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Year 11 boy Saruthan Seelan achieved a top-50 finish among élite mathematicians in his age group in this year’s nationwide Intermediate Olympiad, with four other pupils coming in the top 100.

Like Saruthan, Year 10 pupil James Tan and Year 9 boys Athiyan Chandramohan, Abhinav Santhiramohan and Dan Suciu all won medals for their performance, while Andy Kwak, of Year 9, was awarded a distinction certificate for coming in the top 25% nationally.

The six were among 27 boys from Years 9 to 11 who qualified for the Olympiad after performing strongly in the UK Mathematics Trust’s Intermediate Challenge. An additional 21 were awarded merit certificates.

Congratulating all of them, Assistant Head of Mathematics Wendy Fung said: “Solving any one of the problems set is an achievement and those who did more than that deserve corresponding praise.”

Saruthan said afterwards: “The Olympiad questions help me to extend my understanding of Maths.” James found solving the problems “very satisfying”, while Abhinav praised the “interesting maths” involved.

Nearly 1,700 students took part in the Olympiad. In each year group, the top 50 receive book prizes, the top 100 receive a medal and the top 25% receive a certificate of distinction.

In the Intermediate Challenge’s other follow-on round, the European Kangaroo, 135 QE boys from Years 9 to 11 took part, of whom 41 were awarded merit certificates. The Kangaroo’s high scorers in each year group were: Jamie Watkin-Rees (Year 11 – the second consecutive year that he has come top of his year group in this competition); Tanishq Mehta (Year 10), and Beuran Kannan (Year 9). Tanishq said he particularly enjoyed the “logical aspect of the questions”.

This is the 16th year that the UKMT has run the International Mathematical Olympiad and Kangaroo contests. The latter is promoted by Kangourou sans Frontières, an independent association promoting Mathematics among young people around the world: its name reflects the fact that the organisation was inspired by the Australian Mathematics Trust.

Three million students worldwide take part in the Kangaroo, usually including around 5,500 pupils invited to take part after sitting the UK Intermediate Challenge.

Queen Elizabeth’s School has won a national online Mathematics competition, beating off the challenge of hundreds of other schools.

The winning team, made up of four sixth-formers, dropped just one point in the eight rounds of the University of Manchester’s MathsBombe, scoring 119 points out of a possible 120.

Headmaster Neil Enright: “My congratulations go to this team on an almost perfect performance. The competition attracted a large field of teams from leading schools across the state and the independent sectors, and it demanded both speed and deep mathematical understanding. This victory therefore represents a considerable achievement.”

The winning team comprised Year 12 pupils Bashmy Basheer, Kishan Patel, Nico Puthu and Niam Vaishnav. Notwithstanding the team’s name, maiNlyNiam, Kishan was the captain.

Organised by the university’s Mathematics department and supported by the Dame Kathleen Ollerenshaw Trust (a charity named after a mathematician and Lord Mayor of Manchester, who died in 2014 aged 101), the competition attracted entries from more than 600 schools.

From January, every two weeks a new set of problem was released online. The puzzles spanned the whole spectrum, from logic puzzles in pure Mathematics to applications of Mathematics in real-world settings.

The maximum 15 points were available to the first team to solve the problem and to other teams solving it within an hour of the first team. Other points were awarded on a sliding scale, depending on the time taken to solve each problem. The rules forbade any assistance from teachers and also prohibited collaboration between teams.

An online leaderboard enabled teams to keep track of their progress throughout the duration of the event. Kishan said this proved to be a spur to his team’s success: “The competition from the other teams encouraged us to answer the questions as quickly as possible.” Niam added that the four friends had enjoyed the opportunity to tackle challenging problems that differed from those they normally faced in the classroom.

Other teams entered by QE also performed creditably, with one, BombVoyage, taking 43rd place, having solved six of the eight puzzles and scored 70 points.

