The team’s work is a testament to remarkable perseverance (that hallmark of the NLCS spirit), and cross-curricular collaboration at the highest level. Click here to download the app!

Our Year 12 Engineering Team are celebrating the release of the first NLCS App on the Google Play store. Victoria, the app’s lead creator, taught herself to code over the course of Year 12, and alongside the rest of the team investigated the mathematics behind optimal strategies to create a realistic experience against the computer.
The team’s work is a testament to remarkable perseverance (that hallmark of the NLCS spirit), and cross-curricular collaboration at the highest level. Click here to download the app!
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Massive congratulations to 7N, who emerged victorious in the Year 7 Mental Maths Marathon. In a tense final, two girls from each form competed for marks in individual rounds and a team round. All the girls performed exceptionally well with great support from the audience who joined in solving the puzzles!
## Reflections on GDCs from a Year 13 single mathematician who has just finished all of her exams25/6/2018 The GDC is probably one of the most worthwhile investments you could make going into the Sixth Form. Although, it seems very complex to use and some of you may think you will never get used to it, with frequent use in every Mathematics lesson by the time it comes to the examination period you will be well-versed to use the graphical calculator. Having just recently completed my A-Level Mathematics exams, it was very useful whilst completing papers as it enables you to check your answers efficiently and ensure that you are on the right track! One of the most reassuring things in an exam is to be able to check your answers resourcefully on the GDC, thus I highly recommend that you invest into a GDC as throughout your time in the Sixth Form it will be very useful in lessons as well as throughout the examination period.
By Wendy, Year 12 It is widely recognized among scientists that mathematics is an essential accessory in scientific studies, and that mathematical models are employed to predict future trends and sometimes to infer information about our past. For example, it is possible to use logarithms to estimate roughly how long ago the most recent common ancestor of everyone on Earth today appeared; or we could also use differential equations to predict the infection development spread of a particular disease. This article will discuss the applications of one of the most commonly known and used mathematical models in Biology - dynamical systems - and give some very simplified examples of them. Dynamical system is commonly defined as an evolution rule which determines how things change or behave over time, and examples of such application is in modelling the drug resistance in malaria, auto-regulation in kidney, anti-coagulation therapy and Ebola outbreaks. All of these involved extremely long and complex modelling and are therefore beyond the scope of this article. However, we can use a very simple example to help ourselves understand the principles of dynamical systems. For a population of rabbits, the area in which they live in might only support a population size of 1500. If the population is small enough, the population size will increase 1.5 times from current time to next period. Let’s use the well established logistic equation to model the population: A(n+1) = A(n) + r (1-A(n)/L) x A(n), where A(n) represents population of the rabbits at time n, L represents the carrying capacity of the area (which in this case, is 1500) and r is the restricted growth rate (I.e. 1.5). These values could be found from consistent analysis of experimental data during fieldwork investigations. If A(n)<1500, then 1 - A(n)/L > 0, so at time n+1 the population A(n+1) will increase from A(n) at time n+1 E.g. if A(n) = 500 < 1500, A(n+1) = 1000 and the population has grown by 500. Using this to calculate A(n+2) we find the value is 1500, but if we calculate A(n+3), A(n+4) etc, we find that the value will remain constant at 1500. If we have a population of exactly 1500 rabbits, then A(n)/L = 1 and therefore A(n+1) - A (n) will be 0, and thus there would be no change in population from one period to the next, and we have reached an equilibrium point. In contrast, if A(n) > 1500, (1- A(n)/L) < 0 so population would decrease. This is an oversimplified general model for some populations where adults often survive after reproduction, and is obviously not suitable for all populations. For example, when it comes to insect populations, it is often the case that they have non-overlapping generations, meaning all adults lay eggs in a specific time of year and then die, leaving only their offspring. To understand the modelling of insect populations we must first understand the Malthusian equation (in discrete time), which is a simple first order model that states Nn+1=ʎNn , where ʎ is the growth rate of the population and Nn is the population at a discrete