Sunday, 19 June 2016

Maths my father taught me

Today being Father's Day, I thought I would write about a piece of mathematics my father showed me many years ago.

I have been reading Erica Walker's inspiring book Beyond Banneker: Black Mathematicians and the Paths to Excellence, a study of three generations of Black mathematicians in the USA, the obstacles they faced, and the networks and structures which supported them.  It was interesting that, although few of them had any professional mathematicians in their families, many of the mathematicians Walker writes about were stimulated in their early childhood by a family member with an interest in puzzles, or engineering, or some kind of applied, non-academic mathematics.

I've had a very privileged life.  I had access to excellent schools and the best universities and was taught by outstanding teachers.  But why was I interested in mathematics as a child?  I had no close relatives who had studied mathematics (or indeed science) beyond school level to influence me.  (My father and grandfathers studied classics and law, my mother social work, my aunt history, and (unlike me) everyone was very musical.)

As a child I was obsessed by football.  I was no good as a player but I loved games about football.  I played Subbuteo Table Football with my friends (who were all better than I was, but I didn't mind losing: in retrospect my interest was in the modelling.  I used to simulate tournaments with random numbers, trying to get realistic results.  (It really upset me that a football game we had called "Wembley", in which match scores were decided by dice, gave lower division teams playing away in the cup a dice with a 5 on it, while no other team could score more than 4.  A one in six chance of Rochdale scoring five at Old Trafford?  I couldn't take that game seriously.)

And so I needed to create fixture lists for football leagues - n teams each having to play every other team home and away.  How could I draw up the weekly fixtures?  Trial and error wasn't going to work for the 18 teams in the then Scottish First Division,

Clearly if n is even, if teams play every Saturday (only), it requires at least (n-1) Saturdays for every team to play every other once. Perhaps the first sophisticated mathematical question I asked was "Can it always be done in n-1 Saturdays?  I remember thinking it wasn't clear to me that we could always draw up a fixture list that worked, with every team playing every week,

Anyway, I asked my father, and he told me how to do it.  Number the teams 1 to n. In week 1, team 1 plays team 2, team 3 plays team n-1, team 4 plays team n-2, team 5 plays team n-3 and so on.  This pairs all the teams except team n/2 and team n who play each other.  In week 2, team 1 plays team 3, team 4 plays team n-1, team 5 plays team n-2, and so on: team 2 is left to play team n.  In week k, team 1 plays team k, team 2 plays team k-1, team 3 plays team k-2, and so on: then team k+1 plays team n-1, k+2 plays n-2, k+3 plays n-3, and so on.  The unmatched team in the middle of one of these sets of pairings plays team k.  

This algorithm works, and shows that it is always possible to play all the matches in n-1 rounds.  If n is odd, then one team is necessarily idle each week: the algorithm can be modified by adding an extra team called "bye", and we see that a league with 2n-1 teams can play all their fixtures in 2n-1 weeks.  (A league with an odd number of teams is unusual but in the Scottish League of my childhood, Division Two contained 19 teams.)

It's only recently that it has occurred to me that this solution to a childhood problem is a serious combinatorial algorithm.  So where did my father get it from?  Presumably not the mathematical literature! I asked him recently and he said that he worked it out for himself.

So although I may not have had professional mathematicians in my family,  But my father was capable of working out for himself a nice algorithm to solve a tricky combinatorial problem (even if he had no idea he was doing mathematics).  So my own interest in mathematics didn't come from nothing: my father could think mathematically and solve mathematical problems, even although at the time neither he nor I knew that that was what he was doing.

Monday, 30 May 2016

Game Theory and "Beau Geste"

I wrote a post some time ago about Michael Suk-Young Chwe's book Jane Austen, Game Theorist, which argues that Austen's works are a systematic exploration of game theory ideas.  I have to say that I was not entirely convinced.  For me, game theory is about players thinking about their choices and their opponents' choices, and thinking about what opponent is thinking is important.  The examples in Austen of strategic thinking didn't, for me, capture that aspect of game theory.