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Here is an example of one of the problems, with the solution below:

Grobnog the Goblin King was sitting on his throne consulting with Torqmaga the Inquisitor. “Your Majesty, we’ve been infiltrated by a rogue group of Goblins,” said Torqmaga. “They call themselves Nilbogs. Physically they are identical to Goblins, but – unlike true Goblins – they always tell the truth.”

“Our whole society is founded on Goblins being evil and lying whenever they can!” said Grobnog. “We need to identify these interlopers!”

Torqmaga handed over a piece of paper. “I’ve tortured all of your subjects to find out who is a Goblin and who is a Nilbog. I can assure you that under my questioning, everybody was true to their real nature: every Nilbog told the truth and every Goblin lied.”

Grobnog inspected the list. “What does ‘or’ mean here? Does it mean ‘one or the other or both’?” he asked.

Torqmaga nodded. “Yes, your Majesty, it’s the logical meaning of the word ‘or’. It seems that torture turns Goblins and Nilbogs into very logical monsters. I’m sure you can work out from their statements below who is a Goblin and who is a Nilbog.”

Agmiz “Fragdag would definitely say that I’m a Goblin.”
Bord “Exactly one of Iz and Molk is a Nilbog.”
Cherguff “Those good-for-nothing layabouts Dolk and Lold are the same type of monster as Molk.”
Dolk “Stop the torture! Bord and Yobblot are both Nilbogs or both Goblins!”
Erkaz “I may hate his guts, but Toxplok and I are the same type of monster.”
Fragdag “Quonk and Xinik are Nilbogs.”
Gneeg “Zisbut and I are different types of monster.”
Hrunk “Gneeg is most definitely a Goblin.”
Iz “Molk is a Nilbog and deserves everything Grobnog will do to him.”
Jop “Bord would say that Fragdag is a Nilbog.”
Klaatak “Lold is a traitorous Nilbog!”
Lold “Ronx is a loyal Goblin! Will you let me off the rack now?”
Molk “Erkaz never tells me the truth, she’s a typical Goblin.”
Norbet “All I’ll say is that Wizmok is a Goblin or Zisbut is a Nilbog.”
Oinq “Agmiz and Quonk are loyal to Grobnog! They’re both Goblins!”
Plegkurk “Dolk and Hrunk are either both Nilbogs or both Goblins.”
Quonk “Oinq, if he ever stopped eating, would say that I’m a Nilbog.”
Ronx “Xinik and Bord are both evil Nilbogs.”
Squee “Lold is a typical Goblin – he owes me 200 silver pennies!”
Toxplok “That little toerag Cherguff would say I’m a Nilbog.”
Udonk “Iz would say that Ronx was a Goblin.”
Vuird “Ronx would say that Udonk is a Nilbog.”
Wizmok “What can I say? Iz is a Nilbog or Norbet is a Goblin. Will that do?”
Xinik “I know that if you ask Ronx then he’d say Squee is a Nilbog.”
Yobblot “Klaatak and Squee are both Goblins.”
Zisbut “Hrunk is a goblin — the most disgusting I’ve ever met.”

Your task is to work out which of the 26 monsters above are goblins and which are nilbogs.
Enter your answer as a sequence of 26 letters: G (for Goblin), N (for Nilbog) arranged in the order of the 26 goblins/nilbogs listed above. If you think that Agmiz is a Nilbog, Bord is a Nilbog, Cherguff is a Goblin, Dolk is a Nilbog, …, Zisbut is a Goblin then you should enter your answer as NNGN…G.

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Solution:

Refer to each Goblin or Nilbog by the first letter of its name. If a monster is a Goblin then we’ll write that it always lies; if the monster is a Nilbog then we’ll write that it tells the truth. By saying two monsters are the same we mean that they are either both Goblins or both Nilbogs.