time n. If b is the average number of births given by any individual and d is the probability of deaths in the population at a given time n, then we could infer that Nn+1 = Nn + Nnb - Nnd = Nn (1+b-d). In the case with non-overlapping generations, if Nn is the population size at given time n, and R0 is the number of offspring per adult, then, ʎ = 1 + R0 - 1, and Nn+1=Nn x R0.However, you may have realized that this is not a realistic model, as it is unlikely that all offspring will survive to give births to the next generation due to factors such as larvae predation and both intra and inter-specific competitions for limited resources. S(N), the fraction of offspring that survives, is therefore important, and the equation changes to Nn+1 = Nn x S(N) x R0, which for simplicity is generally rewritten as Nn+1 = F(N) x Nn = f(N) . It may be interesting to note that F(N) represents the per capita production of offspring that survive to the next generation of a population size N.(1)If we hypothesize that insect populations are solely affected by intra-specific competitions only, then some further refinement could be done on the model. If there were no competition, then S(N)=1 as all offspring survive. In the second case, if the resources present are not sufficient to support the whole population, then some individuals obtain the units of resources they would need to survive and reproduce, and others die without producing any offspring. Thus in this case S(N) = 1 for Nn < Nc, (where Nc is the critical value above which it would no longer be the case that all individuals survive) and for Nn > Nc, S(N) = Nc/Nn - this type of competition is usually known as contest competitions. The other type of a commonly-known competition is called scramble competition, where every individual is given an equal share of the resources and if Nn < Nc, the critical value, S(N)=1 as all individuals obtain sufficient resources. If Nn > Nc, S(N) = 0 as the amount of resources gained by each individual is not enough to help them survive til the next generation. It is nevertheless very rarely the case that a population, especially a large one, would crash to 0, therefore the behaviour of S(N), and consequently F(N), is always treated as asymptotic. Idealized contest competition exhibits exact compensation, where the increase in mortality compensates exactly for any increase in population between time n and time n+1. This occurs if F(N)-->c/N as N -->infinity, (where c is a constant) and consequently f(N) -->c, according to . This means that over time, the population size would stop fluctuating and remain at value c (as seen in figure 3). Other behaviour could be modelled by F(N)-->c/(N)^b, which infers f(N)-->(N x c)/(N)^b, and if 0 < b < 1, then under-compensation is observed where f(N) will increase over time as number of mortality is always lower than the increase in population. The population at a given time n will be smaller than population at time n+1 after reaching a critical value (as seen in figure 1). On the other hand, if b>1, then the opposite shall be observed where over-compensation takes place and the population at the time n is going to be larger than population at the time n+1 after population has reached the critical value (as seen in figure 2). If F(N) decreases with N, then a compensatory behaviour is said to be observed. As stated above, contest competition happens when b=1, and scramble competition when b>1 so both of these types of competition are compensatory. The non-linear (1)Hassell equation gives a more realistic representation of the population, and could be used to summarise all kinds of compensatory behaviour: Nn+1 = (R0Nn) / (1+aNn)^b where a and R0 are positive constants and b is larger than or equal to 0. When b = 0, there is no competition; when b = 1, there is contest competition and when b>1, there is scramble competition.These are just a few small glimpses into how populations are monitored. Obviously, there are still limitations to these models, as they do not take into account other factors such as inter-specific competitions, presence of predators etc., however, scientists do hope that the models correspond roughly with real-life data. If that were the case, the models could then become very useful tools when monitoring and predicting the scale of species population trends in the future. Computer algorithms are the maths from which computer programs were made. Algorithm comes from the name of the man who also named algebra. It comes from the medieval version of the name of a man called Aljabr. It is now specifically used to mean a mathematical process to solve a problem. Mathematicians considered a problem to be solved if they could write and algorithm to solve it. Appyling an algorithm wasn’t important because the formula already was the answer but computers changed that view because complicated formulas could easily be solved. An algorithm was like a formula that would lead you to the answer and it could also estimate the time it would take to solve a problem.
