Fermat and Problem Solving

Back in tenth grade when we were just starting out learning algebra, my maths teacher set us homework that included this little pearl:

Prove that there are no whole number values for x, y and z that are solutions to that x3+ y3 = z3.

Yes, he really did.

For the uninitiated, this is an instance of Fermat’s ‘last theorem’. The brilliant mathematician Pierre de Fermat had in 1637 asserted that he had a marvellous proof for this (and for the generic xn+ yn = zn version where n is also a whole number) but he died before he was able to explain it.

Although Leonard Euler (another spectacular mathematical luminary) proved this for n = 3 and n = 4 in 1753, the general proof for all n eluded generations of mathematicians until the great Andrew Wiles achieved this in 1994. Simon Singh wrote a brilliant book about this entitled ‘Fermat’s Last Theorem’ and Andrew’s achievement featured in a BBC documentary.

Back to the homework – unsurprisingly, no one had completed the problem when we handed in our books the next day. And when our teacher revealed the history of the problem to us we just filed the information and the problem as a curiosity and forgot about it.

Recently, decades later, I decided to have another crack at it (the n= 3 version only).

This was not out of a belief that I could solve it but because I wanted to see if I could make any headway at all.

I suspected that I would be a tourist in territory that I would not be able to explore fully but I wanted to see if I would have sufficient resources to make it the trip worthwhile.

Overall, I am very happy with the results - I learned a fair amount in a short time.

My excursion off the beaten track brought back into focus various thoughts about problem solving in general.

Problem solving ability and the ability to successfully overcome challenges is often measured by a very harsh yardstick – by success or failure, absolute or relative, and in real life failure is not as cost free as failing to solve a maths problem; the stakes are higher.

Those who are successful in the given time and with given levels of resource generally are successful on two fronts:

1. Analysis of the data relating to the problem and the environment, formulating viable solution approaches to achieve the requirement in an acceptable time and manner;
2. Executing against plan, overcoming risks and seizing opportunities along the way.

These attributes are learned through sustained, earnest effort and will in time be superior to what they were at the outset.

So if you are undertaking a new challenge with a lot that is outside your comfort zone, how can you give due deference to the gap in your experience and in your default approach compared to what these would have been if you were seasoned in managing the task at hand?

Possibly through self awareness. This is something that one can consciously apply without needing much experience and without needing competence in a given field. It comprises simply making sure that the level at which you are appraising each step along the way is appropriate.

As an example, when building financial models or testing calculations or code, it is customary to sprinkle them with tests – to know how to check the outputs of your code against the expected outputs.

Often it is the people who are newer to the field who underestimate the value of such tests and checks, but this is something which does not take much learning to apply. So work out how you can check deviation from expected outcomes. When I was working through my Fermat attempt, I used a number of checks and reviews, so I knew I was able to confirm that I was falling short of the mark – I did not need someone to call me up on it.

In a real life scenario it would be reality that bites and this can be really unpleasant.

Of course, the ability to be successful at solving a problem or overcoming a challenge is not all about having a wise, sagacious approach. A lot of it is about approaching the challenge with real energy and intent. You need to be prepared to move fast and to make mistakes, provided that you can keep these small.

Trying my hand at Fermat was a good exercise because there was no cost for failure – it’s not like I was sky diving.

Looking back at my math teacher setting us the problem, I can see now that in some ways it was not an unfair one to set, depending on what he wanted us students to get from the exercise. Perhaps we could have learned some general principles of a more meticulous, more creative approach from the exercise.

When I tried it this time, my initial approach to it did actually overlap with the first steps taken by Euler, and these are not beyond high school pupils (coprime numbers, even vs odd, etc; ; you can read Euler's proof on this blog).

But Euler showed deductive leaps of genius (substitutions which are not very complicated but do leave you asking ‘how did you think of that?’), and was also able to apply techniques that I and most newbies will not think of.

So what changed in me that I was able to actually make it up the first few rungs of the problem now? The theoretical tools I had to hand had changed, but not the ones which mattered.

But the way in which I was able to appreciate them and to use them had changed, to some extent at least if not to the degree shown by Euler.

I viewed the exercise now as a puzzle which needed application of logic – not as a mechanical application of complex techniques. For me it was now more about using my brain than applying a series of rote learned techniques.

Overall, I think that the exercise for me refreshed my opinion of what it takes to deliver against problems and challenges. There are many more to come and I hope it serves me in good stead.