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.