SPS Mathematics Blog

I found myself experiencing the strangest sense of déjà vu in the run-up to the second knockout round of the Hans Woyda competition this year. Once again, we were due to host Queen Elizabeth’s School for the quarter final, and once again, when finding a suitable date for the match I had to navigate Olympiads, Chess tournaments and Adavya’s return to the Romanian Master of Mathematics competition after remedy. With few options left before the 24 th  February deadline, we ended up being forced to schedule the match for the last day of half term, and to avoid keeping everyone from their holiday we came to the unorthodox conclusion that the best time to host the match would be during lunch break. So it was that the SPS team of Haoming (4 th ), Rafael (6 th ), Lucas (L8 th ) and Adavya (U8 th ) took their seats straight after period 4 to kick off the match. Last year we ended up winning against Queen Elizabeth’s, gaining an early lead and maintaining it until the end of the match, ...

This talk took us through one of the most counterintuitive ideas: that not every polynomial can be factorised in terms of its solutions and that the barrier for that is so low as to be the cubic. Starting with the quadratic, a base of knowledge we all know, Toller's exceptional talk then brought us to the cubic, explaining its quirks and how to reduce it to a form we all know, with a neat trick to turn it into a depressed cubic. He then used a clever method to split x into u+v and then made everything a quadratic. After following through the algebra, we transitioned into the quartic, having solved a few examples with both real and complex roots. For the quartic, the method was more complicated but doable all the same. Yet he posed the idea that the quintic cannot be solved with a formula. This took us into a tour of group theory, starting with what a group is and its symmetries, then moving to the symmetric group of n things. Then he posed a clever game where, knowing the roots of ...

The start of the spring term is always a bittersweet moment in the Hans Woyda calendar. The excitement of moving through to the knockout rounds is tempered by the knowledge that I will soon have to pick the final team of 4 from the superb squad of 12 pupils who took part in last term’s group stage matches. As ever, the selection trials were fiercely contested, and any of the 81 different possible teams would have stood us in good stead for the remainder of the competition, but eventually Haoming (4 th ), Rafael (6 th ), Lucas (L8 th ) and Adavya (U8 th ) emerged victorious. I had provided each of the team members with a set of practice questions for the journey which they eagerly worked through, swapping sheets with each other and comparing methods for the hardest questions, and at one point Adavya started loudly listing all of the squares from  16 2 to  29 2 to ensure he had them to hand in case he needed them later. The west London traffic and soporific drizzle had dampene...

This talk took us on a tour through one of the most quietly powerful equations in mathematics: the heat equation. On the surface, it’s “just” a formula describing how temperature spreads through space over time. In reality, it turns out to be a kind of mathematical celebrity, popping up everywhere from physics to finance. The talk began by unpacking what the equation says. Temperature at a point changes depending on how curved the temperature is around it – in other words, heat flows from hotter regions to colder ones, smoothing things out. Partial derivatives made a cameo appearance here, framed intuitively as rates of change and curvature rather than scary symbols. Then came some history. Long before the equation became standard, heat was thought of as a weightless fluid (“caloric”). That changed thanks to Joseph Fourier, whose radical claim that any function – even jagged ones – could be written as a sum of sines and cosines initially got his work rejected. Awkward, given how fo...