An example of a fast algorithm is knowing if a number is odd or even: Is the last digit 0? If yes then STOP. The number is even. Is the last digit 2? If yes then STOP. The number is even. Is the last digit 4? If yes then STOP. The number is even. Is the last digit 6? If yes then STOP. The number is even. Is the last digit 8? If yes then STOP. The number is even. STOP. The number is odd. So in conclusion an algorithm is a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer. By Tia, 8L ## School multiplies its praise to boost girls’ maths skillsTeachers are writing positive comments in the maths books of all pupils at a top girls’ schools to improve their confidence in the subject.
Outstanding results are already achieved by pupils at North London Collegiate but positive comments must now be included on all pieces of work to increase self-confidence and promote a “can-do” attitude. The decision was inspired by research from Harvard, which showed that confidence was critical to success. The school also sends postcards to pupils to celebrate small achievements and teachers are asked to email students, and copy in form tutors, when small milestones are reached. One pupil said: “During class, teachers constantly provide positive reinforcement which really helps to keep us motivated and push through the tough and seemingly impossible parts. “Whenever we get tests and homework back, our sheets are always showered with stickers and this adds another incentive to keep on top of homework and makes the experience of receiving marks a much less uneasy experience.” The school has introduced “low-stakes” testing to build pupils’ resilience to exam conditions. Regular ten-minute assessments have been introduced, to help to reward pupils with small victories and to track their progress. According to an international study of gender equality in schools by the Organisation for Economic Co-operation and Development, girls lack confidence in their ability to solve maths and science problems and achieve worse results than they otherwise would, despite outperforming boys overall. At A level more than 60 per cent of maths candidates are boys and only 14 per cent of people working in science, technology, engineering and maths are women, short of the goal of 30 per cent. The school is adopting a number of maths initiatives to inject the subject with fun. These have included “maths-off” contests and musical talent shows. Girls also mentor pupils two years below them in advanced mathematics and sixth-formers have been trying to make maths clubs more appealing. Teachers have given lectures on the philosophy of science, the links between maths and art, and the maths of Alice’s Adventures in Wonderland.Last year 98 per cent of pupils at the school achieved an A or A* at International GCSE maths. Well done to all the girls who sat the paper this week.
All are welcome to have a go! Click here for the paper. Answers to mcopin@nlcs.org.uk by the end of the weekend. Family efforts are encouraged! On Wednesday this week, Year 5 did us a huge favour and were mathematical guinea pigs for the afternoon. They took part in a trial version of a new maths competition EUREKA @ NLCS. The girls worked in teams of 4. After taking part in some activities in the hall, they set off on a treasure hunt around the school. Armed with maps and co-ordinates, they had to find their allocated station and complete a maths puzzle within a ten minute time frame before being given another set of co-ordinates to continue their trail. At the end of the event the winners were announced with Lettie, Kayla, Sofia, Ella coming second and Lily, Misha, Amy and Mary coming first.
Many thanks to the many teachers from both the Junior and Senior School who volunteered to man the stations and help run this event. We hope the Y5s enjoyed their maths with a twist! Three take-aways1. Mathematics never hurt anyone, except when it did. It is incomparably powerful when trying to understand the modern world, but mathematical ideas need to exist alongside thinking from other disciplines. The world is complex and interconnected. Reject bubbles of specialism. 2. Miss Buss made NLCS so that young women would be able to play the same games as their brothers, and with the same advantages. Though today's society is fairer than ever before, it has been a man's world since the beginning of time and women are still woefully under-represented in public life and leadership roles. Society teaches women to underestimate their powers. You have the opportunity to lead the way. By winning in whatever field you enter, you can change the rules of the game for the better. 3. Don't play life as though it is a game of Monopoly. Monopoly makes you a bad person. Furthermore, Mathematics tells us that cooperation is better for individuals as well as for the collective. Why not think of upcoming exams as an opportunity to work together? Individual lives are full of struggle: "success is going from failure to failure with no loss of enthusiasm." So you should learn to rejoice - to really rejoice - in the success of other people.Because that way, you get to win all the time. Further Points - thank you for these suggestions## The Maths Society Share their Fountain of Knowledge and Fountain of Chocolate with Years 7 and 88/3/2018 In preparation for the Junior Maths Challenge in April, members of the Maths Society are running how-to-solve-it sessions every week for middle school girls. Sharing our favourite techniques to solve the challenges quickly and efficiently is very enjoyable for the sixth form and hopefully very useful for girls. This week, we were delighted to see 25 enthusiastic girls turn up to the session, and their insight and ingenuity impressed us all! Letting them indulge in the maths society chocolate fountain is therefore a perfect way to reward them for their hard work and a sweet way of finishing off! The poem read out in assembly is below:
