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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.

A Year 10 team’s hi-tech lockdown project was placed third in an international competition aimed at stemming the global tide of plastic pollution.

The Prata Neptunia team combined their skills in Technology, Mathematics and Chemistry and also produced a slick video presentation to promote their design for an autonomous hovercraft robot.

Competing against teams from more than 40 countries, Ashwin Sridhar, Anish Rana and Merwan Singh impressed judges from the British International Education Association with their use of artificial intelligence to tackle plastic waste in rivers and canals, reducing its harmful effects on flora and fauna.

A second QE Year 10 team, called Ocean, won the Best Effort prize in their category in the competition, which was launched in January.

Head of Technology Michael Noonan said: “My heartfelt congratulations go to the boys, who began their projects when we were deep in lockdown and thus had to overcome some significant obstacles in putting their entry together. Although narrowly missing out on the grand prize, the team are proud to have had their project acknowledged on an international scale and to have learned countless new skills along the way.”

The BIEA International STEM Innovation Challenge invited young people from the age of nine to 21 to research, write a report and design a solution to Save our shores from plastic waste through STEM (Science, Technology, Engineering, Mathematics). In its brief, the BIEA pointed out that one lorryload of plastic is dumped every minute worldwide – the annual equivalent in weight of 40,000 blue whales or 1.6 million elephants. The competition drew entries from schools in countries including China, the United States, Argentina, Norway and Indonesia.

Ashwin took on the role of Project Manager and Lead Scientist for Prata Neptunia, while Merwan was Lead Researcher and Anish the Lead Robotics Designer.

By using hovercraft technology informed by artificial intelligence, the trio were able to devise a design that could travel across multiple terrains, both land and water, and target different types of plastic. These notably included microplastics, which have become a huge problem worldwide because of their devastating effects on marine life.

The team learned project-management skills in order to optimise their time effectively, from the use of Gantt charts to task delegation. They designed prototypes at home, building and testing parts, and investigating processes to remove microplastics in order to determine the feasibility of their design.

As part of the overall design process, they applied skills acquired in Technology lessons before finally designing their solution on CAD software.

Their work led to an invitation from BIEA to participate in the virtual international finals, where they were awarded their third place in the 15-17 category.

Anish said: “We started our journey back in March and were quite behind, compared to other teams, which started earlier. However, through thoughtful planning and hard work, we were able to pull together to create a product we were proud of in time for the due date.”

Unable to meet up freely or access all the resources of The Queen’s Library, the boys worked from home and used technology including Zoom calls to co-ordinate their work.

“We all saw plastic pollution as a big problem all over the world: the BIEA competition has targeted a global crisis that needs fixing.”

The competition gave him and his teammates the opportunity to deploy their skills and knowledge to tackle this crisis, which, he said, has shown him “how we can all work together to solve it”.

Anish added: “Of course, we had our ups and downs, but overall the competition was a great experience with a satisfying conclusion.”

The trio’s project required some fairly advanced Science, as they investigated methods of removing plastics, which led to their inclusion of PETase, an enzyme which catalyses the hydrolysis of Polyethylene terephthalate (PET) to monomeric mono-2-hydroxyethyl terephthalate (MHET). MHET is then broken down into Ethylene glycol and Terephthalic acid (Benzene-1,4-dicarboxylic acid) using the enzyme MHETase.  The team also delved into fluid dynamics – encompassing Mathematics and Physics – to optimise their design’s motion and efficiency.

The Ocean team, Jashwanth Parimi, Utkarsh Bhamidimarri and Siddarth Jana, also started their project relatively late and had only about a month to complete it.

Jashwanth said: “During multiple Zoom calls, we learned much more about plastic pollution and, eventually, we designed an idea that we thought was suitable for solving the problem. Then we each split into our specialised areas to fulfill the requirements of the project, but we still all helped each other in each of our project areas until we finally finished.”

The team designed a multi-terrain vehicle that used a net in order to collect macro-plastics on both the ocean and the mudflats. “Our project was innovative since we tried to consider all the wildlife on all the terrains, such as fish and snails, and so on.”

The number of QE pupils receiving top awards in the UKMT Intermediate Maths Challenge has increased again this year, with one boy achieving a perfect score.

The IMC competition, run by the UK Mathematics Trust, is for pupils in Years 9 to 11; 317 boys from QE took part – 174 were awarded gold certificates (up from 172 last year), while 103 were awarded silver (compared to 91 last year), with a further 31 receiving bronze.

Ansh Jassra from Year 10 was awarded Best in School, scoring a maximum-possible 135 points.

With only 500 places available nationally across all schools for the highest scorers for the Intermediate Olympiad, Ansh and 22 other QE boys qualified. A further 174 QE pupils secured entry into the Intermediate Kangaroo, the competition’s other follow-on round.

Ansh said: “With its many challenging yet intriguing maths problems, sitting the IMC was fun, testing and overall a great experience. I am looking forward to the Olympiad!” said Ansh.

Assistant Head of Mathematics Wendy Fung said: “We are delighted with how well the boys have done and extremely pleased with the continued increase in the proportion of boys reaching the follow-on rounds. As the recently introduced 9-1 GCSE has a strong focus on problem-solving, success in the IMC will stand the boys in good stead for their examinations.”

Maxwell Johnson, who was named Best in Year 9 with a score of 130, said: “I hope that I will be able to improve on my score in the [Junior] Olympiad from last year. It will be challenging, but I’m sure I will enjoy it.”

Shimaq Sakeel Mohamed, who also scored 130 and was named Best in Year 11, said: “I am proud to be part of a School where I can achieve great things and the IMC is a great way to do this.”

Sample question:
The Knave of Hearts stole some tarts. He ate half of them, and half a tart more. The Knave of Diamonds ate half of what was left, and half a tart more. Then the Knave of Clubs ate half of what remained, and half a tart more. This left just one tart for the Knave of Spades.

How many tarts did the Knave of Hearts steal? A. 63  B. 31  C. 19  D. 17  E. 15

Solution:
Suppose that at a particular stage there are m tarts available for a Knave to eat and that there are n left after he has finished eating. Then n = m − ( ½ m+ ½ ) = ½ m – ½ . Therefore, m = 2n +1. As the Knave of Spades received one tart, then the number of tarts which the Knave of Clubs was given was 2×1+1 = 3. Similarly, the number of tarts which the Knave of Diamonds was given was 2×3+1 = 7. Finally, the number of tarts which the Knave of Hearts stole was 2×7+1. So the correct answer is: E. 15.