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

]]>One problem: that’s Wolfram’s Rule 135, not the Game of Life. You can tell because of the pixels.

Rule 135 is a 1-dimensional automaton: you start with a row of black or white pixels, and the rule tells you how the colour of each pixel changes based on the colours of the neighbouring pixels. The Cambridge North design shows the evolution of a rule 135 pattern as a distinct row of pixels for each time step. Conway’s Game of Life follows the same idea but in two dimensions – a pixel’s colour changes depending on the nearby pixels in every compass direction.

Either way, it’s a lovely pattern. I suspect the designers went with Rule 135 instead of the Game of Life so that they’d get a roughly even mix of white and black pixels, which is hard to achieve under Conway’s rules.

Just in case gawping at train stations is your cup of tea, here’s a promotional video with lots of lovely panning shots of the design:

Delayed £50m Cambridge North railway station opens on BBC News.

Cambridge North Station information from Atkins Group, the design consultancy responsible for the station building.

Press release from Greater Anglia trains.

The Game of Life: a beginner’s guide by Alex Bellos in the Guardian.

*Brought to our attention by @Quendus on Twitter.*

Mastodon is a new social network, heavily inspired by Twitter but with a few differences: tweets are called toots, it’s populated by tusksome mammals instead of little birds, and it’s designed to run in a decentralised manner – anyone can set up their own ‘instance’ and connect to everyone else using the GNU Social protocol.

Colin Wright and I both jumped on the bandwagon fairly early on, and realised it might be just the thing for mathematicians who want to be social: the 500 character limit leaves plenty of room for good thinkin’, and the open-source software means you can finally achieve the ultimate dream of maths on the web: LaTeX rendering!

So, we’ve set up our own Mastodon instance, and given it the nifty name of mathstodon.xyz. I’ve got to say I’m pretty pleased with that bit of punning.

I added MathJax straight away so we could toot notation, and changed our web interface to use the Computer Modern fonts for that extra mathsy touch. I also set up a bot to toot a daily entry from my Interesting Esoterica collection, taking full advantage of the larger character limit to include abstractss as well as links to the papers.

However, a social network lives and dies by the network effect, so now we need to attract other people to join our glorious mathematical chat-topia. So, to get to the point of this post, to provide an easy way in to Mathstodon we’ve come up with a challenge: write the best #proofinatoot. With 500 characters of space and all the typesetting power of LaTeX, the world is your very oyster.

Here’s a classic of the form, to show you what’s possible:

Get your thinking hat on, head over to mathstodon.xyz, and wow us with your most concisely persuasive proof in a toot. In the spirit of free software we don’t have a real prize to give away, but the proof we judge to be most tootematical will go on the instance’s front page to greet new users.

If you do join in, this guide to Mastodon will help you get your head round the slightly different way of working, and here’s a list of accounts on mathstodon.xyz that you might want to follow:

- @christianp – Me!
- @ColinTheMathmo – Colin Wright
- @aperiodical – The Aperiodical
- @henryseg – Henry Segerman
- @esoterica – Daily Interesting Esoterica toots
- @pecnut – Adam Townsend

See you there!

]]>News from France, where the family of the late Alexandre Grothendieck, legend of basically all maths, have finally reached an agreement with the academic community about his huge archive of written notes. Discussions have been ongoing for a while but it’s finally been agreed that the notes can be released online for the community at large to take advantage of.

The notes comprise over 100,000 pages of mathematics, diagrams and letters to collaborators, and an initial chunk of over 18,000 pages will be online from 10th May on the University of Montpellier’s website. It’s expected that many undiscovered mathematical treasures might be found within, although the challenge of reading through and deciphering it all may take a Polymath-style mass effort.

The notes of the mathematician Alexandre Grothendieck arrive on the net, at Libération (in French)

]]>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.

]]>

*The Mathematics of Secrets* kicks off with a pretty decent chunk of introductory linear algebra in the service of basic substitution ciphers, preceded by a few pages of terminology. This is introduced apologetically as an unfortunate necessity, but some of the explanation could be handled better. Part of the rundown is handed off to the following quote from David Kahn, none of the terms in which have previously been defined (the ellipses are the book’s, not mine):^{2}

A code consists of thousands of words, phrases, letters, and syllables with the codewords or codenumbers…that replace these plaintext elements….In ciphers, on the other hand, the basic unit is the letter, sometimes the letter-pair…, very rarely larger groups of letters….

