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A Year 12 team have been presented with their trophy after seeing off competition from 1,200 teams from Slovenia to Singapore to triumph in the international Ritangle Mathematics competition.

The five-strong team submitted the best answer to the final question, thus winning them the title in the competition run by education charity MEI (Mathematics, Education, Innovation).

Competition organiser Bernard Murphy, a Maths Education Support Specialist for MEI, visited the School to present the trophy and a hamper in a special assembly.

Headmaster Neil Enright said: “My congratulations go to the five on a splendid victory in a competition that is designed to stretch young people’s Mathematics knowledge to the limit.”

Ritangle is a free competition for teams of students aged 16-18 studying either A-level Mathematics or equivalent qualifications. Fifteen QE teams took part in the competition.

Harik Sodhi, captain of the winning team, said: “My favourite part of Ritangle was collaborating with friends on solving an interesting Maths problem. I definitely enjoyed the final question the most as it was a very open question and required lots of steps and a combination of Python and Maths and Excel programming, which made it very fun.”

His fellow team members were Anshul Nema, Koustuv Bhowmick, Shreyas Mone, Joel Swedensky.

For the first four weeks, one question was released every Monday. For the following five weeks, three questions were released, on Mondays, Wednesdays and Fridays. Every correct answer revealed a piece of information that helped solve the final task.

Shreyas said he particularly enjoyed the period in which the number of questions started to increase each week.

The final problem, involving the pairings for a ‘jamboree’-format chess tournament, was released on a Tuesday evening, with even those who set it not knowing what the best answer would be.

Mathematics teacher and Head of Academic Administration Wendy Fung said: “It essentially involved finding the smallest possible number that met a set of criteria. The question setters came up with an answer of 56, which they expected to be beaten by someone, but not by a great deal.

“The first correct answers submitted were above 100. On the Thursday afternoon, the winning QE team submitted the figure of 0.16. MEI initially assumed it must be wrong, but it proved to be correct once checked, and was never beaten – though some other teams got quite close. It was a remarkable performance by the boys, who have vowed to try and win again in 2024!”

The winning team, The Flex Angles, worked independently to solve the questions and needed to get to grips with using various forms of technology, from spreadsheets to graphing software.

Joel particularly enjoyed the early stages of Ritangle, when large numbers of boys at QE were taking part. “Racing to complete each question first after it was released was its own mini-competition.”

Anshul added: “Beside the final question, which will always be special to me, I really enjoyed all the prior questions in which I could use my technical expertise in plotting advance or interactive graphs to optimise functions. I’d have to say that my favourite part was at the very end, coming back to my computer after taking a break from the frustration to see my code announcing that a solution had been found, and rushing to tell Harik and the team.”

Of all the teams who entered, only 153 submitted correct answers to the final question. A second Year 12 QE team – The Left Angles, captained by Uday Dash – was among these and was therefore listed among the “commendable” teams by MEI. Uday’s fellow team members were: Saim Khan; Kovid Gothi; Adyan Shahid; Vijay Lehto and Dinuk Dissanayake.

QE expanded its series of popular challenges for local primary school children this year, adding a humanities day to the programme.

The events, which are part of QE’s partnerships work with the local community, are aimed at giving Year 5 girls and boys an early taste of secondary school education.

Headmaster Neil Enright said: “We are pleased to support local primary schools in this way.

“I know that our staff and pupils involved in running these enrichment activities greatly enjoy the opportunity to meet the visiting children.”

The first of the three days was the ever-popular Primary Forensics Workshop. The visitors were tasked with completing a number of experiments and analyses to work out who had murdered the Headmaster!

There were stations where the children could undertake: hair and fibre analysis using a microscope; fingerprint analysis, and blood spatter analysis (with a blood substitute).

The pupils worked to solve the ‘crime’, using the evidence they collected to build a case, while also weighing up the respective motives of the suspects.

Boys from Year 12 helped staff run this workshop, engaging with the children at each station.

