Saturday, 15 October 2016

Puzzles - to be solved, or to be admired?

The composer Howard Skempton once said to me that there are three kinds of piano music: music to be read, music to be played, and music to be listened to.  I think something similar is true of mathematical puzzles.

Certainly one ought to attempt most mathematical puzzles for oneself, before looking at the solution. But I think there are some examples where one can admire the solution without attempting the puzzle oneself.  One example might be the 100 prisoners problem - the solution is beautiful and I don't think that I would have gained anything by spending a long time thinking about the problem before looking it up.  I don't feel too bad about looking up how to solve the 5x5x5 Rubik cube - I did work out how to solve the 3x3x3 one by myself (albeit almost 40 years ago: I might not be able to do that now) so I didn't feel that I had to prove anything to myself, and I felt that I had better things to do with my time.  (On the other hand, the fact that I am writing this self-justifying post may suggest that I do feel some guilt about this!)

Anyway, here is one problem that is certainly in the "to be solved for oneself" category.  It was knew to me: I came across it, surprisingly, in a literary novel - Ethan Canin's A Doubter's Almanac, one of the small category of novels in which the principal character is a Fields Medallist.  (The only other one I can immediately think of is Peter Buwalda's Bonita Avenue - if you know of any others, please tell me!)
A mathematician buys a lottery ticket, choosing six different integers between 1 and 46.  She (it's "he" in the book) chooses her numbers so that the sum of their base-ten logarithms is an integer.  How many possible choices are there?

Sunday, 24 July 2016

The Murdered Mathematician

One of my interests is mathematicians in fiction - fictitious mathematicians tell us something about how the extra-mathematical world views us (of course there have been great novels by mathematicians too).  There is crime fiction in which mathematicians are murderers or detectives. I particularly like Hector Hawton's Murder by Mathematics (HT John Sharp who told me about it), in which the Head of the Mathematics Department in a London university gets murdered, and it turns out (implausibly, I hope) that everybody in his professional and personal life wanted him dead.

Perhaps the strangest such novel is Harry Stephen Keeler's The Murdered Mathematician.  Keeler (1890 - 1967) was an eccentric novelist many of whose (decidedly unusual) works are, wonderfully, available from Ramble House Press ( whose service is excellent.  There is an entertaining Wikipedia article.

In The Murdered Mathematician the victim is an eccentric professor, "Radical Luke" (whose radicalness is exemplified by his refusal to use Greek letters in doing mathematics).  The murder is solved by Quiribus Brown, who is 7 foot 6 tall, and has been taught higher mathematics by his father. The book contains an exam paper of Radical Luke's, and Brown uses some fairly sophisticated mathematics to identify the murderer.  It is certainly unlike any other book I have ever read!  I'm now reading the further adventures of Quiribus Brown in The Case of the Flying Hands.

Saturday, 9 July 2016

Maths and Tennis (Again)

I wrote about mathematics and tennis at this time of year four years ago: specifically on the decisions about reviews and about the curious fact that, against a stronger opponent, you are more likely to break serve from 30-15 down than from love all.  I had completely forgotten about that post, but was stimulated to return to the topic by a comment on the BBC web coverage of yesterday's semifinal between Raonic and Federer.  Raonic was 40-love up on his service, and according to the report, rather than follow the standard tactic of playing a safe second service that is almost certain to avoid a double fault, Raonic chose to attempt to serve aces on first and second serves, backing himself to be successful in one of his six opportunities.

So is this a good strategy when you are 40-love up on your service?  How much better than your second serve does your first serve have to be to make this tactic optimal?

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.