Mathematical Angles the official magazine of the MA President Prof Nira Chamberlain Mathematics and the Value of Representation News from Committees Mathematical PiE No. 220 Simply solve 14 How many can you solve? October 2023 Issue 1
• Get a degree and recommendation for qualified teacher status in just three years. • School placement experience integrated through the course • Delivered by experienced subject experts. Apply by Wednesday 31 January to join Edge Hill University in September 2024. We also offer PGCE Maths, Masters, EdD and PhD qualifications. A great teacher can show their pupils that mathematics is a way of communicating and understanding the world. BSc (Hons) Secondary Mathematics Education with QTS Scan or visit ehu.ac.uk/X1GC
3 A word from the Editors Welcome to the new look members’ magazine, Mathematical Angles! This single, larger magazine is now the flagship magazine for the Mathematical Association and brings their strengths together, with greater flexibility. It is an exciting time to be involved in these changes and an opportunity to try new things out without losing the aspects of SYMmetryplus, Mathematical Pie and MA News that you know and enjoy. Our readers’ contributions have always been at the heart of our respective magazines, and we are grateful to everyone who has ever written in, especially those who do so regularly. So, one new addition is that we are starting a letters page, for readers who want to comment on articles, give feedback or share an idea, however long or short, or just supply gems of information you think others would like to read. As always, this relies on you, so please send your letters to angles@m-a.org.uk. We’d love to hear from you. This first edition includes articles from Paul Stephenson, who writes on the recent Einstein monotile discovery and on a problem by Arsalan Wares published in SYMmetryplus a year ago. Coralie Colmez makes her debut, writing about p-adic numbers. Neil Walker provides a Simply Solved set of problems as well as a first lot of Systems of Equations puzzles, whilst Erick Gooding provides a mathematical magic trick as well as an out-of-this-world barn dance! Jenny Ramsden delves into the life of Reverend Thomas Bayes and Peter Ransom writes about trapezoidal doors. There is a brief report on how the UK team got on in the recent International Maths Olympiad, and a sample problem from the competition. We also celebrate Halloween with some Haunted puzzles and Christmas through a crossnumber from Mike Rose. There is news from our committees to keep you updated, and our most recent events, whether they are from the Professional Development committee or Branches committee, and words from the President of the MA, Prof. Nira Chamberlain. And, if that’s not enough, there are puzzles aplenty in Mathematical Pie to keep you occupied! Many thanks to all our contributors in this issue. Answers to questions, errata, notes and resource sheets can be found at www.m-a.org.uk/ mathematical-angles Oli Saunders and Dave Pountney angles@m-a.org.uk News from HQ New Membership Administrators At the begining of September MA HQ welcomed Jennifer Moore and Lucy Watson, who are job sharing the role of Membership Administrator. Jen’s working days will be Mondays, Tuesdays and Wednesday mornings, with Lucy’s working days being Wednesday afternoons, Thursdays and Fridays. Jen and Lucy will be supporting Challenges and Membership Officer Alice Hall and the wider team when required. My name is Jen! My role before joining the MA was as a baby and toddler swimming teacher and my previous roles have been within Sport Development and Apprenticeships, where I consistently worked within customer service. In my spare time I enjoy spending time with my family and visiting the cinema and theatre. I also enjoy most sports and spend a lot of my time at swimming, gymnastics and toddler rugby with my children! I am looking forward to working with and meeting everyone over the coming months. My name is Lucy! For the past 8 years I have worked as a Marketing Administrator within the printing & safety industries. Here I have provided support to a range of departments from technical to sales. I also have a degree in Visual Communications and experience of Adobe Design packages. It is these skills which I hope to utilise in my new role at the MA. In my spare time I enjoy spending time with my family, visiting new places and taking photographs. At the end of May we packed up our old office, said our goodbyes and moved to our new office at Loughborough University Science and Enterprise Park. The university campus has fab views, a duck pond (see pic) and plenty of green spaces for a walk during lunch. Our team has settled in well and finally unpacked the last of the moving boxes, so the space looks more like home. The office currently houses the library’s duplicates of the special collection, members are welcome to visit by prior arrangement, just get in touch. We have also welcomed two new members of staff in September, Jennifer Moore and Lucy Watson. Meet them here …. Mathematical Angles | October 2023 | Issue 1
