Dara's Big Mistake

I've just watched the latest (and final) episode of Dara O'Briain's School of Hard Sums, with Andi Osho again demonstrating a remarkable ability to get straight to the solution by what appeared to be inspired guesswork.  If I were doing undergraduate groupwork, I'd want her in my team!

SPOILER ALERT: If you're planning to watch this programme (it's repeated over the next few days) you may not want to read this yet as I give some hints as to what happens (though I don't give away the maths)..

The final problem involves runners going round a race-track at different speeds and asks how long it is after the start before all three runners are again together.  Dara produces reams of incomprehensible algebra to get an answer in which he doesn't have full confidence - he presents it reluctantly, convinced that he has gone wrong.  His delight when he finds his answer is correct was the highlight of the series.

I think Dara got it badly wrong.  He didn't check his answer!  His algebra was complex but it gave a solution: it would have taken only a moment to see where the three runners were at the time in question and establish that they were all together.  ((OK, only a partial check: it might not have been the first time, but it would at least have shown that his answer was plausible.)

In any maths problem, if you can check your answer is valid, it gives you confidence in your solution.  I worked for many years in the mathematical modelling of industrial plant.  There were essentially two kinds of problems.  Engineers wanted either to find the steady state of the model - the normal plant operating condition when nothing is happening to disturb it - or to find out what happened dynamically when some disturbance was applied (say if a pipe broke and the cooling system failed).

The models were essentially defined by a set of differential equations.  Finding the steady state involved finding values such that all the derivatives were zero - that is, finding a zero of a system of a couple of thousand non-linear equations.  Not easy  - sometimes the equation-solving algorithms failed and we were reduced to starting from a random initial state and running the model for a very long time in the hope that it would naturally arrive at the steady state.  But if the algorithm gave a solution, you could test it by plugging the numbers into the equations and making sure they all gave you zero.

But running dynamically involved numerical integration, and there is no simple way to check your answer!  Since numerical integration can give completely incorrect results (try integrating dx/dt = -1000x by Euler's method with step-size 0.01, say, to see output which is not only quantitatively but also qualitatively completely wrong) this is a serious problem.  (There are ways one establishes confidence in the output from numerical integration, but they aren't as simple as plugging the answers in to the equations to check that you get zero!)

Dara's algebra may or may not have been right.  But if he had checked where the runners were after the time he had deduced, he could have had a lot more confidence in his calculations.  Of course, it wouldn't have made such good TV.