But I recently reread Beau Geste, P.C. Wren's 1920s adventure story of the French Foreign Legion with a tragic denouement at Fort Zinderneuf.  Let me say straight away that the book shows all the unpleasant racism of its time.  It also presents an old-fashioned view of the code of duty which I fear, as a child, I took more seriously than it deserves.  (Wren's sequel, Beau Sabreur, is much more ambivalent in this regard, or perhaps I just missed the irony in Beau Geste.)

In these matters Beau Geste is very much of its time.  In its own terms, it is a rattling good adventure story.  And, unlike Austen, it presents real game theory dilemmas (I am trying not to give significant spoilers) .  The problems the heroes face require them to think through the consequences of their actions and how others will react.  Both in the matter of the theft of the jewel which sets up the adventure, and in taking sides in the potential mutiny at Fort Zinderneuf, they are thinking not only about their own actions but about the other parties'.  The mutiny is interesting because, thanks to the characters' interpretation of the demands of their duty and loyalties to their comrades, everyone on both sides has full information, so it really is a nice example of strategic game theory thinking.

Sunday, 13 December 2015

The geometrical artwork of José de Almade Negreiros

As always, the annual MathsJam conference was full of wonderful things, and if I ever find time to post on this shamefully neglected blog I may return to some of these topics.  But one of the special highlights of the 2015 MathsJam was Pedro Freitas's talk about the geometrical art of the Portuguese painter José de Almada Negreiros (1893-1970).  Almada Negreiros (who also wrote novels and poems) made a collection of drawings "Language of the Square" which mathematicians will find fascinating.  Happily, Pedro and Simao Palmeirim Costa have written a book Livro de Problemas de Almada Negeriros (Sociedade Portuguesa de Matemática, November 2015) which contains excellent colour reproductions of twenty-nine drawings.  The book is available from

Sadly for me, however, the text is in Portuguese, but an essay by the same authors, in English, can be found at

I'm delighted to have discovered these mathematical artworks - yet another MathsJam discovery!

Sunday, 1 November 2015

Remembering Lisa Jardine

I was very sorry to hear of the death last week of the historian Lisa Jardine.  Although it wasn't her main focus, she made a big contribution to our understanding of early modern mathematics and especially of key figures like Robert Hooke and Christopher Wren.  Her books are wonderful - readable, full of insights, and giving a vivid picture of intellectual life in the seventeenth century.

I was lucky enough to hear her talk, less than a year ago, at the BSHM Christmas meeting last December when she gave an inspiring talk about women in twentieth century mathematics -in particular Hertha Ayrton, Mary Cartwright and Emmy Noether.

Jardine;s scholarship was important, but so was her encouragement of others.  I believe she was an exceptional research supervisor, and her writing certainly inspired many, myself included.  I experienced her kindness several times, and enjoyed a few conversations with her in coffee breaks at conference.  Twice I consulted her by email, and although she can have had no idea who I was, she replied quickly, enthusiastically and helpfully.  (On the first occasion I was seeking clarification of a view attributed to her in someone else's book, and on the second I was hoping to persuade her to talk about the novelist Robert Musil at a conference I was organising - she agreed in principle but sadly the dates didn't work out.)

Her contribution to the history of science, direct and indirect, is immense.  She is a great loss to the history of mathematics.

Sunday, 25 October 2015

Outreach: Simon Singh's comments

The Times Higher has reported a conference talk by the writer Simon Singh about public engagement in science under the heading "Simon Singh criticises wasteful science outreach".  This has even led to the ultimate distinction of a mention in Laurie Taylor's Poppletonian, the comic column which is consistently the highlight of my week.