Clear? Once the book gets going, it offers a pretty good explanation of a variety of different encryption techniques from simple substitution ciphers through to modern stream ciphers and key exchange systems, as well as the strategies used to attack them. In one of the most accessible sections, Holden explains a particularly elegant system for cracking ‘polyalphabetic’ substitution ciphers: ones where the encoded ‘ciphertext’ is produced by switching to a different substitution cipher ‘alphabet’ after each letter, in particular the case where a relatively small number of alphabets are used in rotation. The first thing to work out is the length of the cycle: how long before the encoding alphabets repeat? One method of finding this is to compute the “index of coincidence”: the probability that two randomly chosen letters of the ciphertext are the same. For a normal ‘monoaphabetic’ cipher this would be the same as for unencoded text: about $0.066$ for English. The more alphabets are used, the nearer this number gets to $1/26=0.038$, the value for a string of random letters. The value your ciphertext gives you suggests a rough value for the number of alphabets, give or take one or two. A second approach is to look for short strings of three or four letters that show up twice in the ciphertext. Some of these will be coincidental, but more will arise from repetitions in the message that happen to get encoded using the same sequence of alphabets. In that case, the distance between the repetitions must be a multiple of the number of alphabets. So any particularly prevalent factor among the distances-between-repeats is a good bet for the number you’re looking for. The two methods combine perfectly, since the second will give you candidates that are unlikely to be close in value, so the approximate value gained from the first should settle which is the correct one. Once you know the number of alphabets, you can split the message up and attack the parts with normal frequency analysis. Holden explains this clearly, with the full details and a couple of examples in case you want to have a go yourself. (I don’t know how challenging these informal exercises are — I had a review to get on with — but I’m guessing they’re less intense than *The Code Book*‘s cipher challenge, which stood unsolved for thirteen months despite a £10,000 prize.)

Towards the middle of the book, some of the discussion of real-world implementations of modern cipher systems towards can get a little mystifying. A flurry of ‘key schedules’, ‘S-boxes’ and ‘P-boxes’ seem to be chained together in specific arrangements it’s hard to muster up much enthusiasm for. Perhaps some of these details could have been relegated to appendices and more time spent on the general aims of these systems, which I think I am still a little hazy on. The version of the book the publishers sent us was also marred in a couple of places by typographical annoyances. Ciphertext is marked out in the main prose in small caps, but plaintext (unencoded messages) are not distinguished at all, producing the momentarily baffling sentence on page 46 “The plaintext letters at repeat 4 times”. And in one unfortunate instance, factorial signs have been omitted leading to the startling claim that $12 = 479,001,600$.

Things get going again when it’s time for the fun stuff. The seemingly-impossible shenanigans of public-key encryption, where the two parties can concoct some secret numbers that only the two of them know despite all their communication being entirely public, is well-explained. As are the newfangled elliptic-curve-based systems that might one day replace all that current mucking about with massive semiprimes, and the exciting world of quantum cryptography, where the preposterous properties of protons are corralled into a fundamentally unbreakable code system. This more theoretical stuff seems to me a lot more interesting than worrying about why a P-box was added at the beginning and end of the DES standard (even the author seems unsure: “Apparently, the P-boxes are merely there to make the data easier to handle on the original chip”). Unless you have a particular interest in the gnarlier details, a few pages might prove skippable, but otherwise this is a decent tour of cryptography for anyone who wants to go a bit deeper than Simon Singh took them.

Joshua Holden: The secrets behind secret messages: press release from the publishers with an interview with the author

The Mathematics of Secrets at Princeton University Press

- I will in this review unapologetically make no attempt to maintain any distinction between the terms code and cipher; cryptography, cryptanalysis and codebreaking, etc.
- Khan is introduced as “author of perhaps the definitive account of the history of cryptography”. Obviously, I beg to differ.

Bonus challenge: See if you can count how many times Katie accidentally says ‘compass’ instead of ‘pair of compasses’ during the video.

