion, which is a flipping superb property for a number to have – it’s equal to the sum of the factorials of its digits, that is to say

$145 = 1! + 4! +5!$

The only numbers (at all, out of all the numbers) which have this property are $1, 2, 145$ and $40585$ (OEIS A014080). How cool!

Anyway, you’re not here to learn about interesting properties of numbers! You’re here for maths blog posts, so here’s one about a number with interesting properties: Sam Shah, over at **Continuous Everywhere But Differentiable Nowhere**, has posted about Graham’s number and how to teach 9th-12th graders (KS3/4) about it. Relatedly, a fun post from Tim Urban over at Wait But Why takes us up to Graham’s number from 1,000,000, with some cool facts about the quantities you meet along the way.

Interesting properties of numbers continue, with a post from Dave Richeson on **Division By Zero**, comprising Seventeen facts about the number seventeen (n of them will shock you!). James Propp over at **Mathematical Enchantments** has some more ways to convince people that $0.999\ldots = 1$, and why our innate understanding of how numbers work might be to blame for any doubt.

If you’d like to remind yourself it’s possible to find mathematics amazing and beautiful, even if you’re not a full-time card-carrying professional mathematician, read this **New York Times** article by psychiatrist Richard A. Friedman describing his fascination with the beauty of mathematics. Speaking of beautiful mathematics, Alex Bellos and Edmund Harriss’s new mathematical colouring book gets a feature post over at **The Guardian, **with plenty of pictures.

The wonderful Evelyn Lamb, on her Scientific American column **Roots of Unity** has blogged about her discovery of a second (yes, there’s two) female mathematician who has a street named after her in Paris, Marie-Louise Dubreil-Jacotin. She’s also discovered a gorgeous octagonal tiling, which you’ve almost certainly seen before but never noticed.

**Bow Tie Teacher** has documented an effort to find correctly-named mathematical shapes in home decor shops, and the interesting names some shops have come up with for the objects – since when is this a hexagon (right)‽

Over at **The Conversation**, USyd’s Stephen Woodcock writes about some statistical paradoxes and fallacies, accompanied by some truly magnificent cumulonumbers. Karl Kruszelnicki at Australia’s **ABC Science** blog recalls Joseph Keller and his wonderfully simple real-world maths applications, including why ponytails swing from side to side if you’re running forwards.

The **Cambridge Mathematics** blog has a nice post about teaching sequences using coding, and **Colin Wright** has written up a nice mathematical game he discovered at an event recently. Meanwhile, Brian at **Bit Player** has been messing around with factorials and squares and Tony at **Tony’s Maths blog** has found something in a cupboard.

Now on to some videos! While Matt Parker’s Standupmaths YouTube channel has been mysteriously quiet this month, some new videos have gone up on **Numberphile**, including one featuring Matt on The 10,958 Problem (the challenge is to write an expression for 10,958 using the digits 1-9 in order and some operations) along with a solution video, and one featuring Twin Prime conjecture maestro Dr James Maynard, describing some recent developments on the problem.

I’ve managed to continue **my YouTube efforts** with an Easter special – on how to construct an egg shape using arcs of circles. Arguments about what shape an egg actually is, or banter about how many times I’ve said ‘compass’ instead of ‘pair of compasses’ during the course of the video, in the comments, please.

And finally, enjoy this wonderful Twitter thread from geophysicist and science writer **Mika McKinnon** on the geometry and engineering that went into some of the outfits at this month’s Met Gala:

The engineering casually on display at #MetGala never fails to impress me. I do nearly every textile craft . That’s mindblowingly hard: https://t.co/pGzmp6u2h9

— Mika McKinnon (@mikamckinnon) May 2, 2017

That’s it for this month! Next month’s Carnival will be hosted by Peter at Boole’s Rings, and you can submit your favourite blog posts/videos/content from the month of May. If you’d like to host an upcoming post, please get in touch.

]]>While we’re not massively bothered by the pricing, the articles do raise, and then completely fail to address, an interesting point: an oval pizza is harder to cut into equally sized pieces! Luckily, maths is here to save the day. I found a nice method and made a video explaining how it works:

Take a look and improve your future pizza cutting technique!

Cheesed off! Families roundly cheated by trendy oval pizzas as experts warn they are smaller in size… And how CAN you slice them into equal portions, anyway? in The Daily Mail

Why those trendy new oval pizzas you see in major supermarkets may not be worth it, in The Mirror

(Sadly, all proper newspapers have declined to comment)

]]>The film is a painstaking and at times brutally realistic depiction of the struggles faced by African-Americans, and by women, during the era of the early space missions.