In the Maths and English Challenge, the girls and boys had to solve a series of games and puzzles that ranged from a cross-number round to a session looking at composing and performing poetry.

There was a focus on teamwork and collaboration. Each team had the support of a QE Year 7 pupil.

Special plaudits went to Foulds School pupils, who achieved a near-clean sweep of the prizes, having impressed across the various disciplines on the day.

The new humanities day hosted by the History, Geography and Economics departments comprised two separate activities.

Firstly, teams were given the challenge of designing a castle on paper. They had to base their design on a certain set of criteria and follow a budget, requiring them to decide which features they wanted to prioritise.

They then faced a number of scenarios, presenting both challenges and opportunities for their fortifications. Could their castle and kingdom survive?

“This was a way of exploring history and strategy in a fun and engaging way,” said Mr Enright. “The Year 5 pupils also had to adapt their plans as the scenarios unfolded, which meant teams had to communicate well and quickly make decisions.”

There was then a Sustainability Challenge run jointly by Geography and Economics. The children had to work in groups and devise a sustainable product. They designed their product, chose a logo and decided on their target market. Then each group presented to the other children in attendance. Among the ideas generated were: a mobile phone where the case is a solar panel and charges the phone, and a ‘plastic’ bottle where the bottle itself is biodegradable.

“Our staff were really impressed with the confidence shown by the children in their presentations and by the creativity they brought to bear in designing their products,” said the Headmaster.

Participating Barnet primary schools this year included: Underhill, Whitings Hill, Christchurch, and Foulds.

Four Year 10 boys won every round when they took on other regional schools in the Maths Feast competition.

After being selected to represent the School in the event, the four emerged with a perfect score of 121 out of 121.

Mathematics teacher Kirtan Shah said: “This was the first time I’ve seen full marks in every round of the Maths Feast competition since I started working here in 2018. So they did really, really well: definitely something to be proud of!

“They worked so well together as a team; by building on each other’s arguments, they were able to successfully reach sensible solutions to some challenging problems. They knew what each member’s strengths were and that really helped them gain their clean sweep.”

Hadi Al-Esia, Kovid Gothi, Saim Khan and Shreyaas Sandeep travelled to St Dominic’s Sixth Form College in Harrow on the Hill for the competition run by Advanced Mathematics Support Programme – a Government-funded national initiative.

They faced four rounds: team captain Hadi said each involved “intriguing puzzles that stretched our knowledge and problem-solving skills”.

The rounds were as follows:

1. What No Words? All teams were given a series of problems to solve, with the catch that they were only given diagrams. Not only did they have to work out the answer; they first had to work out what the question was!
2. Four for Forty: Students were given four long problems, including logic puzzles, which all required outside-the-box problem-solving, including logic puzzles. “They were able to deftly negotiate this round by each member of the team taking the problem which suited their strengths the most,” said Mr Shah.
3. Card Sort: Competitors had to reimagine every 3D shape (such as cubes) to try to unravel the shortest way to pass through or over them. “This was by far the most challenging round for the team to tackle as it involved a new dimension of geometry for them – a combination of Pythagoras and 3D visualisation,” Mr Shah added. “They finished the round with less than 20 seconds to spare.” The team’s favourite problem came from this round (see picture right): competitors were asked to calculate the distance from A to B if the net [what a 3D shape looks like if opened out flat] were open for the cone.Saim said: I particularly enjoyed the card sort round – trying to reimagine and visualise the shapes in a new way was challenging but immensely rewarding too!”
4. Four in a Row: A relay round, with teams splitting into two pairs to solve two separate sets of questions. “Our boys were able to comfortably finish the round, with eight minutes to spare,” said Mr Shah.

Hadi said: “I’m proud of our teamwork and the dedication we showed on the day,” while his teammate Saim added: “The Maths Feast was a fantastic opportunity; the problem-solving and lateral thinking the rounds called for was a refreshing invigorating experience.”

Year 12’s Shankar Vallinayagam took his place on the stage alongside professional mathematicians as a speaker at the 2022 Maths Fest.