The clues are then:
A: F says A always lies
B: Exactly one of I, M tells the truth
C: D and L are the same as M
D: B = Y
E: T = E
F: X and Q tells the truth
G: Z != G
H: G always lies
I: M tells the truth
J: B says F tells the truth
K: L tells the truth
L: R always lies
M: E always lies
N: W always lies or Z tells the truth
O: A and Q both always lie
P: D = H
Q: O would say Q tells the truth
R: X and B both tell the truth
S: L always lies
T: C would say T tells the truth
U: I would say R always lies
V: R would say U tells the truth
W: I tells the truth or N lies
X: R would say S tells the truth
Y: K and S both always lie
Z: H always lies

1. Consider clue I. If I is telling the truth then M always tells the truth. If I is lying then M is lying. Hence I = M (but we don’t know whether they both lie or both tell the truth).
2. Clue B says that I != M. Hence B is lying.
3. Clue R says that B tells the truth. Hence R must be lying. (Note that we can’t say anything about X from clue R.)
4. Clue L says that L must be telling the truth. Hence K is also telling the truth (K’s clue) and S is lying (S’s clue).
5. Clue Y says that both K and S both lie. But K tells the truth. So Y is lying. As both B and Y are lying, Clue D is true; hence D tells the truth.
6. Consider clue X. Suppose that X lies. If X is lying then R would actually say that S lies. We know that R lies, this would actually mean that S tells the truth. But we know S lies, so our assumption that X lies is wrong. Hence X tells the truth.
7. Consider clue F. Suppose that F is telling the truth. Then clue F tells us that X tells the truth (we already knew this) and Q tells the truth. Clue Q then tells us that O would say that Q is telling the truth (which indeed Q is), so O must also be telling the truth. Clue O tells us that both A and Q both lie. But this contradicts the fact that we’ve just argued that Q is telling the truth. Hence our assumption that F is telling the truth is wrong, so F must be lying.
8. As F is lying, it’s not true that both X and Q tell the truth. We know that X does tell the truth. So this tells us that Q must be lying.
9. Knowing that Q is lying, clue Q tells us that O would actually say that Q lies. This is indeed the case, hence O is telling the truth.
10. Clue O now tells us that A lies.
11. Consider Clue J. We know B lies. As F lies, B would indeed say that F told the truth. Hence J is making a true statement, so is telling the truth.
12. Consider Clue M. We’ll consider the two cases (M tells the truth, M lies) separately. First suppose that M tells the truth. Then E must lie. Clue E says that T and E are different, hence T must tell the truth. Now consider the other case where M lies. In this case, clue M says that E is telling the truth; it then follows from clue E that T is also telling the truth. Hence, no matter whether M is telling the truth or lying, we must have that T is telling the truth.
13. Clue T tells us that C is making a true statement. Hence C tells the truth.
14. Clue C tells us that M is the same as D and L (who are both telling the truth). Hence M is telling the truth. Clue M then tells us that E is lying.
15. Clue I is making a true statement about M. Hence I tells the truth.
16. Consider clue U. Monster I tells the truth, and R does indeed lie. Hence U is telling the truth.
17. Consider clue V. Suppose V tells the truth. Then R would indeed say that U tells the truth. We know that R lies, so this would mean that U lies. But U tells the truth, a contradiction. Hence V must lie.
18. Consider clue Z. Suppose Z tells the truth. Then H lies. Clue H then tells us that G tells the truth. Clue G tells us that Z and G are different. But we’ve just argued that both Z and G tell the truth, a contradiction. Hence Z must lie.
19. Clue Z then tells us that H tells the truth.
20. Clue H then tells us that G lies. (Just to check: G lies, so clue G tells us that both Z and G are the same, which indeed they are.)
21. As both D and H tell the truth, clue P implies that P tells the truth.
22. Consider clue W. Suppose W always lies. Then clue W tells us that monster I always lies and N tells the truth. But we already know that monster I tells the truth, a contradiction. Hence W must tell the truth. (Note that, even though we know W tells the truth, clue W doesn’t tell us anything about whether N lies or not.)
23. Finally, consider clue N. If N is telling the truth then either W lies or Z tells the truth. But W tells the truth and Z lies, so neither of these possibilities can happen. Hence N must be lying.
Hence (denoting T for ‘telling the truth’ and L for ‘lying’) we can assign