It’s the first film I’ve ever seen where the BBFC certificate at the start warns, where you’d usually see “scenes of violence/nudity/strong language”, a warning for “discrimination theme” – and it’s justified, as you find yourself battered by the repeatedly unfair and horrible treatment received by the clearly brilliant and lovely main characters, and worst of all how that is completely normal and widespread for the time. Langley, where NASA HQ is located, is in Virginia, a state where segregation continued in many aspects until the Civil Rights Act of 1964 – this meant ‘coloureds’ had to use separate bathrooms, drink from separate water fountains, sit in a separate section at the back of the bus and attend separate schools, all of which (and their horrible demoralising effects) are seen clearly in the film.

It also doesn’t shy away from the discrimination received by women, which is another layer of problems these characters face – even within their own community, they encounter people with a ‘do they let women do that?’ attitude. When one character arrives in her new department to provide her mathematical expertise and is immediately handed a bin which needs emptying, it’s not clear whether it’s because she’s female or black (or both).

But you’re here for the maths, right? This is at its core a film about a mathematician, an engineer and a computer scientist and how they helped America put a man into space. The maths and science is well incorporated into the story and doesn’t feel awkward, and serves to display the brilliance of the main characters, as well as the difficulty of the work everyone at NASA was doing – it was groundbreaking, and in some cases involved inventing new mathematical techniques to solve problems nobody imagined they’d have to solve (and this aspect of the space race is even mentioned as a justification for the huge cost and effort of the programme).

In one early scene, Katherine Johnson (as a child) demonstrates her mathematical brilliance by, having skipped two grades in school, being called up to the blackboard in front of a room of older students. She demonstrates that the product of two quadratics, set equal to zero on the board, can easily be solved by noting that if a product of two things equals zero, it means one of the two terms must equal zero so you can factorise each of the two quadratics, set each of the four resulting components to zero and find four solutions. It’s a beautiful piece of maths, well explained and displayed in full, front and centre in the scene. While some filmmakers might worry about this, it completely works and demonstrates her brilliance – the looks on the other students’ faces tell the story too.

Elsewhere in the film, Katherine is required to do Analytic Geometry, studying the trajectories of the space flights and their take-off and landing. The initial calculations for the earlier low-Earth orbit flights were simpler, and it’s when they start considering the more complex orbital flights – in which the trajectory of the capsule changes from an elliptical orbit of the Earth to a parabolic trajectory as it comes down, where they needed to work out something new. Together the team realises they can use numerical integration – Euler’s Method, which has been known for a long time but hadn’t been used at NASA for years – to obtain a solution which works numerically, without having to actually solve the differential equations. Johnson’s brilliance in helping come to this realisation is clear, and anyone watching who didn’t understand the maths can maybe even get a sense of the beauty and satisfaction mathematicians find in their work, especially when it has such amazing applications.

The other main characters demonstrate their excellence in their own fields too. Mary Jackson, clearly a brilliant engineer, is encouraged by her boss to join the NASA engineer training programme, but finds her bachelor’s degree in maths and physics is insufficient; the administrator’s veiled joy in telling her she doesn’t qualify could just be a true bureaucrat relishing in the rules, but as with many of the things people do in the film it’s in line with the standards of the era, and advancement seems an impossible dream.

The third amazing role model in this film is Dorothy Vaughan, who runs the computing department (at the time, a team of human ‘computers’ who perform all the calculations by hand). She notices that the new IBM supercomputer coming in to do the calculations will make all their jobs obsolete, so she educates herself and her team, getting books on Fortran from the library and using her skills with machinery and numbers to understand how it works.

I’m glad this film exists, and I’m double glad that it’s such a well-made piece of cinema, with well-developed characters, a gripping story and emotional tugs which work regardless of whether you’re interested in maths – because it means lots of people will see it. I hope that as well as telling these women’s stories – something badly needed – and reminding us of how painfully recent these attitudes held sway (and warning us not to let them take hold again now!) this film will also help people to see the wonder of maths and science, and show that people who do these things struggle the same as everyone else, and are all part of the same world.

It looks like the film is having this effect – many people are now talking about it, and Katherine Johnson also features in this new LEGO set, recently approved, of Women Pioneers at NASA. I’d recommend going to see it, but also taking along some tissues and being prepared to come out energised and angry that people were treated this way, and, I’d hope, ready to take on injustice now.

]]>“Life moves very fast. It rushes from Heaven to Hell in a matter of seconds.”

― Paulo Coelho

This week, I was suddenly reminded of a fact I’d been meaning to keep track of, and I was disappointed to discover that even though I always endeavour to remember birthdays and holidays (mainly due to a system of elaborate reminders, notes and excessive list-making), I’d missed a hugely significant anniversary. Shortly after the clock struck midnight on New Year’s eve, I had passed one billion seconds old.