Shankar was among 45 Year 12 mathematicians from QE to attend the annual series of lectures at The Royal Institution in London.

He was selected as a speaker after his video submission for a related Mathematics competition – the Maths Slam – was picked as one of the winning entries by the judges. He was one of four winners who gave presentations on the day.

Mathematics teacher Kirtan Shah said: “I know that many of our boys not only enjoyed the day, but also really relished the opportunity to learn about fascinating aspects of maths and its applications in the real world. As one of our students, Haipei Jiang, put it afterwards ‘It was great to be in an environment where so many other students appreciated really cool maths.’”

As in previous years, the day was chaired by Mathematics YouTuber and ‘stand up mathematician’ Matt Parker.

The day began with Professor Jennifer Rogers, Vice President for Statistical Research and Consultancy at PHASTAR, the London-based international biometrics contract research organisation, giving a talk on Stats to Save the World. As the lead statistician on a treatment trial for Covid-19, she  explained the importance of sample size for clinical trials and the factors which help to determine how large a sample size should be.

Next was internet mathematician and public speaker James Grime, who talked about hidden Mathematics behind the digital world, from looking at how wifi signals are communicated to how cryptanalysis was used to break codes from Germany’s Lorenz cipher machines during World War II.

Host Matt Parker explained that our brains naturally think of numbers using a logarithmic scale, yet the modern world uses a linear system of numbers – which can easily confuse people in understanding how big large numbers actually are.

Puzzle expert and author Rob Eastaway looked at ‘fairness’ and ‘guilt’, telling the audience that chimpanzees have been seen to grasp the concept of fairness, refusing a treat if they felt they were receiving preferential treatment over their fellow chimps.

Mathematics teacher and examinations expert Nicole Cozens shared her top five tips for exam success from her experience of marking papers for 15 years, starting with: Always quote the formula first when using it in a question – this is to show the examiner that you know the formula, even if you end up making a mistake in how you use it.

Ben Sparks, musician and star of the educational YouTube channel, Numberphile, ended the day by explaining the Mathematics of the notes in an octave and sharing how sine waves, trigonometry and complex numbers are used to make noise-cancelling headphones work.

Shankar’s short talk was entitled The Alexander Horned Sphere: he came across the sphere, an object found in the branch of Mathematics known as topology, during his research for his Extended Project Qualification (EPQ) dissertation. Mr Shah said: “His talk interested the audience and got them thinking about why we shouldn’t say something is ‘obviously true’ in maths without proof.”

During breaks, the boys visited the event’s Maths Village, where they could enjoy mini-mathematical activities and meet people who use Mathematics every day at university and in commerce.

The boys were accompanied by Mr Shah and his fellow Mathematics teachers, Deljoo Mahdmina and Heena Haq.

Reflecting on the Maths Fest afterwards, pupil Rajveer Mukherjee said: “I particularly enjoyed it as although the talks were incredibly interesting, they also proved to be accessible to all, while leaving room for further research into the topics.” For his part, Abir Mohammed loved the opportunity to meet “renowned maths celebrities”.

Author, scientist and QE parent Dr Ananyo Bhattacharya gave a talk to senior pupils on his book about John von Neumann, the brilliant Hungarian-American polymath who made breakthroughs in fields ranging from nuclear energy to economics.

Dr Bhattacharya’s book, entitled The Man from the Future: The Visionary Life of John von Neumann, was named a Financial Times and Times Literary Supplement Book of the Year in 2021.

His lunchtime talk to A-level Mathematics, Physics and Economics students explored how von Neumann’s advances in mathematics 70–80 years ago continue to inform the science of today.

Headmaster Neil Enright said: “We are grateful to Dr Bhattacharya, as a QE parent, for coming in to School to share his expertise and to inspire our senior boys. It is great that we can draw upon different constituencies within the Elizabethan community, including parents and alumni, to enhance the educational experience offered here.”