]]>I’d half-remembered Katie’s friend’s Dad’s golf tournament problem and made a guess about the root of the difficulty she was having, but on closer inspection it wasn’t quite the same. I’m going to try to recount the process of coming up with an answer as it happened, with wrong turns and half-baked ideas included.

Here’s a summary of what she told me:

- There are 7 groups, each with a leader (who must be in charge of the choreography or something)
- There are five timetable periods available, each an hour long.
- Each dancer should get at least 3 hours’ rehearsal time.
- Some students belong to more than one group.

She also gave me the list of dancers in each group. (Names have been changed to make the maths easier)

I could’ve pretended these are their real names, but then this post would have to be about the astronomically unlikely coincidence of being presented with a real-world maths problem with such helpfully ready-made notation.

1: Amelia | Bmelia | Cmelia | Dmelia | |
---|---|---|---|---|

2: Emelia | Dmelia | Fmelia | Gmelia | |

3: Hmelia | Cmelia | Imelia | Jmelia | Kmelia |

4: Lmelia | Mmelia | Nmelia | Omelia | |

5: Pmelia | Gmelia | Qmelia | Rmelia | |

6: Smelia | Rmelia | Tmelia | Umelia | |

7: Vmelia | Smelia | Pmelia | Tmelia | Jmelia |

SIL was concerned that among the people in two groups were the leaders of two other groups – Smelia and Pmelia. I wasn’t sure that this made any difference, but it was an extra bit of information to bear in mind in case it later led to a complication I hadn’t thought of.

My first instinct was that I needed to come up with an individual timetable each dancer. SIL’s original email wasn’t very clear about what exactly she needed, so I thought that giving each dancer 3 hours would turn out to be difficult – maybe the room can only accommodate so many people at once, for example. With that in mind, I set about looking for ways to assign dancers to periods while maximising the length of time groups had together.

First: if every dancer gets exactly 3 hours’ rehearsal, is it possible to look at every permutation of timetables and rank them by how well they fit the constraints? In other words, **can I brute force it**?

There are 20 dancers, each of whom need to be timetabled in 3 of 5 periods, so 12 dancers in each period. That’s ${20 \choose 12} = 125970$ choices of dancers for each period, and a total of

\[125970^5 = 31720180268697990275700000 \]

timetables for the whole class. Most of those will be invalid, giving some dancers more or less rehearsal time than they need, but any number wider than your field of vision is a bad candidate for brute forcing.

Alternately, each dancer is assigned 3 out of 5 periods, and that happens for each of the 20 dancers. Looking at it this way, there are

\[ {5 \choose 3}^{20} = 10^{20} = 100000000000000000000 \]

timetables to consider. While that’s a million times smaller than my first guess and the least significant digit is starting to hove into my peripheral vision, that’s still far too big. so I think brute forcing is out, and I’ll have to **do some thinking**.

I’m not very good at thinking.

Strategy 2: **get someone else to do it, preferably a computer**.

This is some kind of linear programming problem, so I did a quick google and ended up at GLPK: the GNU Linear Programming Kit. I’d never heard of it before, but further googling led to this program to solve the knapsack problem on wikibooks.org which looked enticingly gnomic:

# en.wikipedia.org offers the following definition: # The knapsack problem or rucksack problem is a problem in combinatorial optimization: # Given a set of items, each with a weight and a value, determine the number of each # item to include in a collection so that the total weight is less than a given limit # and the total value is as large as possible. # # This file shows how to model a knapsack problem in GMPL. # Size of knapsack param c; # Items: index, size, profit set I, dimen 3; # Indices set J := setof{(i,s,p) in I} i; # Assignment var a{J}, binary; maximize obj : sum{(i,s,p) in I} p*a[i]; s.t. size : sum{(i,s,p) in I} s*a[i] <= c; solve; printf "The knapsack contains:\n"; printf {(i,s,p) in I: a[i] == 1} " %i", i; printf "\n"; data; # Size of the knapsack param c := 100; # Items: index, size, profit set I := 1 10 10 2 10 10 3 15 15 4 20 20 5 20 20 6 24 24 7 24 24 8 50 50; end;

Unfortunately, the documentation for GMPL, the language used by GLPK, is hard both to find and to read.

