After school on Tuesday 5th March, while other classrooms throughout St Paul’s echoed with the encouragement of the 5th Form Parents’ Evening, room 228 played host to the semi-final of the 2023-24 Hans Woyda Mathematics Competition.
It was with great delight that I read the email announcing that we would be on home turf for our match against Haberdashers’ Aske’s Boys’ School; the final is always held at a neutral venue, so regardless of the outcome this would be my last opportunity to arrange my desks into the patented Battle Chevron formation for this academic year. With the room in order and the SPS team of Zane Kumar, Anango Prabhat, Eason Shao and Benjamin Atkinson ready to take their seats, I headed down to reception to greet the opposing side and welcome them into the school. I was caught by surprise when, after introducing myself, their Year 12 remarked that he was a fan of my blog; having never written a blog it eventually transpired that he had stumbled across my Hans Woyda match reports when researching the competition online! However, I couldn’t be sure that this wasn’t all a ruse designed to stoke my ego and distract me from the task at hand, so I refused to contemplate the implications of my newfound fame and brought my focus back to the semi-final itself.
It was a shaky start from both sides on the first four questions, but some handy surds work from Ben gave us an early advantage which we maintained throughout the rest of the starter section. Next up there were some fiddly geometry questions to contend with, and unfortunately a grazing goat tethered to a wonky railing got the better of Zane and Anango and allowed HABS to equalise. The next few questions were closely related to Moser’s circle problem, and while both sides seemed familiar with the context a slip by the HABS Year 11 gave Anango the opportunity to reclaim the lead for SPS. The team question was up next, and was based upon the surprising result that every positive integer has a multiple which is a Fibonacci number. Both sides ploughed through the required list of examples at speed, but unfortunately the SPS side made a few small mistakes in their final answers (including missing the fact that 13 is already a Fibonacci number itself!). However, even though HABS won that section it wasn’t enough to catch up completely and they were still trailing by a single point as we entered the sandwich break. The second half resumed with a number-crunching set of calculator questions on cyclic quadrilaterals. These proved straightforward for all involved, leaving the margin between the two teams at a single point going into the penultimate section. Here we had an increasingly challenging set of questions on the factorial function, where another slip by the HABS Year 11 gave us the opportunity to extend our lead from one point to four.
With two points per correct answer in the race section we had a buffer of exactly two questions going into the final stage of the match. This would be nerve-wracking at the best of times as such an upset is commonplace in this competition, but the tension was heightened further still by a quick parity check. Parity, for those unfamiliar with the mathematical usage of the term, refers to whether a given number is even or odd. If the difference in scores is even going into the race then it is theoretically possible to end with a perfect tie, and this far through the competition that would necessitate the use of an additional round of tie-breaker questions. This is something I have never previously experienced in a Hans Woyda match, and given the effect a normal race section has on my heartrate I have little doubt that the stress of having to go through a tie-breaker would trim years off my life. The closeness of the match so far suggested that a tie was very much on the cards, and as such the stakes felt especially high as the room went quiet for what were at least notionally the last eight questions of the match.
The first question was a classic bit of misdirection, where the right method wasn’t to embark on the lengthy calculation suggested but instead to spot a nifty shortcut and write the answer down immediately. Unfortunately, HABS noticed the trick first, and so the gap was narrowed to two points. The next question concerned bounds and rounding, and both sides succumbed to the pressure, failing to give correct answers before the sixty seconds had elapsed. Up next were the two Year 12s and a question that tested their knowledge of prime factors. Both put their hands up almost simultaneously, but both had also failed to spot the common factor of 37 that needed to be cancelled in their answers. I was ready to move on to question 4, but suddenly there was an interjection from my HABS counterpart, who up until now had been keeping score and imposing the time limits from the back of the classroom. Since Eason had offered his incorrect answer first, he suggested that his opposite number should be able to use the entirety of the remaining time to check his own answer before submitting it to scrutiny. He seemed well-versed in the minutiae of the Hans Woyda rulebook, and with the benefit of hindsight I’m sure that he was correct, but with the match on the line I couldn’t help but feel personally affronted by this suggestion. As the seconds ticked down I could sense my indignity rising, but luckily I was saved from an unprofessional outburst when the HABS pupil failed to simplify the fraction in time. A symmetric integral of an even function rounded out the first set of race questions, and a sign error from the HABS side gave Ben a clear run to methodically work through the required steps and find the correct answer.
Once again we were two questions ahead, with each year group facing one last question before the end of the race. The Year 9s had a relative ages word problem to solve, and while both sides furiously transcribed the problem into algebra it was HABS that landed on the correct answer first. With the buffer reduced to one question, the Year 11s had to wrestle with some unit conversion and ratio simplification; Anango answered first, but he was off by a factor of ten, and while the HABS Year 11 almost made the same mistake he crossed out the extra zero just before submitting his answer. With the two teams now neck and neck, the Year 12s tackled an unexpected bit of geometry, converting a sphere into a cylinder and finding the resulting cross-sectional area. HABS beat us to the punch yet again, and while I rather desperately checked whether or not we were accepting equivalent answers (as he had forgotten to extract the extra factor of four from the square root) I knew deep down that we would. For the first time this match, HABS had taken the lead, and with one question left our only hope was to go for the tie-breaker. Everyone collectively held their breath as the Year 13 question appeared, and as both sides started to work on this final trigonometric equation it felt like time stood still. Ben diligently worked on the algebraic approach, using the double angle formulae to expand the left hand side and setting himself on a path that would eventually lead to a quartic, but my heart sank as I realised that the HABS Year 13 had spotted the much faster approach using the symmetry of the cosine function, and as his hand went up all I could do was hope that he had made an arithmetic slip along the way. There it was;2π/5, and with it our Hans Woyda journey had come to an end, with a final score of 49 – 53.
It was an extremely impressive performance against what unfortunately turned out to be an equally impressive opposition, and they should be immensely proud of how incredibly well they have done throughout the competition so far. As ever, I have attached the questions, and below you can try your hand at the only geometry question that evaded both teams; could you solve it in 90 seconds? The competition will pick back up for St Paul’s in October, and I look forward to resuming my coverage then.
Written by Samuel Cullen-Hewitt, Teacher of Mathematics