A notice from Kureha, Saachi and Mengyao in the Maths Society. Hey there middle school you better listen to this It’s Maths Soc news that will fill you with bliss. In 223 every Wednesday, first half of lunch It’s how-to-solve it sessions with the loveliest bunch For year 7 and 8 of ALL maths abilities To develop your Junior Maths Challenge capabilities Getting Bronze, Silver, Gold is really very easy If you use all our tricks from the proper to the sleazy. So bring a pen and a friend and be brave, don’t be shy And this April we’ll beat the school’s all-time high Of 90% getting medals because We’ve got a chocolate fountain and some jokes about cos. 223. Every Wednesday, during lunch (the first half) For some maths and some food, and a bit of a laugh. On Wednesday 28th February, a group of NLCS girls had a friendly competition with our partner school NLCS Jeju. The girls found the experience amazing and were excited to get in the competitive spirit for this competition. We had seven competitors from a range of years on each side and it was a close match with the score being 44 to 48 for NLCS. Even though we had only lost by 4 points, it was still a fantastic experience for the girls and we were delighted that we had this opportunity to get to know our partner school in Jeju even more. Some special thanks go to the Maths Department, especially Dr Hearmon and Dr Van Beek who organised this event and gave this opportunity to the girls. Gaya, Year 9 We video-called Jeju and everyone sat around a desk ready to win. It as tough and everyone was under pressure. The first question was given and straight away everyone began working fast. Throughout, everyone was tense. London and Jeju were side by side then suddenly London took the lead. Many year groups participated to keep us four points ahead. Soon the harder questions kicked in and Jeju were confident. They took the lead and eventually took the crown.
They won but London will be back! We put a lot of effort in and it was fun to talk and compete with our sister school. Tia, 8L The Further Maths support program has published a report on Girls Participation in Further Mathematics. 18% of girls nationwide do A Level and 2% do Further Mathematics. At NLCS the figures are over 70% and 20% respectively. We are very lucky to be free from some of the gender-bias affecting young women in other schools.
On the Tuesday 12th December 2017, Mr Linscott accompanied the three of us to the Institute of Mechanical Engineering in Westminster, London.
We set of from North London Collegiate at 5pm and took the Jubilee line to Westminster station. When we got there, we were amazed by the sight before us. There were so many people inside and the building looked beautiful. Inside, it had intricately carved wood paneling and a very elaborate staircase. During the evening, we had the opportunity to partake in a range of activities including: building and racing models of the 1000 mph Bloodhound World Land Speed record car and a model of a Rolls Royce jet engine, designing a robot hand and talking to engineers. We were also able to debate cutting-edge research ideas with engineers from Imperial College and Kingston University, explore aerodynamics in a special wind tunnel with model aeroplanes and drones and chat to RAF Cadets. Two lectures took place throughout the evening; one on aerospace engineering and the other on robotics engineering from Imperial College, which we really enjoyed. In addition, we won a competition of who could create the fastest and most efficient model of a car using certain materials in order to travel the furthest after being launched using pressurised gas. Our design theory was that we should create the car with the most streamline, least amount of mass, reinforced foundations, strong bonding and eventually a triangle shape. When we won, we were very surprised as we were not sure our design theory would pay off, but it did, and we were awarded our certificates and a kit to build our own Bloodhound World Land Speed record car. Overall, the evening was very interesting and we feel like we gained more of an insight into the world of engineering and especially mechanical engineering. We would definitely recommend it to anyone interested in engineering, not just mechanical engineering, and maths. By Isabella Menéndez, Alyssa Quinney and Visharlya Vijayakumar. The Year 12s who went on this trip have reflected on their day...
