Numbers, Keiths and Keith numbers

Those of you who are Mike Leigh fans will know that the unglamorous reputation of the name, Keith, received a further kick when the unbelievably annoying protagonist of his early film,  Nuts in May, married to Candice Marie (almost as tedious) was given the Keith moniker. “K-e-e-ith” she would say in her dreary voice, as he indulged his officious enthusiasm for knowing the codes of individual trunk roads during their ill-fated camping holiday in Devon. I suspect that being a Keith was moderately unfashionable when I became one. Mike Leigh helped consolidate the process. When a Keith (say, this one) meets another, he will often share a moment or two of Keith kinship – along the lines of “How is it for you, being a Keith?” Sympathetic looks are exchanged amongst the fraternity.

Well, imagine my delight when, during the presentation being given by the Mathematics department to the Bedales  Schools’ Governors on Friday afternoon, I discover that there are things called Keith numbers. I have been so dazzled by the Italianate splendours of the Fibonacci sequence, not to mention the homely Grecian charm of Pythagoras and the wallowing of Archimedes in his bath that I have overlooked the genius of Keith numbers.  OK, I need to let slip now that the Keith whose eponymous numbers caused me such excitement has a surname which is Keith but author (Mike) Keith- can still count an achievement for the clan of Keiths.

Most of you will no doubt be familiar with Keith numbers but for those who aren’t, here goes – and thanks to Darran Kettle (Head of Blocks’ Maths) and Su Robinson (Head of Groups’ Maths) at Bedales Prep, Dunhurst, for the worksheet we were all given on Friday afternoon so that we could keep doing some Maths over the weekend.

Keith numbers have a property which is like this. To see if an n-digit number is a Keith number, write out the sequence that starts with the n-digits of the number; then, to get each new term, add the previous n terms.

Here is an example: 197 is a 3 digit (Keith) number, so we form the sequence:

1, 9, 7,     1+9+7= 17,   9+7+17=33,   7+17+33 = 57, 107, 197.

You will notice that we had to work out the last two, but this will help you appreciate the magic that leads to this unusual property whereby you start with 1, 9, 7 and end up with 197. So, now have a go at proving that 47 is a Keith number…

Right, smarty pants, you have done that, so now find all the other Keith numbers less than 100.  (Clue: there are five.)

Discovering things like this can be called recreational maths. How widely that term is used, I don’t know, but this chance encounter has got me thinking about Maths and Maths teaching: it seems to me the great feat of good Maths teaching is to show people that all Maths can be recreational – i.e. fun and useful. Since I started teaching, I have always tried to see as much good Maths teaching as possible. In my year’s teaching exchange amongst the lotus-eaters of California in Pebble Beach, I quickly learnt that there was a legendary Maths teacher by the name of Senuta. I went to see him in action. It was a brilliant revelation – students were laughing, learning, questioning and discovering. Senuta walked round the class helping people – with a smile on his face. The fear and paralysing solitariness of too many Maths classrooms was nowhere to be seen.

My father was a history graduate and started off teaching that subject, but early on in his teaching career started teaching Maths to some of the less quick Maths sets at a senior school. I remember him explaining to me that because Maths hadn’t come naturally to him at school but he had worked away at it and become quite competent, he felt in a good position to help those who didn’t find it easy to master it. My first bursar here, Bruce Moore, was taught by him and produced one of my father’s splendidly brief and spidery reports for me to see.

So, were reincarnation to be a possibility and were I able to express a preference, then learning how to make Maths as approachable as I now often see it taught would be up there on my preferences.