Dr Bhattacharya, whose son, Callistus, is in Year 7, is a science writer who has worked for The Economist and Nature, the weekly multi-disciplinary scientific journal. Prior to that, he worked as a medical researcher at the Burnham Institute in San Diego. He has a degree in Physics from Oxford and a PhD, also in Physics, from Imperial College London.

The subject of his book, von Neumann, was born in 1903 to a wealthy Jewish family in Budapest. A child prodigy, he had published two major mathematical papers by the age of 19.

After an early career in German academia during the late 1920s, he took up an invitation to Princeton University in October 1929, becoming a naturalised citizen of the USA in 1937.

In a life of only 53 years – he died of cancer in February 1957 – he made major contributions in subjects including mathematics, physics, economics, computing and statistics.

During World War II, he worked on the Manhattan Project – the research and development that produced the first nuclear weapons – and after the war, he served on the General Advisory Committee of the United States Atomic Energy Commission.

In his talk, Dr Bhattacharya mentioned the Manhattan Project as well as, inter alia, von Neumann’s contributions to set theory, game theory and the development of the first programmable digital computer.

Head of Library Services Jenni Blackford said: “Dr Bhattacharya delivered a friendly, accessible and vastly informative talk about the life and accomplishments of von Neumann.”

A seven-strong team from Harrisons’ House won a Year 9 careers competition with their carefully thought-through design for a ‘smart’ water bottle that would use UV light to kill bacteria, viruses and fungi.

Team 4’s Water UVC bottle could thus benefit many millions across the developing world without access to safe drinking water, the boys explained in their richly illustrated, 31-page PowerPoint presentation. They even included an option for the UV lamp to be solar-powered to make the bottle viable for people who could not afford mains electricity.

The aim of the activity was to encourage boys both to develop their entrepreneurial skills and to make use of Science, Mathematics and Technology as they prepare to make their subject option choices.

The teams were given a choice of four briefs to work on, with Team 4 choosing the fourth  – “design and make a ‘smart’ water bottle that has at least one other function”.

Making the announcement that Team 4 had won, Assistant Head (Pupil Progress) Sarah Westcott said: “During last term’s lockdown, our usual face-to-face careers activities for Year 9 in this important period of their School careers had to be reimagined. We amended our plans so that boys could work from home, while still developing important work-related skills such as creativity, teamwork, independence and the ability to communicate their ideas.”

Dr Westcott judged the entries together with Head of Year 9 Sean Kelly, who is a Technology teacher.

“Both Mr Kelly and I loved the creativity and simplicity behind the winning design: all members of the team worked equally to realise the design brief and develop the marketing materials – which included a website!”

The boys’ PowerPoint presentation featured hand-drawn illustrations, as well as photography and computer graphics.

After an introduction, the presentation set out the whole process of turning the stainless steel bottle into a mass-produced product, with major sections entitled Design and Creativity, Manufacturing, Marketing and Pecuniary Matters (finance).

• Team 4 comprised: Shivam Vyas; Rohan Varia; Jenarth Thavapalan; Manthan Thakkar; Shrey Tater; Abyan Shah and Shreyaas Sandeep.

Year 13’s James Tan sealed his long and glittering record of success in Mathematics competitions at QE with a perfect score in this year’s Senior Maths Challenge (SMC).

He was one of nine Sixth Form mathematicians who performed so strongly in the challenge that they qualified for the élite British Mathematical Olympiad.

James’s tally of 125 out of 125 secured him the Best in School title, while Abhinav Santhiramohan, with a score of 112 out of 125 was Best in Year 12. To qualify for the Olympiad, candidates had to score at least 108 points.

Assistant Head of Mathematics Wendy Fung said: “James has scored perfect, or near-perfect, marks in every Maths Challenge he has sat, from Year 7 to Year 13. He has done phenomenally well throughout his School career and is so unassuming about his successes.”

In addition to the Olympiad successes, a further 29 boys qualified for the challenge’s other follow-on round, the Senior Kangaroo, which required a score of at least 91 points.

A total of 136 QE sixth-formers sat the challenge, which involved answering 25 multiple-choice questions in 90 minutes.