"It was a very interesting and informative - I am a mathematician and I am proud of it!" "A truly fun day! Bobby Seagull and Dr Rock were entertaining and inspirational. I had a great time!" "It was really inspirational and interactive" "Maths in Action was educational and thought provoking." "I really enjoyed Maths in Action as the speakers were so lively and engaging. Listening about problems and theories I had never heard about before was very inspiring." "A day planned with a variety of fascinating talks by very different teachers." This summer, I went on a residential course on Computing and Microelectronics at the University of Southhampton with the Smallpeice Trust.
The course involved building a substantial autonomous robot, capable of navigating its way around an arena, collecting objects and avoiding confrontation with rival robots in the process. It required a great deal of electrical, structural and mechanical engineering, through which we were able to learn about rapid prototyping, soldering, circuit design and fabrication, debugging, and wood-working. Programming was most challenging for me, as although I had some experience with coding on the Arduino in C++, I was yet to programme a Raspberry Pi in Python. After teams had miraculously scrambled together a functioning robot in less than 5 days, we put them to the test in an arena appropriately named “The Cube” to compete against other teams. Some robots had speed but displayed no signs of intelligence, others were blessed with amazing code yet lacked stealth. In the practice round, my team’s robot did a lap clockwise only to reverse anticlockwise, erasing any points gained in the first lap. What followed was an intense 10 mins of tremendous team-work to modify our code, which subsequently ensured our success in the tournament rounds (at least until the semi-finals!). During the course, we also gained first-hand experience of university life, mingling with academics and students from the Electronics and Computer Science department at the University of Southampton; had the opportunity to visit ECS’ nanofabrication cleanrooms and an impressive High Voltage Laboratory; and took part in engaging talks on computer vision, cyber security and robotics. I learnt so many things during this course, including the importance of simplicity in design and coding, and was lucky enough to meet the most amazing people who all shared the same love of engineering as me. I strongly recommend the Smallpeice Trust courses to any aspiring scientist reading this because it’s an amazing opportunity to explore your passion through hands-on learning. By Kureha Yamaguchi (13MFW) Our School won Round 1 of Hans Woyda Competition on the 4th of October 2017! Hans Woyda is a team competition in the form of two teams of 4 from separate schools competing against each other. Although there was pressure to win, I feel extremely privileged to have represented Year 12 in the NLCS team; and the experience was worth it.
Most part of the competition was spent one to one which means that I was up against the year 12 student in the opponent team on my own. To answer each question within 30 - 90 seconds was definitely a challenge for me. Not only should I spot a quick way of solving the questions, but I also needed to be familiar with all topics so that the time taken to solve the question can be significantly reduced. For example, one of the year 12 questions was: ‘Given that 0° < x < 90° and 13 sin x = 5, find the value of 5/tan x.’ To obtain the answer, 12, one must be aware of trigonometry functions and their use on a right-angled triangle. Additionally, having the knowledge of the side lengths of 5, 12, 13 of a right-angled triangle would greatly decrease the time taken to work out the solution of the question. Overall, I am really glad that I took part in such a fun and exciting experience (including little disappointment when I made a stupid mistake). Our team members have all done an amazing job which results in the victory. Good luck with Round 2! The first session of Pi Bar was a huge success, making conic sections out of clay. It was great to explore the 3D shape and use it to print shapes onto paper.
Thanks for having us Art Department! On Monday 25th September, a group of Sixth Form students attended a maths lecture run by the UK Mathematics Trust at the Science Museum, London.