4 Mathematical Angles | October 2023 | Issue 1 SUBMIT A SESSION PROPOSAL
5 CONTENTS October 2023 | Issue 1 3 News from HQ New Membership Administrators 6 Mathematics and the Value of Representation Professor Nira Chamberlain 8 News from Joint ATM/MA Primary Group Sue Lowndes 10 Teaching Committee News David Miles 10 Branches Committee Cindy Hamill 11 “Don’t let me influence you!” Erick Gooding 11 p-adic numbers and how to get from Hereford to Peterborough Coralie Colmez Mathematical PiE 15 The einstein Paul Stephenson 19 Japanese triangles Oli Saunders 20 It’s a Barn Dance, Jim! Erick Gooding 21 What shape is your door? Peter Ransom 22 The Reverend Thomas Bayes Jenny Ramsden 26 Arsalan Wares’ problem Paul Stephenson 29 Systems of equations 1 Neil Walker 30 Haunted Oli Saunders 31 Crossnumber 7 Mike Rose 32 Simply solve 14 Neil Walker 34 Forthcoming Branch Events 35 Branch Contacts 36 Letters Mathematical Association Charnwood Building, Holywell Park Loughborough University Science and Enterprise Park Leicestershire LE11 3AQ Tel: 0116 221 0016 Email: office@m-a.org.uk Website: www.m-a.org.uk Please note that the copyright of all material published in Mathematical Angles is held by the Mathematical Association. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Requests for permission to copy material should be sent to the Editor-in-Chief via email: editor-in-chief@m-a.org.uk or by post to the address to the left. The Mathematical Association produces a number of journals including Primary Mathematics and Mathematics in School. For more information visit: https://www.m-a.org.uk/ma-journals The views expressed in this publication are not necessarily those of the Mathematical Association. The inclusion of any advertisement in the journal does not imply endorsement by the Mathematical Association. Mathematical Angles is produced by Media Shed on behalf of the Mathematical Association. ISSN 2977-182X Mathematical Angles | October 2023 | Issue 1
Under-representation At last year’s Black Heroes of Mathematics conference – Dr Aris Wenger CEO of Mathematical Enrichment for Diversity and Learning, USA, presented this statement: You can’t say that mathematics comprehension and aptitude is so important to health of a nation and have so many under-representation populations and not label it a crisis! For a long time of my life, I thought that ‘mathematics is mathematics’. Whether is be shapes, numbers, algebra, trigonometry, mathematics is mathematics. It was the world that I could escape into and just enjoy being a mathematician. During both my undergraduate and Masters degrees, I would wrongly assume that all mathematical theorems were invented by men of European origin. Little did I know that this same view held by others would impact me later in life. This article documents why I changed my mind. Before the Black Heroes I am of the opinion that any child that wants to be a mathematician should be allowed to be a mathematician. Let them escape into that mathematical world of numerical joy of wonder. However, for many that take this freedom of choice for granted, such doors are not open for everyone. Most people are familiar with my story detailed in my talk, The Black Heroes of Mathematics [1]. That back in the day, my career teacher tried to persuade me to become a Boxer despite my love of mathematics, which led my Dad to give me my life long motto “You don’t need anybody’s permission to be a great mathematician” However, those motivational words didn’t impact me very much later in life. Growing up in the education system, mathematics was very much mathematics, diversity had no influence on the subject. Looking back how many of my maths lecturers from A-level to masters were female? One. How many of my maths lecturers from A-level to masters were Black? None. How many of my maths lecturers from A-level to masters were Asian? One. When I was at school and I was being taught the Pythagoras Theorem, I was told that this was invented in Europe. The next lesson would be History where I was taught about the Pyramids built by the Egyptians. There was no one telling me that these two things could possibly be related. The idea that there was underrepresentation in those that taught me maths, or that the subject itself lacked diversity never occurred to me. Mathematics and the Value of Representation 6 Mathematical Angles | October 2023 | issue 1