The top 40% of SMC entrants nationally in the country receive certificates with gold, silver and bronze awarded in the ratio of 1:2:3. At QE, however, there were 38 gold certificates, 65 silver and 22 bronze, which means that 92% of the School’s participants gained certificates.

“We are very pleased with the boys’ success at the SMC,” Miss Fung. “The challenge provides an opportunity for our senior boys to hone their problem-solving skills with fun, yet challenging, questions, and we are grateful to the UK Maths Trust for providing these opportunities.  Many congratulations to Years 12 & 13 – we look forward to receiving the Olympiad and Kangaroo results in due course.”

She added that Abhinav had said that he particularly enjoyed solving the following question in the challenge (answer below):

• Question: Two congruent pentagons are each formed by removing a right-angled isosceles triangle from a square of side-length 1.
  The two pentagons are then fitted together as shown. What is the length of the perimeter of the octagon formed?
  A: 4
  B : 4 + 2 √ 2
  C: 5
  D: 6 − 2 √ 2
  E: 6

• Answer: E: 6
  Explanation: The perimeter of the octagon is made from four long sides, two medium-length sides and two short sides. The long sides are given to be of length 1. The medium-length sides have length 1 √ 2 , using Pythagoras’ Theorem on the right-angled triangle which was removed from the original square. Therefore the length of each short side is 1 − 1 √ 2 . In total the perimeter has length 4 × 1 + 2 × 1 √ 2 + 2 × (1 − 1 √ 2 ) = 6.

One of the UK’s leading mathematicians explained to QE’s Year 11 how Mathematics is at the forefront of the battle against the coronavirus.

In a special lecture delivered via Zoom, Chris Budd, Professor of Applied Mathematics at Bath, first explained to the whole year group what mathematical modelling is, with contributions also coming from a number of his PhD students.  In a highly illustrated presentation, he then set out modelling’s crucial role in determining the best strategy for fighting the pandemic, even drilling down into issues such as how shopping can be made safer in a pandemic.

Assistant Head of Mathematics Wendy Fung said: “This was a detailed look at how Mathematics has been, and continues to be, at the heart of tackling the biggest national and international issue of our day. I know that the boys found the presentation engaging and enjoyed the opportunity to considerably deepen their understanding.”

The focus on mathematical modelling struck a chord with many of the Year 11 audience, including Theo Mama-Kahn, who enjoyed discovering “an area I haven’t learnt about before. I liked how he showed us the real applications of the theory he was talking about.”

In addition to his position at the University of Bath, Chris Budd is Professor of Mathematics at the Royal Institution of Great Britain and Director of Knowledge Exchange for The Bath Institute for Mathematical Innovation – a role which involves him in finding innovative ways of applying Mathematics to real-world problems.

He was one of the authors of the Vorderman 2011 report on the current state and future of Mathematics education in the UK.

And, said Miss Fung, “Professor Budd was also the chair of the UK Mathematics Trust from 2016–2019 and, as such, has signed all of the many, many Maths Challenge certificates received by QE pupils every year.”

Professor Budd was awarded an OBE in the Queen’s Birthday Honours List in 2015 for “services to science and maths education”. In lieu of charging a fee to deliver the talk for the QE boys, he requested that a donation be made to Maths World UK – a charity aiming to establish the UK’s first national Mathematics discovery centre.

The professor started his talk by revealing that, although not an Elizabethan, he had been born in Friern Barnet and had moved to Harrow Weald at primary school age, before going up to Cambridge to read Mathematics.

He moved to the University of Bath in 1995 and has remained there ever since.

In introducing the idea of modelling, he explained how a mathematical model should balance being simple enough to analyse with being complex enough to be realistic: he quoted Einstein, who said: “A model should be as simple as possible, and no simpler”.

Next, Professor Budd showed how a model could be used to determine whether it is possible to save a dog from a speeding car.

For this, he took boys through the modelling cycle, asking:

1. What are the variables?
2. Which formulae can you make/use?
3. Can you make a prediction using this information
4. How can you make the model more realistic?