For reference, here’s the GMPL language reference. It’s a breezy 70 pages, but my eyes go fuzzy when I look at it.

While I could probably get it to find a solution for me, I’d probably spend longer learning how to write the program than I would have spent coming up with the answer on my own. So, I made a mental note to learn all about GMPL at a later date, because it does look rather marvellous, and decided to phone SIL and fish for more hints.

On talking to SIL, it turned out **I was solving the wrong problem**! I’d assumed there’d be a constraint on the number of dancers who could rehearse at once, and that dancers wouldn’t want to be assigned more than 3 periods, so the problem would be putting the right people in the room at the same time. That wasn’t the problem at all: everybody can rehearse at the same time, and SIL was perfectly happy to work her charges like dogs for the full 5 periods if required. The real problem was how to make sure that each group leader got at least 2 hours with their whole group. So even if everybody’s in the room for all 5 sessions, the timetable should tell them which group’s dance they should be rehearsing at any time.

Suddenly, the problem looked a lot more manageable: for the sake of simplicity, I can begin by assuming everyone’s in the room, and I just need to assign periods to groups, making sure not to timetable two groups sharing a dancer in the same period.

I immediately returned to a little doodle I’d made while thinking about how to avoid thinking earlier.

While some dancers belong to two groups, no dancer belongs to three groups. That’s good: if a dancer belongs to three groups, then it’s impossible to give all three of those groups two hours rehearsing with all their members, since there are only 5 hours available.

Next I thought about the timetable from the perspective of a dancer who belongs to two groups: for two hours they’ll be with one group, for two different hours they’ll be with the other, and there’s another hour spare. If I can split the groups into two sets so that none of the groups in each set have a dancer in common, I’m done: I can timetable the groups in the first set to dance together for the first two hours, and then the groups in the second set will dance together in the two following hours.

So **I needed the graph I’d drawn above to be bipartite**. Unfortunately, it isn’t:

No matter how I shuffle things about, that triangle involving groups 5,6 and 7 will scupper any attempt to draw a bipartite graph. It also became clear at this point that group 4 is either really unpopular or really snooty.

So, it was impossible. But at least **now I know it’s impossible!** I rang SIL to tell her the good news: she hadn’t wasted hours stupidly, it really isn’t possible to timetable her groups fairly. SIL’s sister (aka my wife) told me not to bother her with the details, but I felt that my findings were only good news if I explained my reasoning. At least I didn’t use the word ‘bipartite’. To offer extra comfort, I worked out that the only spanner in the works was **Pmelia**: she belongs to groups 5 and 7, and only removing her edge from the graph would make it bipartite.

I asked if it would be possible to swap Pmelia out of group 7 for one of the members of group 4, maybe Nmelia. Unfortunately, SIL said that wasn’t possible. So, having realised that Pmelia was the problem (should’ve guessed: alliteration is a powerful epistemological tool), I came up with a “least-bad” timetable:

- In the first two periods, groups 2, 3 and 6 should rehearse with all of their dancers.
- In the following two periods, groups 1, 5 and 7 should rehearse with all of their dancers, but 7 will have to make do without Pmelia.
- In the fifth period, group 7 should rehearse for another hour, with Pmelia included this time.
- Group 4 can do whatever the heck they like.

Now, having had all this marvellous insight, there was still the chance that my timetable would be unacceptable for some reason SIL hadn’t yet mentioned, but I got lucky and she was very happy with it. She did ask what the dancers should do when they’re not timetabled. Rather than get bogged down in more unwelcome thinking, and since SIL didn’t have any particular aims for them, I told her to get them to sort themselves out: there’s no point making up a rule unless you really need to.

So, that’s the story of how I accidentally got my graph theory wedged in my sister-in-law’s dance class. I quite like the simplicity of my final answer, though I expect I’ll carry a grudge against people called Pmelia for a long while.

]]>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.

]]>(Apologies that the last few seconds of the video cut off – apparently the 10-minute limit is actually a 9:49 limit and YouTube declined to notify me of this, or that they’d cut the end off.)

]]>