SPS Mathematics Blog

Tuesday was the Toller Prize group round. It was an exhibition of work done by L8th Further Mathematicians over the last two weeks. The aim of the competition is to create an 8-minute presentation on an interesting area of Mathematics and its applications. These were presented to 8 groups with the winner of each groups going through to the semi final. There is also a separate task to create a poster. The best of these will be displayed around the school next academic year. Well done to everyone who took part. These students made it to the semi-final: Ben Ettore Max Jake Kaysan Max Harris Harry It is great to see students from a range of sets making it through to the semi-final. This is being held today, where we find out which presentations will make it to the grand final.

This June, we welcomed students for our annual Maths Partnership Day, running across Thursday 18 and Friday 19 June. Spread across two cohorts and two days of activities, the event brought together a rich mix of hands-on challenges, deep mathematical thinking, and collaborative problem-solving. Both days kicked off with Puzzles and Problems during the arrival slot, setting the tone nicely for what was to follow. From there, students moved into four structured sessions across the day, covering a diverse range of topics. Thursday's highlights included an exploration of lattice methods and area, a perennial favourite in the form of Sticky Tape and Spaghetti (always a reliable test of structural intuition), and a session on Chomp and Other Games, which gave students the chance to think about combinatorial strategy. A Cypher Challenge in the Physics Corridor added a codebreaking flavour to the early afternoon. Friday offered a different mathematical menu. Students tackled Modular Ari...

Students at St Paul's first learned about the OCR Large Dataset in L8th. Today, however, Ms Fickling led a lecture for the U8th to refresh this knowledge with its applications for the A level statistics exam. The session started by defining the key differences between categories of the dataset such as Unitary and Metropolitan Authorities. Then exam questions were given and strategies for approaching them were discussed. An excellent session just in time for the A level exam (fancy dress was optional!)

Déjà vu was a feature of the last Hans Woyda match report, and fittingly enough it reared its head again in Tuesday’s Semi Final against Haberdashers’ Boys’ School. This time, we have to go back to the 2023-24 Semi Final, when we welcomed Habs into St Paul’s for an exceedingly close game. In that match, St Paul’s were either tied or in the lead throughout, until Habs edged in front for the first time in the penultimate question and held off our U8th on the last question to extend their lead even further, knocking us out of the competition with a final score of 53 – 49. Two significant changes gave me hope that history wasn’t about to repeat itself. First, we were the ones making the long journey to the outskirts of Watford this time, as hosting duties had fallen to Habs for this match. Second, none of the current SPS team of Haoming (4 th ), Rafael (6 th ), Lucas (L8 th ) and Adavya (U8 th ) had been involved in that tragic loss. As the voyage northwards came to an end and we made our ...

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...

To put a cap on a brilliant first term of MathsSoc, Yidong came up with a fantastic Christmas Quiz including a greatly enjoyed maths race. Adrenaline was flowing throughout this speed-based event, and while the difference was only made up of a few points, Team 1 (a name they accurately picked for themselves) took home a well-deserved win and a mountain of chocolates. A big thank you goes out to Yidong for entertaining us with this quiz, as well as all the people that have helped run and organise MathsSoc this term. It has been absolutely incredible, and I hope to see everyone there in 2026. -  Arié

It was another successful year for the students who qualified for the British Mathematical Olympiad Round 1 (BMO1). Of the 85 students who took part, 35 received merits, 25 received distinctions, and 7 students qualified for BMO2. Having 7 students qualify for BMO2 is a fantastic achievement, especially given that only around 100–120 students nationwide reach this stage each year. This is also the highest number of merits and distinctions the school has ever attained in BMO1. Well done to all the students who took part — your hard work is truly paying off! Here is a selection of questions from the six problems on the BMO1 2025 paper, courtesy of UKMT. The full paper and mark scheme are available here :

SPS held a maths partnership event with a variety of the partnership schools in the local area. The event included a series of UKMT Team Challenge problems where groups were composed of one student from each school. The event kicked off with a 10 question quick fire quiz, followed up by a maths crossword, called the crossnumber. The event finished with a unique maths relay, in which each group was split in half and the teams competed in finishing a 20 question quiz, where the questions had to be completed sequentially with one half doing odd numbered questions and the other half doing the even ones.  There were two competitions those with 3 team members and those with 4. The scores were very tight and so huge congratulations to the winning team in both category. The top team was outstanding, almost scoring almost full marks. Overall, it was a great day where everyone enjoyed themselves and had their fill of maths. - Shorta  

The final group stage match of this year’s Hans Woyda Competition took place against The London Oratory School at home. The remaining squad members of Ryan (4 th ), Misha (6 th ), Lucas (L8 th ) and Adavya (U8 th ) helped me arrange my classroom into the battle chevron formation, after which a tube delay on our opponent’s side offered up plenty of time for some warm-up questions. Luckily, we were due for an early start regardless, and as such we were still making good time after The London Oratory had managed to overcome the challenges of the Piccadilly line. With both teams settled into their seats we prepared to start the match that would determine whether or not we make it through to the main knockout competition next term or end up competing for one of the plates instead. We were off to a strong start on the starter questions, and although an order of magnitude slip and a challenging three-way HCF question prevented us from getting full marks, Ryan managed to best his opponent ...