While not one of the usual anniversaries to celebrate, I’d been looking forward to this one – it turns out that one billion seconds works out to somewhere between 31 and 32 years (my ‘just-after-midnight’ statement assumes I know the exact time I was born, which I don’t, but I have a reasonable estimate) . If you’d like proof, here’s a breakdown:

\[ 1000000000\ \mathrm{seconds} = 1000000000 \div 60\ \mathrm{minutes}\\

16666666.\dot{6}\ \mathrm{minutes} =16666666.\dot{6} \div 60\ \mathrm{hours}\\

277777.\dot{7}\ \mathrm{hours} =277777.\dot{7} \div 24\ \mathrm{days}\\

11574.\overline{074}\ \mathrm{days} =11574.\overline{074} \div 7\ \mathrm{weeks}\\

1653.\overline{43915}\ \mathrm{weeks} =1653.\overline{43915} \div 52\ \mathrm{years}\\

= 31.\overline{796906} \ \mathrm{years} \]

This quantity may mildly surprise you – partly because humans in general can be quite bad at interpreting numbers like a million and a billion. We know what the number means, and can calculate with it, but intuition can fail us when trying to put it into context.

It turns out that a second is quite a nice way to contextualise large numbers – for example, here’s an interesting fact I heard about the number of seconds in six weeks:

\[ \begin{eqnarray}

6 \ \mathrm{weeks} &=& 6 \times 7 \ \mathrm{days}\\

&=& 6 \times 7 \times 24 \ \mathrm{hours}\\

&=& 6 \times 7 \times (8 \times 3) \ \mathrm{hours}\\

&=& 6 \times 7 \times (8 \times 3) \times 60 \ \mathrm{minutes}\\

&=& 6 \times 7 \times (8 \times 3) \times (10 \times 3 \times 2) \ \mathrm{minutes}\\

&=& 6 \times 7 \times (8 \times 3) \times (10 \times 3 \times 2) \times 60 \ \mathrm{seconds}\\

&=& 6 \times 7 \times (8 \times 3) \times (10 \times 3 \times 2) \times (3 \times 5 \times 4) \ \mathrm{seconds}\\

&=& 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times (3 \times 3) \times 10 \ \mathrm{seconds}\\

&=& 10! \ \mathrm{seconds}\\

\end{eqnarray} \]

The number of seconds in six weeks can be expressed as a product of the numbers one to ten – that is to say, there are 10! seconds in six weeks. Large factorials like this ($10! = 3,628,800$) are similarly difficult to quantify, so this is a nice fact to have in your pocket.

A million is a more manageable number; a million seconds is just over 11 and a half days, which might be the length of a single short project you work on in your lifetime, or how long a holiday lasts, or somewhere at the long end of how long you might reasonably expect a banana to keep for (if it was really fresh when you got it).

So my 1 billion seconds = 31 years milestone makes a nice distinction between a million and a billion – a couple of weeks versus a good chunk of my life. Another reason I’m disappointed not to have properly celebrated (I mean, I was celebrating, but not necessarily this) is because this is probably the biggest power of ten I’ll reach in my lifetime. I’ll probably survive to 2 billion seconds, and if I’m lucky maybe even 3 billion, but there’s no way I’ll make it to 10 billion and certainly not a trillion.

But here’s some you might manage:

- 1 year on the planet Jupiter is about 11.86 years
- 10 million minutes (aka 10 MEGAMINUTES) is about 19.01 years
- 1000 fortnights is about 38.33 years
- 1000 months is about 83.4 years, if you’re lucky!

So raise a billion glasses for me, and celebrate your milestones in seconds not years (as long as it doesn’t make you feel too old).

]]>Having spoken at the MathsJam annual conference in November 2016 about my previous phone spreadsheet on multiples of nine, I was contacted by a member of the audience with another interesting number fact they’d used a phone spreadsheet to investigate: my use of `=MID()`

to pick out individual digits had inspired them, and I thought I’d share it here in another of these columns (LOL spreadsheet jokes).

Twin primes, which are pairs of prime numbers with a difference of two, are well-studied. It’s not known whether there are infinitely many such pairs (although we’ve made some progress on that), but it’s suspected. The gaps between pairs get larger as you go up the number line, and there’s a few interesting bits known about them, including Brun’s Theorem. Today I’m going to investigate a thing about their products.

We can start by populating a spreadsheet with a list of pairs of twin primes, starting with the classic \((3,\!5)\) and proceeding from there. This data was obtained from the amazing resource at primes.utm.edu, maintained by known prime-basher Chris Caldwell. Putting this list of primes in the first column, and populating the second column with the number two higher gives us a lovely list of twin prime pairs.