One of the boys watching, Aran Ismail, said later: “I enjoyed the relation of the lecture to motor vehicles and the ability of maths to be used to help calculate braking distances.”

Professor Budd went on to explain how modelling has been used in the case of the Covid pandemic. There are three basic questions, he said. Firstly, how will the epidemic grow if the authorities do nothing (which was the case for the 1918 Spanish ‘flu epidemic)? Secondly, how can we stop the number of cases growing? And third, how should we change our behaviour to keep safe?

He explained that the ‘r number’ often mentioned in the media is found by considering the rate of transmission and the size of the population who are susceptible.

In order to reduce the r number, there are three strategies – achieving herd immunity, instigating lockdown and using a vaccine.

Herd immunity relies on the r number eventually decreasing; lockdowns spread out the rate at which people get infected, which helps the NHS to cope with cases, and a vaccine will reduce the number of people who are susceptible, but will need 60% of the population to be vaccinated in order to be effective.

Modifying behaviour patterns is simply the most effective way to prevent the spread of Covid, which is why wearing masks, staying in closed bubbles and keeping 2m away from other people are the strategies the Government has been promoting most, he said.

The lecture also demonstrated mathematically why it is usually safer in terms of virus transmission risk for supermarkets to allow ‘random’ shopping, rather than implementing guided movement, such as a one-way system – a conclusion which certainly caught the attention of pupil Abir Mohammed. “It was very insightful to learn about the various ways mathematical models can help simplify difficult situations and come up with solutions – I would have definitely thought having a one-way system is best,” he said.

The PhD students taking part related a little of their own experiences as young mathematicians and explained how they use modelling in everything from climate modelling to investigating the solar system to modelling traffic flow.

Pupil Sid Dutta said: “I was very interested in finding out about the lives of the students who were studying Mathematics PhDs and their daily routine.”

• More information about Professor Budd’s work, publications and interests may be found on his web page.

Year 11 pupil Ashwin Sridhar has crowned a series of wins in competitions he entered during lockdown with outstanding international success in the prestigious Microsoft Imagine Cup Junior.

He was named among just three winners from across the vast EMEA (Europe, Middle East and Africa) area after designing an artificial intelligence-powered device to help tackle the crisis in care for the elderly. Ashwin was one of only nine winners across the whole world and was the sole UK winner.

The same design also brought him success in another competition – the Connect the Community: Design Challenge – where it was named among the 10 winning entries in phase 1 of the challenge.

Congratulating him, QE’s Head of Technology Michael Noonan said: “Ashwin is an outstanding Technology student who has had a tremendous year. Despite the challenges of the school closure, or perhaps even taking advantage of them, he threw himself into many competitions using his vast technological experience. He was successful in eight competitions on a local, national and, with his latest win, international level. He should be extremely proud of his achievements this year, and he undoubtedly has a bright future ahead of him!”

Like the Imagine Cup, Microsoft’s sister competition for older students, Imagine Cup Junior provides those aged 13 to 18 with the opportunity to learn about technology and about how it can be used to positively change the world. In 2020, the competition was focused on artificial intelligence (AI), with participants challenged to come up with ideas to solve social, cultural and environmental issues.

Ashwin’s design, named AI RetroMate, is an all-in-one solution to help the elderly and carers with their everyday lives. An Internet-connected hub that dispenses, chats, and detects loneliness, AI RetroMate is controlled by a virtual caregiver and aims to support independence for elderly people who require care but want to stay at home.

Its features include:

• A remote connection that uses cellular IOT (Internet of Things) technology to keep carers and patients connected reliably and securely, thus helping reduce the cost and strain of full-time care
• A ‘chatbot’
• A remote hub with a built-in a pill dispenser, incorporating facial recognition for additional safety
• An attractive retro design.

After first researching online, Ashwin entered the cup competition, using AI to develop and prototype the device. As part of the project, he had to delve into advanced Mathematics to help enhance the prototype, using, for example, ‘nearest neighbour’ algorithms and linear regression models.