  The talk “Behind the Scenes of Retirement Dreams” was given by Ms. French, a former teacher who became an actuary. She explained how actuaries use maths to help people plan for retirement and make sure pension schemes are fair and sustainable. Ms. French described different types of pensions, like Defined Benefit (DB) and Defined Contribution (DC), and showed how both employees and employers contribute to them. She also explained how actuaries calculate the value of future pension payments using interest rates and life expectancy. I found it really interesting how much maths and problem-solving goes into making sure people have enough money after they retire. Ms. French also shared how to get into actuarial science, mentioning online courses, podcasts, and professional exams. Her talk showed how a strong background in teaching and maths can lead to an exciting and meaningful career as an actuary. - Shlok

A match report by Yidong from Cloete house: The First Round & Semi Final : In the first round, Cloete knocked out Nilsson by 34 points to 26, despite the scores being tied with just four race questions left. Gilks narrowly edged out Harrison by 24 points to 22, a gap of just one question, whilst Langley and Blurton won against Stewart and Warner respectively. Cloete beat Langley 34-26 in a replay of the semi-final that had gone the other way last year, and Blurton were victorious against Gilks to seal their spot in the final. The Final : The usual buzz spread throughout the school on the day of the final, and a large queue had even formed outside room 220 before the match! Unfortunately, most filtered into Chess Club in room 221, but a large crowd of four people came to watch, nevertheless. The line-ups for the match were Aryan (4th), Yidong (5th), Lucien (6th), Ashok (L8th) and Terry (U8th) of Cloete House, against Blurton’s team of Haoming (4th), Aron (5th) Kevin (6th), Richard (...

At the end of last Friday 7 th   November, I met the second iteration of the SPS Hans Woyda team in the atrium as we prepared for the unusually short journey to our first away match of the season. Unfortunately this meant that there was barely any time for me to pester Haoming (4 th ), Ehan (6 th ), Andrew (L8 th ) and William (U8 th ) with warmup questions on the trip over, and before I had even managed to get through three sections’ worth we had arrived at Harrodian. After a brief walk through the site to the prepared classroom (during which I admitted, to my embarrassment, that after eight years at St Paul’s I had never stopped to wonder where the name of our neighbouring school comes from!) the team took their seats and the match got underway. Both teams had a strong start, getting three out of the first four questions correct, with a counting slip on our part made up for by a summation slip on theirs. The next four starter questions caused a few more problems on both sides, no...

We were given a riveting talk by an OP from 10 Downing Street, Mr Conrad Bannister. MathSoc saw a record-breaking number of questions asked, especially about some of the more sensitive areas of government policy, as well as insight on the use of Maths and Statistics in the "Misfits and Weirdos" Club of 10 DS. It was an incredibly engaging and unique experience as we learnt about how concepts such as Winters Triple Smoothings and prisons were linked, truly a once in a lifetime opportunity! -  By Arié

On Tuesday 30 th  September, we welcomed Emanuel School for the first official match of the 2025-26  Hans   Woyda  competition. The SPS team of Aryan (4 th ), Rafael (6 th ), Zicheng (L8 th ) and Aman (U8 th ) were already warmed up from the friendly against SPGS, and they were further buoyed by the support of Adavya, William and Andrew who came to spectate (along with the inimitable Dr Stoyanov). The team assisted me in arranging the classroom into my famous Battle Chevron formation (patent pending) before taking their seats to await their opponents. Once the Emanuel team had arrived and settled into their places we had a momentary pause while one of the tables was adjusted to avoid a stream of sunlight going directly into their Year 12’s eyes (which I insisted was not a deliberate act of sabotage!), after which I projected the slides, hushed the audience and handed out the first set of questions.   SPS hit the ground running with a perfect set of answers to th...

House Mathematics started this week and it already looks like a fiercely fought contest. With house points on the line, who can blame them! It is true that Paulines love competition but House Mathematics is certainly one of the highlights of the year. The competition is formed of three rounds, the first round, semi finals and finals. In these, a student from each year group partakes in quickfire questions, similar to those of the Hans Woyda, in a head to head battle with another house. All 8 houses played in the first round:  Langley 32 Stewart 26 Cloete 34 Nilsson 26 Gilks 24 Harrison 22 Blurton 34 Warner 26 These were hosted in individual classrooms. I personal thought what was most impressive, was the crowd of supporters who came to cheer on their houses. In some rooms it was hard to find a place to sit. Well done to all who were involved and a special thanks to the 8 Maths teachers who gave up their time to run the first round of the competition. The Semi Finals are up ne...

This week, Louis gave a fascinating talk to the Maths Society all about fractals; those strange, endlessly repeating patterns that you find in nature. Whether you’re zooming into a fern, a snowflake, or a Sierpiński triangle, the same shapes keep appearing. Louis explained that fractals were originally developed to help describe rough, irregular shapes in nature – things that traditional geometry couldn’t handle. He introduced us to the Hausdorff dimension, a tool used to measure the 'size' of irregular shapes. Unlike traditional dimensions, it exists on a continuous scale. If you apply the Hausdorff measure to a shape for all dimensions lower than its true Hausdorff dimension, the result becomes infinite, a sign of its complexity. He demonstrated this using the triangle fractal. We also looked at the box-counting dimension, a more practical method for measuring how a shape scales as you zoom in. This helped us apply ideas to the Sierpiński triangle and understand how a for...

This Friday at the weekly meeting of MathsSoc, the members participated in a Maths quiz. Different teams competed against each other, tackling questions from an inconspicuous Japanese textbook, as well as IMO preparation papers. Over twenty minutes, teams divided the questions and attempted to conquer. Some rushed straight into the algebra-heavy sections, confident that their speed would pay off, while others carefully picked apart the geometry problems, preferring precision over pace. The atmosphere quickly became one of focused intensity: pages filled with half-finished calculations, muttered debates over which substitution to try next, and the occasional groan when an apparently simple problem turned out to have a hidden twist. Although the competition was light-hearted, there was a real sense of excitement in the room. The questions were unlike those found in the average textbook: they required creativity as much as method, forcing participants to think beyond routine exam prac...