The talk, given by an Oxford lecturer on the gaps between prime numbers, was extremely interesting and insightful. The girls also had the opportunity to hear about those who had competed for the United Kingdom at international maths competitions, inspiring a drive to join them on the teams. Thank you to the Maths Department for organising such a great trip, it was extremely enjoyable for all involved. By Imogen Woods-Wilford, Year 12In July, Anusha and I attended a week-long
Inspire Headstart course at Queen Mary University to gain a better understanding of the various STEM fields, meet some like-minded people with similar aspirations and get a sense of university life and it gave us the opportunity to do just those things.In the earlier days, we were given various talks and corresponding activities to complete as a group, along with a set of instructions and equipment we’d never seen before. We were shown around the university, tried several meals around the campus, were able to interact with the creations of the university’s robotics students and with the end of the week as a deadline, were given a topic to present to a panel of expert judges; ours was ‘Electrolysis in cars’. Our presentation, although maybe not the most informative, was definitely the most fun as it consisted of an in-depth role play demonstration of Electrolysis. Alongside friends, this experience was something I am really grateful for and I’m sure they would recommend it as well. Routes into STEM was three days spread out over a few weeks where we went to different institutions and were taught about the courses and opportunities they offered primarily involving engineering.
We went to a college, a university and then a company. I want to highlight that this course doesn’t teach you STEM topics but shows you different paths you can take to a future and career in STEM such as apprenticeships. I thoroughly enjoyed my experience and got to make friends with people from other schools as well as coming out of the experience well-informed of my options in pursuing a STEM future. This course would be good for people who want to know what other directions they can take into STEM education and employment instead of following the traditional six form and university route. It turns out that the most impressive-sounding work experiences are not necessarily the most rewarding. Chisato, ex-big-six with a rather splendid scientific mind, immersed herself in real-world work this summer. She was immediately promoted to a maternity cover as she proved so much more impressive than the average intern. Hurray for the confidence and maturity that NLCS instils! Here is what she writes about her 8 weeks of hard graft in the real world of work. Sixthformers heed her words - Excel skills are useful!!!
"Through a personal contact through the Maths Department, I worked at the recruitment company Total Jobs Group, located in the snazzy Blue Fin Building in Southwark. I helped match account records on Salesforce (which is a Customer Relationship Platform) and enrich registration and contact details of companies so they could be passed on as leads to the sales team. I also helped fill in the trackers for managers to analyse their team’s performance and build a custom report on Salesforce. It was a great opportunity to practise some self-taught Excel skills and gain experience navigating through Salesforce. The working environment was very pleasant and I enjoyed having lunch with my friendly colleagues as well!" The Dragonfly Engineering course at Oxford was a rewarding and insightful experience. Having the opportunity to build our own circuit boards, and use Computer Aided Design software, in Oxford’s own Engineering department, enabled us to get a real appreciation for the many facets of modern engineering.
The day started off with a brainstorm of what we thought engineering was, and our instinctive ideas on how to tackle problems, such as ‘How to stop a car from speeding around bends in the road’, and ‘How to increase visibility of oncoming traffic’. The questions demanded the consideration of safety, reliability, accuracy and more. This got our brains fired up for the day ahead, putting the role of and engineer into perspective. My group’s first activity involved a short quiz to found out what kind of person we were. The options included: investigator, persuader, communicator, entrepreneur and manager. We were each given badges of our top two results, and were encouraged to explore which jobs might suit our interests and personality traits, opening up options besides engineering. I thought this was a fun way to inspire people my age to see what might be open to them and seek a profession that they will enjoy now, and for many years to come. Next we were given the task of creating a display of flashing LED lights on a circuit board, to use as a decoration. In preparation, we were taught how to use formulae and apparatus to measure the time taken for capacitors to reach a certain voltage, and how this would effect the speed at which our lights flashed. We discovered the relationship between capacitors, resistors and and integrated circuit, before assembling our devices, and having the opportunity to decorate them with pipe cleaners and glitter! This activity demonstrated how the knowledge of physics aids engineers and how it is vital in the work they do. After a generous lunch break, we were escorted to one of the computer labs, where we were going to use a Computer Aided Design software, in order to build a rotor. There was a short demonstration by one of the tutors, which made the whole thing look very easy. When is came to trying this out ourselves, it took a lot of getting use to. While managing to do the majority of the work ourselves, I think it is fair to say we would not have gotten through it if it weren’t for the help of our excellent tutors and the older Headstart pupils. With some of us having finished with time to spare, we were given the treat of creating our own key-ring design to be printed by a 3D printer, which we could then take home. Personally, this was the highlight of the day. Being a fairly artistic person, I enjoyed the fusion of the creativity of design, and the reliability of maths and physics, and I think that’s what being an engineer is all about; being able to think inventively to solve problems, but managing to execute those ideas with the utmost precision. It was a truly amazing experience, which sparked in me a greater passion for engineering, and I loved every aspect of it. I was lucky enough to be given such an opportunity, and I hope those of you reading this will be inspired to take a course like this, to widen horizons and just have fun. ## Click here for the paperImagine a maths exam where you have as much paper as you like AND as much time as you like (almost) AND if you get absolutely no marks whatsoever it doesn't matter in the slightest. That's my type of maths exam.