ABCDEFGHIJKLM NOPQRSTUVWXYZ
LLTTLLLTTTTTT LTTLLLTTLTTLL

Reverting back to ‘Goblins always lie’ and ‘Nilbogs always tell the truth’ this gives

ABCDEFGHIJKLM NOPQRSTUVWXYZ
GGNNGGGNNNNNN GNNGGGNNGNNGG

so the required answer is GGNNGGGNNNNNNGNNGGGNNGNNGG

QE pupils beat off competition from 30 other schools to win the regional round of the Team Maths Challenge.

The four boys from Years 8 and 9 secured victory over Merchant Taylors’, in second place, and Haberdashers’ Aske’s Boys’, who came third. They now go through to the national finals in London’s Royal Horticultural Halls in June – the third time that a QE team has reached this stage in the prestigious UK Mathematics Trust contest.

Headmaster Neil Enright said: “I congratulate our boys on a resounding success, which demonstrated not only their mathematical prowess and their ability to think clearly under pressure, but also skills in communication and teamwork.”

The team was led by Year 9 pupil Dan Suciu and comprised Shimaq-Ahamed Sakeel Mohamed, also of Year 9, together with Year 8 boys Bhunit Santhiramoulesan and Agrim Sharma. They scored a winning total of 223 points out of 236 in the event, which was hosted by Haberdashers’ Aske’s School for Girls in Elstree.

The competition aims to offer pupils a means of expressing and developing their enjoyment of Mathematics, with problems that are mostly accessible, yet still challenge those with more experience. The event involves four rounds:

• Crossnumber – one pair of contestants is given the ‘across’ clues and the other pair the ‘down’ clues
• Shuttle – pairs solve problems where the answer to the previous question feeds into the next question
• Relay – again working in pairs to solve problems, but also involves movement around the room in a race against the clock
• Group round – working as a team of four to solve ten problems.

Captain Dan said after the event: “We were delighted to win and really pleased that our hard work paid off, especially in the Shuttle Round. We’re all really looking forward to the next round.”

Sixth Form mathematicians saw a genuine Enigma machine at work at Maths Fest 2018, where the wartime device was the star of the show.

Thirty-eight Year 12 mathematicians attended the event at the Piccadilly Theatre in the West End and saw the demonstration of encoding and decoding. Another memorable moment came when one of the speakers burst into song!
Now in its third year, Maths Fest 2018 was set up by long-established Mathematics speakers, Matt Parker and Rob Eastaway, who “thought it was time there was a Maths festival for schools run entirely by passionate mathematicians”.

The first speaker, James Grimes, who is part of the Millennium Mathematics Project at the University of Cambridge, spoke about codes and ciphers and about the importance of cracking codes throughout history, from Julius Caesar to internet security, with the Enigma machine providing an exciting climax to his presentation.

Aoife Hunt, an industrial modelling expert, demonstrated how she uses statistics and crowd-flow models to make sure large venues are safe. She showed how three particular graphs (reciprocal, quadratic and ‘normal’) are vital in her work.

Ben Sparks from the Further Maths Support Programme at the University of Bath gave an insight into spirals and circular motion inspired by the 1968 Michel Legrand song Windmills of Your Mind.

Round, like a circle in a spiral
Like a wheel within a wheel
Never ending or beginning
On an ever-spinning reel

Mathematics teacher Phil Brady, who accompanied the boys to the event said: “He ended his session by singing for us – a real treat. The festival, which was hosted by Matt Parker in his usual witty manner, was both impressive and great fun.”

The event also featured Maths Slam, an opportunity for some of the students to go on to the stage to talk for three minutes about an interesting aspect of maths. QE boys Binu Perera and Uday Kataria gave a presentation on ‘How to hold a pizza’.

“Apart from the various talks scheduled for the day, Maths fest was a fantastic opportunity for us to speak about an interesting mathematical idea in front of over a thousand Year 12 students,” said Binu. “We decided to talk about Gauss’s theorema egregium because, whilst being fun to both present and research, it is an understated, yet simple, mathematical concept that is fundamental to our everyday lives,” added Uday.