In the next column, we calculate the product of each pair:

And in the fourth column, we can use a similar `MID()`

/ `VALUE()`

based construction to find the sum of the digits in this number:

Giving the not-especially-interesting result:

Or is it? In the fifth column, we can take `=MOD(`

and see what happens:*this value*,9)

With the exception of the first row, and let’s be honest the first one’s always weird anyway, you’ll see that all of these numbers have the value $8$ modulo $9$.

For an explanation of this, done in a bar after a few pints of course, we can thank sometime Aperiodical contributor and general maths guy Colin Beveridge, who contributes the following; if you’d like to try and work it out yourself, we’ve made these bullet points appear one at a time when you click, so you can use as many as you need and work the rest out.

- All twin prime pairs other than $(3,5)$ are of the form $6n +1$ and $6n – 1$ (it says it on Wikipedia so it must be true; left as a pleasing exercise for the reader)
- The product will therefore be of the form $36n^2 – 1$ (classic difference of two squares)
- This means the first part of this number will be divisible by $9$, as $36$ is divisible by $9$
- The whole thing must therefore be congruent to $8$ modulo $9$, as it’s a multiple of $9$ minus one.
- This means the sum of the digits of the number will also be $8$ modulo $9$.

A fun diversion for everyone. Thanks to Simon Allen, who sent me the email, and to Colin for his neat explanation, relayed via Simon.

If you’ve seen any nice number facts I can investigate using a spreadsheet on my phone, please send them in!

]]>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.

]]>Another name for a Rubik’s cube is ‘the Magic Cube’ – and Dr James Grime wondered if you could make a Magic Cube which incorporates its 2D friend, the Magic Square.

Several designs were considered for such a puzzle cube – early ideas included a cube with the numbers 1-54, each face being a normal magic square using the numbers 1-9, with a different multiple of 9 added to give a range for each face. After some searching and testing James settled on one particular type of magic square to make it work.

The Grime Cube is based around the **perimeter magic square** – an arrangement of the numbers 1-8 such that the three numbers on each edge sum to the same total. For example:

The square above has all four edges adding to 12 (and of course any rotation or reflection of this arrangement will also have this property). One nice interesting maths fact about such constructions is that only six valid perimeter magic squares exist for the numbers 1-8 (obviously, you can just add the same thing to each value to get another one, but that’s not as elegant).

What James Grime discovered was that you could use a Rubik’s cube, where the centres are fixed, to implement a puzzle involving such squares – since the centre of each face is fixed and only the edges move, you could try to arrange the pieces so that they add up to a number given in the centre. There only being six makes this an ideal thing to do on the faces of a cube, and the six squares pair up with each other as complements – so if you overlay two and sum them, the squares in the same place all add to 9.

This nice complement property meant that the cube could be designed to have opposite individual cubies (single chunks of cube) have their faces complement each other – so where there’s a 4 on one face of the cube, the opposite square will be a 5. Dr Grime also worked out a way to arrange six colours on the cube, so that each pair of mini-faces not only sum to 9, but are the same colour – and only one instance of each number/colour (e.g. red 5) occurs anywhere on the cube.

The final coup de grace for James was discovering a way to arrange the faces on the cube so that these colours could also be solved as a normal Rubik’s cube – an alternative arrangement where each face contains a single colour. This makes it a double puzzle – it can be solved so that either the magic squares work, and the opposite pairs match up, (left) or so that the colours on each face are all the same and there’s still 1-8 on each face (right).

As part of the testing process, I’ve had several goes at solving the cube and found it actually pretty pleasing – with my knowledge of how to solve a Rubik’s cube, I know how to put a particular edge or corner piece in a particular place, so I’ve found that very useful. I could find a pair of edge cubies that sit opposite each other (because e.g. the red 4 is always opposite the red 5) and then try other pairs until I get an arrangement that doesn’t use anything disallowed and lets me arrange the rest of the face so the totals are all the same. Then I can figure out which face I’ve just made, arrange it around the right centre face, and start working on the other faces around it. Others may have their own methods but I like how much it makes me think – not just blindly solving the colours, but also having to do sums and deduction along the way.

There’s another outstanding question about the cube, which Grime doesn’t know a good answer to yet – from the solved-for-magic-squares state, there should be a sequence of moves which translates it to the solved-for-colours state (and the maths tells us that this can be done in fewer than 20 moves, aka God’s Number). So far, James has found a path which takes 70 moves, but who will be the first to find the optimum path?

**The Grime Cube is on sale at MathsGear.co.uk for £14.99, and limited stock is available. **The last ordering date for delivery before Christmas in the UK is **19th December**, and more stock should be available in the New Year.

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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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