In the MOG - international maths olympiad for girls - there are 5 questions and 2.5 hours, which is the longest time you will ever spend at school doing one single activity!! I enjoy the sense of freedom- to either get right to the bottom of one puzzle or to flit between all 5 of them - working away at the first until the next one starts to look more interesting... Something else that marks out the MOG from the other ukmt challenges (eg multiple-choice junior/senior maths challenges where you've just got to get as many answers right as fast as possible) is the focus on presentation and demonstration of complete solutions. When I am writing my workings, I imagine it like a conversation with my piece of paper or a performance of what I am thinking. If you get a question about a real life situation, then why not try it out? This year's question 3 was about some kind of sudoku. Quinn was told by her friend how many counters were in each row and column of a 5x5 square board (all of these totals had to be different). Quinn's friend could place any number of counters on a board square- from zero to, I suppose, infinity. Were there any situations in which Quinn would not be able to guess correctly how many counters in each square? I created my own 5x5 board of numbers on my paper and gave myself a set of 10 totals. Then i tried to re-create my sudoku using only the 10 totals. Soon, two things became apparent: Quinn's intellect and patience were streets ahead of my own, and the only board Quinn wouldn't be able to work out would be one where two boards have the same set of 10 row totals. Questions 1, 2, and 4a made you prove statements about shapes. Sometimes, in the proofs questions, it feels like they've told you the answer in the question, and the examiner knows that it's the right answer so why should you have to tell them, and there's no way that you would have worked out the statement in the question unless you'd been told, so how ironic that you're the one doing the explaining! However, particularly in question 2 it was really useful to have the statement to prove, because it provided an idea of where to start, working out the maths going on behind the shape. Question 5 was a lot less fun than it looked. 4 girls all had models for different fractions - Bella had "(6p-5)/(3p+6)", Isabella had "(n+1)/n", Christine had something else. Using different values of their constants, each girl could multiply examples of her fraction model together as many times as she liked with the aim of getting an integer answer. Drawn in by the word "integer" (a clean whole number is always pleasant) I thought I'd have a go. However, After working out that Isabella could obtain all integers (bigger than 2) [because 2/(2-1) = 2 and multiplying that by 3/2 = 3, and that X (4/3) = 4 and so on], two successive disasters happened. Firstly, I realised that I'd done the whole next stage of working calling Bella "Christine" (so I had to do tons of scribbling out - yuck). Then, completely missing the fact that actually, Bella (6p-5)/(3p+6) couldn't make any integers at all [because there aren't any values of p that could make a multiple of (6p-5) divisible by a multiple of (3p+6)], my prospects for Bella's integers just kept getting more and more disconcertingly un-integer-like!! I didn't try 4b until the very end because the diagram, drawn in the question, looked like someone had taken a chessboard and smashed it onto the floor, breaking the black squares but not the white ones. Crazy. Fortunately (thank you UKMT) it was made out of squares and equilateral triangles, all with sides of length 2. X and Y were two points in the middle of two of the squares, and you had to find the distance between them. The first thing that I thought of doing was to draw circles around the squares (regular shapes fit really nicely inside circles - and outside of them, I think that's because tangents to a point are equal). It turned out that length XY was equal to 6 circle radii. So the MOG ended happily ever after, and I'm now really excited to go through the answers. It's always a true enlightenment to see the clear solutions that your tangled workings could have looked like. However, I'd still say that getting the answers wrong is just as much fun as getting them right - if anything, it's more creative!! -Anoushka Sharp, Year 12 |
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