Ashwin developed his project late in lockdown, deploying CAD (Computer-aided Design) and electronics to create a prototype, using skills that he had learned in Design and Technology and in Physics.

Speaking on behalf of the judging panel, Tina Jones, Business Strategy Lead, Azure Skills and Employability, said: “The judges were thoroughly impressed by AI RetroMate, especially the research [Ashwin] had undertaken into the difficulties faced by the elderly and by carers and how to create something to improve the quality of their lives.

“We particularly liked how [he] added a chatbot following initial product feedback, and the video, and how [he] brought the product to life with [his] CAD drawing was incredible.  [Ashwin’s] concept, ethics and use of AI was thorough, well thought-through, and it was clear how much effort [he] had put into [his] project.”

Ashwin, who won a trophy as well as a prize of Microsoft’s Surface Go tablet computer and case, said: “This project has helped me to explore STEM [Science, Technology, Engineering & Mathematics], using and developing skills from class to help solve real-world problems.”

In the Connect the Community: Design Challenge (run by RS Components, Nordic Semiconductor and Cadent), having been chosen as one of the international winners of phase 1, Ashwin is now working towards a final prototype, in time for the second phase, where he could receive the funding to help to bring his product to life.

Three QE boys scored 135 out of 135 in the 2020 Junior Mathematical Challenge, as the School recorded an exceptional number of strong performances in the annual competition.

Hisham Khan, now of Year 8, and current Year 9 boys Jothusan Jeevakaran and Saim Kahn were among 117 QE pupils to be awarded gold certificates in the national challenge, which this year was held online only and entered by pupils from home.

All 384 boys in Year 7 and 8 were invited to take part earlier in the year, and 318 of them – 83% – won either a gold, silver or bronze certificate, even though such certificates are given nationally to only the top 40% of entrants, to whom gold, silver and bronze are awarded in the ratio 1:2:3.

Assistant Head of Mathematics Wendy Fung said: “It was another very good performance this year, showing the strength in depth of Mathematics in the lower years at the School. My congratulations go especially to Hisham, Jothusan and Saim for their outstanding achievement.

“Much of the success achieved by our youngest boys in the challenge stems from the excellent guidance and help given to Year 8 by the Years 10 and 11 mentors at our Élite Maths (Junior) group: we are very grateful to them for giving up their time and passing on their wisdom.”

To win gold certificates this year, entrants had to score more than 102 points; for silver, the threshold was 86, and for bronze, 70.

The annual event is run by the UK Mathematics Trust. The usual follow-on rounds for successful entrants – the Junior Olympiad and Junior Kangaroo – are not taking place this year.

Here are two sample questions from this year’s Junior Mathematical Challenge – answers and explanations below.

1. The mean of four positive integers is 5. The median of the four integers is 6. What is the mean of the largest and smallest of the integers?

A 3   B 4   C 5   D 6   E 8

2. A group of 42 children all play tennis or football, or both sports. The same number play tennis as play just football. Twice as many play both tennis and football as play just tennis. How many of the children play football?

A 7   B 14   C 21   D 28   E 35

Solutions & explanations

1. The mean of four positive integers is 5. Therefore. the sum of the four integers is 4 × 5 = 20. The median of the integers is the mean of the two middle integers. Since this median is 6, the sum of the two middle integers is 2 × 6 = 12. Hence the sum of the smallest and largest of the four integers is 20 − 12 = 8. Therefore, the mean of the largest and smallest of the integers is 8 ÷ 2 = 4.

2. Let the number of children who play only football be f, the number of children who play only tennis be t and the number of children who play both sports be b. Since there are 42 children, f + t + b = 42. Also, since the number of children who play tennis is equal to the number of children who play only football, t + b = f . Therefore f + f = 42. So f = 21 and t + b = 21. Finally, twice as many play both tennis and football as play just tennis. Therefore b = 2t. Substituting for b, gives t + 2t = 21. Hence t = 7. Therefore, the number of children who play football is 42 − t = 42 − 7 = 35.