Now, not only did I not hear Singh's talk, so that I am relying on the THE report, but Singh is one of my heroes for his contributions to mathematics and its public reputation.  For one thing, his documentary film on Andrew Wiles's proof of Fermat's last theorem conveyed to the general public the emotion and the joy of doing mathematics, and what the profession is about, in a way which was almost unprecedented.  His subsequent writings - the book on Fermat's Last Theorem, the book on codes (and the wonderful CD-rom he made with Nicholas Mee), his recent book on the maths of The Simpsons and others - have continued to inform the public and inspire young mathematicians. And his creation of the undergraduate ambassador scheme, which puts maths undergraduates into schools and colleges, is a hugely important contribution to maths education in the UK, directly benefiting school and university students and motivating many outstanding graduates to become maths teachers.  So his views on public engagement certainly deserve to be taken seriously.

But on this I disagree with Singh's comments (at least as they are reported).  He says that in his view the best science outreach is "largely dirt cheap".  Well, there is certainly a lot of excellent public engagement work on mathematics that is done on the cheap (such as the Royal Institution Masterclasses and the British Science Festival, the power of both of which I have seen at first hand), and a large number of people doing it more or less in their spare time for no reward other than the joy of communicating mathematics.  But it shouldn't be like that, and it isn't only like that,

Singh is critical of the funding of a ballet about relativity.  "People hate physics, they hate ballet, all you've done is allowed people to hate things more efficiently."  Well, I believe that science and mathematics are so important that they should feature prominently in art.  There should be ballets, novels, operas about mathematics.  Happily, thanks to people like Marcus du Sautoy, Scarlett Thomas, Dorothy Ker and a great many others, there are.  Not everyone hates ballet: a dance piece about science is potentially reaching a valuable audience.  When Singh says of the "Faces of Mathematics" project of portraits of mathematicians that "I don't quite understand how this is really going to have an impact", he is surely not using his imagination.  We need to show the world the diverse nature of mathematicians to encourage the aspirations of the potential mathematicians of the future,

Of course Singh is right to suggest that the value for money of any outreach project should be compared with the cost of a teacher.  And we certainly need more teachers, and to pay them better.  But imaginative (and expensive) public engagement projects are also important.  They won't all succeed,  Singh's TV programme about Wiles was of inestimable value in showing what mathematics is about.  Dance and photography projects promoting public engagement with mathematics and science have similar potential.  Even when they fail, they are not wasteful.

Thursday, 8 October 2015

The Mpemba Paradox

As a mathematician I love mathematical paradoxes because they are disturbing and thought-provoking.   For example, Parrondo's Paradox tells us something counter-intuitive about probabilistic games; Simpson's paradox reminds us that we have to think carefully about statistics; and Curry's paradox is just mind-bending.

Paradoxes in the sciences are important because they make us think about our theories and where they don't quite match reality, driving new scientific ideas.  My favourites include Olbers' Paradox (why is the sky dark at night?) and the EPR Paradox which shows us just how surprising the world is.

So I was delighted to come across, in an article by Oliver Southwick in the excellent magazine Chalkdust, a paradox that was new to me, the Mpemba Paradox.  "If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?"  (The background story is wonderful - read the article!)  Not only is there no agreement on the answer, but it gives insights into the mathematical equations involved, and mathematical modelling may help us understand the effect.

Like all the best paradoxes, this is amusing but tells us something surprising about our world.

Sunday, 24 May 2015

Greenwich's mathematician in residence

Mathematician in Residence show

As I write, the mathematician Katie Steckles is in residence at the Stephen Lawrence Gallery of the University of Greenwich, where she is demonstrating some fun maths to the public and getting people to take part in various mathematical activities.  It's a fascinating show, covering many different kinds of mathematics, from tessellation to random walks, fractals to Benford's Law, and some graph theory.

For me, what has emerged is the visual connections.  My photo above shows to the left images of blackboards relating to some of the maths research at Greenwich: even without a full explanation of the context the blackboards viewers are intrigued by the look of the mathematics.  The photo below shows a wall of "doodles" by visitors, showing that all such doodles (formed by closed curves) can be coloured in only two colours so that regions with a common boundary are different colours.

graph theory doodles

Katie has curated a marvellous collection full of visual and mathematical interest.  Hopefully the show will persuade people that there is much more to maths than sums!