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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.

Ninety-nine Year 12 boys were entertained, amazed and inspired at a special series of lectures on the application of Mathematics.

Fifty of the sixth-formers went to one set of lectures, while the remaining 49 went on a later day to hear a second set. The Maths in Action lectures were organised by The Training Partnership (the UK’s leading provider of educational study days) at the Emmanuel Centre in Westminster.

Assistant Head of Mathematics Wendy Fung said: “Each lecture was inspiring in its own way and has encouraged the boys to delve deeper into the topics they found most engaging. These lectures are a very good way of introducing branches of Mathematics and ways of mathematical thinking which are not covered as part of the A-level syllabus, and of showing the range of applications to which the subject can be applied.”

Both groups heard lectures on statistics. On the first day, Michael Blastland, creator of BBC Radio 4’s More or Less programme, spoke on Bad Stats: what they don’t tell you on the news. The second group heard from economist and journalist Tim Harford. He counselled that if used well, statistics can help people learn about the world and he emphasised the paramount importance of using statistics in a responsible way.

Pupil Sachin Sarin said: “Tim Harford’s thorough explanation of how statistical findings were being used by politicians and firms to manipulate the general public into believing certain ideologies allowed us to gain a deeper understanding as to how powerful statistics are when trying to persuade or argue a point. I learnt that statistics can often be cherry-picked and even distorted by these individuals to achieve their motive.”

Beker Shah enjoyed Michael Blastland’s talk, in which he similarly demonstrated “how manipulating data could serve a political agenda or purpose, as shown by the increase in cancer deaths and the increased pregnancy rate”.

Common to both days was a lecture by author and broadcaster Simon Singh on Fermat’s Last Theorem, which he began by introducing 17th century French mathematician Fermat and the concept of a mathematical proof. In Fermat’s spare time, he would find mathematical statements and see if he could prove whether they were true or not. Over time, mathematicians proved all of Fermat’s theorems except one, which hence became known as ‘Fermat’s Last Theorem’. Simon took the audience through the inspiring story which culminated in its proof in 1993.

Simon Singh was, said Zidane Akbar, “a great speaker”, while Janujan Satchi added: “Learning about the story of [British mathematician] Andrew Wiles and how his perseverance led him to prove Fermat’s last theorem was really interesting.”

Cambridge mathematician Matthew Scroggs’ lecture on the Mathematics of Video Games impressed Charan Kumararuban, who said: “I was particularly amazed by his demonstration of using Mathematics in order to predict the shortest possible routes to complete a game of Pacman in the shortest possible time.”

Oxford University’s David Acheson brought some musical moments to the day with his talk, From Euclid to the Electric Guitar. Ayushman Mukherjee said: “I liked the humour, practical demonstrations and guitar solo!”

The other speakers were:

• Sara Jabbari, from the University of Birmingham, on Fighting disease with Mathematics, who looked at how differential equations are used to understand antibiotic resistance, track the dynamics of bacterial infections and even develop new drugs to tackle disease.
• Ed Southall, author of several books on geometry puzzles and a lecturer at the University of Huddersfield, who led a hands-on session on how problems could be solved in multiple ways. For example, he set students the task of cutting 2D and 3D shapes into pieces of equal area using only a set number of straight lines.
• Award-winning teacher Jamie Frost on How to prepare for exams;
• Jackie Bell, from Imperial College London on Maths in a Space Suit, in which she recounted her journey from Mathematics graduate, to particle physicist and finally to trainee astronaut.

Afterwards, pupil Manas Gaur reflected on the value of the day: “I enjoyed being able to link Mathematics to other fields and seeing how it connects with other subjects.”

Sixth-former James Tan has become the first QE pupil for many years – and possibly ever – to win a prize from the prestigious Mathematical Association.

Further Mathematics AS-level student James, of Year 12, submitted solutions to both of the Student Problems published in the summer issue of the association’s journal, The Mathematical Gazette, and has now heard that he was won first prize.

Congratulating James, Mathematics teacher Phillip Brady said: “Nobody in the Mathematics department can remember any boy here winning this prize before, so James has broken new ground for QE with this achievement.

“The problems are designed to be accessible to students taking Maths A-level, but solving them usually requires careful thought or cunning methods!”

Established in 1871, the MA is the oldest subject association in the UK. Its journal, which is published three times a year, has a readership that includes teachers and college and university lecturers worldwide.

Entries to the problems come in from bright young mathematicians around the world. A first prize of £25 and second prize of £20 are awarded for the best solutions to the Student Problems; entrants can submit solutions to both of them or just to one.

James heard from puzzle page editor Stan Dolan, who is the author of several books aimed at A-level students, that he had won the first prize.

The letter from Mr Dolan, a Fellow of the Institute of Mathematics and its Applications (FIMA), states: “The most elegant and simplest answer was given by James. The solutions […] were especially good in terms of clarity and a well-expressed generalisation.”

Here is one of the questions James tackled: If a 5-digit number is a multiple of 271 then so are all numbers given by cyclic permutations of the digits of the number. Explain this property and generalise the result.

The second question involved giving the area of a hypotenuse triangle that was surrounded by three circles of varying sizes.

• QE boys and their families can read the current set of Mathematical Association Student Problems, together with a selection of easier puzzles, on the Puzzle of the Fortnight page in the dedicated eQE private web portal.

QE’s youngest boys had to combine individual talent and good teamwork to succeed in the inter-House Year 7 Maths Fair at the end of the summer term.

Inspired by the UK Mathematics Trust’s Team Maths Challenge events, the Mathematics department’s annual morning of activities pits the six Houses against each other.

The boys take part in a carousel of mathematical challenges, some of which are more familiar problem-solving (such as the round entitled A Question of Maths), while others major on the practical (such as tangrams, which involve putting together seven flat shapes to create a specified shape).

One highlight of the event is the relay round, which combines speed in movement around the room with mental speed in solving a mathematical problem.

Assistant Head of Mathematics Wendy Fung said: “The idea is to show boys that mathematical problems come in many different formats as well as to help them to develop team-working skills.”

The overall winners were Pearce, with 767 points; second was Broughton, with 740, and third, Underne, with 688. Pearce were subsequently presented with a trophy – the Scarisbrick Shield – in assembly.

“From electricity to football, Maths is all around us. Winning the Maths Fair is an unforgettable achievement,” said Haris Shahid from Pearce.

This was the second year in which Year 12 pupils have supported the event. Each Year 7 team was supervised by one of the sixth-formers, who also provided essential logistical support for the fair.

Paying tribute to the Year 12 boys for their contribution and noting that they had themselves taken part in a similar Year 7 Maths Fair back in 2014, Miss Fung said: “The event would not have been able to run without their help. The sixth-formers certainly enjoyed issuing red cards to any Year 7 boys who did not follow the relay ‘no-running’ policy!”

All six Houses were also required to create a poster entitled What is Mathematics? Each of the multiple teams within each house had to create a part of the poster. They were asked to prepare in advance by coordinating the different sections so that their poster would encompass the many facets of Mathematics.

Having been chosen as the winning poster, Broughton’s entry was displayed in the Mathematics department.