Seb Lee-Delisle, a creative coder who works on large-scale installations, described how to transform a simple animated point into an impressive multicoloured firework display and showed how this was the basis of his professional laser displays.

The show was closed by Hannah Fry, a senior lecturer in the Mathematics of Cities at the Centre for Advanced Spatial Analysis at UCL. She demonstrated how to generate random (and not-so-random) numbers to produce a four-digit winning lottery number. Mr Brady’s ticket was just four away from being a winner!

Thirty-six QE sixth-formers qualified for the next stages of a national Mathematics competition – twice as many as in 2016 – while the School comfortably out-performed the national average.

Eleven pupils taking part in the Senior Mathematical Challenge qualified for the élite British Mathematical Olympiad – up from eight last year – and 25 reached the other follow-on round, the Senior Kangaroo, representing a 150% increase on 2016’s total of ten.

The Best-in-School title went to Year 12 boy Robert Sarkar (right in photograph), who scored 121 out of a possible 125 marks. Andrew Shamis (left in photograph), top scorer in Year 13, was just behind, on 120.

Of the 123 pupils who entered the competition, 33 were awarded gold certificates (22 in 2016), 53 received silver (39) and 30 (42) won bronze. The certificates are given only to the top 60% of entrants nationally, but at QE the proportion gaining them was 94%. They are awarded gold, silver and bronze in the ratio of 1:2 :3.

Congratulating the successful entrants, Assistant Head of Mathematics Wendy Fung said: “Many of the boys who have qualified through to the follow-on rounds have been members of our Élite Maths mentoring scheme for a number of years and are now passing on their experience and wisdom through mentoring students in Years 9 to 11.”

The competition, run by the UK Mathematics Trust, involves answering 25 multiple-choice questions in 90 minutes. To qualify for the Olympiad, candidates must score at least 104 points and for the Kangaroo, they must score at least 85.

Sixty-two boys from Year 11 headed off to the West End to hear from some of the country’s most engaging Mathematics speakers – including one who hit the headlines with her investigation into Ryanair’s seat allocation policy.

The Maths Inspiration Show at the Piccadilly Theatre featured interactive lectures on a wide range of mathematical topics. It was hosted by Matt Parker, who has the unique distinction of holding the prestigious title of London Mathematical Society Popular Lecturer and of having a sold-out comedy show at the Edinburgh Festival Fringe.

Afterwards, the boys were full of praise for the event. Viraj Mehta said: “I found the lectures intellectually stimulating whilst still being humorous and interactive. I especially enjoyed the variety of topics of maths and how they link to everyday life.” For his part, Joshua Han said the show was “much more interesting and entertaining than I expected.”

Among the speakers were Jennifer Rogers, who gained a PhD at Warwick and is now a research fellow in the Department of Statistics at the University of Oxford. She was widely quoted this summer when she worked with the BBC’s Watchdog programme on investigating Ryanair’s claims that it allocated seats to people who had not paid to reserve seat on an entirely random basis. All the people in her sample were allocated the dreaded middle seats – and the chances of that happening were smaller than the chance of winning the National Lottery jackpot, she found. In her lecture, she explained how she had made the calculations using simple probability and asked the audience to consider whether this meant Ryanair’s claim to random allocation was invalid.

Mathematician and juggler Colin Wright received his maths doctorate in 1990 from Cambridge University. He looked at the importance of spotting patterns – showing that juggling tricks are, in fact, patterns with mathematical properties – but warned that patterns are not always as predictable as they seem.

Comedian and science communicator Steve Mould showed the audience his favourite shape, the Reuleaux triangle – a shape formed from the intersection of three discs. He also gave his name to the Mould Effect: his clip of a 50-meter string of metal balls briefly and mysteriously flowing upwards before plunging to the ground caused a Youtube sensation in 2013.