However, it is those very factors that try to close that mathematical door of opportunity to me and others. Pursuing a career in sports and music seemed to be the default option placed in front of me. Nevertheless, when the same default options were put in front of my children, I had to decide to make a stand to become a professional mathematician. For the first few years of becoming a mathematician, it was challenging as some of my peer group and stakeholders held the same view as I did: That all mathematical theorems were invented by men of European origin Hence, how can I, some one of Jamaican parentage, possible be able to solve a mathematical problem that everybody else is struggling with? What we are taught in schools, whether intention or unintentionally, follows us and has consequences. We teach the children today to become the adults of the future. I remembered what my Dad told me, stuck at being a professional mathematician and the rest is history. Black History Month Black History Month is celebrated in the UK every October and in the last few years I have been pleased to see Black Mathematician featured in the maths classroom. What is sad is that after three weeks or so, those nice posters are pulled down. It would be kind if at least one could be left all year round. Though I am not the only Black Mathematician in the world (smile), I am deeply honoured when I do see drawings of me and my quote being put on the wall. Some schools have even named classrooms after me! May I repeat that I am not the only one, there are many, many more! The Future My journey started with me thinking that mathematic is just mathematics. However, I came to realised that mathematics is stronger when it is more diverse. For everybody, seeing yourself in the future of STEM and mathematics is so important. For this, I do take my hat off to Dr Anne-Marie Imafidon, whose company Stemettes have just celebrated its 10th anniversary and what they have been doing for girls, by far the largest under-represented group, is just amazing. I am also proud that the Black Heroes of Mathematics Conference, which I have the honour of chairing, is going into its fourth year! But let me do a little spin on what my Dad told me: Everybody has the potential to be a great mathematician and, as educators and stakeholders, we must play our part to keep that door firmly open until the said pupil chooses to permanently close it by themselves. End Note [1] The Black Heroes of Mathematics The Black Heroes of Mathematics, Dr. Nira Chamberlain - 07/07/20 - YouTube Professor Nira Chamberlain MA President 2023-4 7 Mathematical Angles | October 2023 | issue 1
News from the Joint ATM/MA Primary Group The Joint Primary Group met on Friday June 9th, 11am – 3pm. We were extremely fortunate to hold this at the Royal Society again and that it was a beautiful sunny day in London. The meeting was chaired by Alison Borthwick. The group now comprises 128 members and 28 members were in attendance. Out of our three annual meetings, two meetings online and one face-to-face meeting seems to be working well at the moment. Feedback from the Easter Conference. This year’s conference was a joint one of five subject associations held at Warwick University and this Primary group led a workshop. Many people were involved in the planning of the session which was entitled ‘Why map making?’. Alison Borthwick, Alison Parish and Liz Woodham summarised the session briefly and thanked the ECMG (Early Childhood Maths Group) whose work featured heavily. Delegates seemed to enjoy the session and rich discussions took place. The group was later invited to write up the session in an article for the Scottish Mathematical Council Journal and Priya Shah is coordinating this. Alison P is also intending to write an article for Primary Mathematics about her work with the Wildlife Trust in Suffolk which formed part of the session. Academy for the Mathematical Sciences. Lynne McClure gave us an overview of the Academy for the Mathematical Sciences. The aim is to bring together all existing individuals working in mathematics, to be ‘an authoritative and persuasive voice for the whole of the mathematical sciences’. The idea is to be able to arrive at balanced views which can feed into policy recommendations. Currently it is operating as a ‘protoacademy’ – during this phase the structures are being put in place to enable the Academy to launch next year. Currently there is one parttime paid Executive Director, fifteen voluntary members on the Executive, and more than 50 others who are involved in particular projects. The proto-Academy workstreams include education, implementation of mathematical sciences, Equality, Diversity & Inclusion (EDI), development and finance, advocacy and policy, and communications. The Academy will be a fellowship organisation and Lynne asked members of this group to send her suggestions about what the profile of a fellow working in mathematics education might look like. The link to the Academy website is https://www. acadmathsci.org.uk/ Lynne also gave a brief update about ACME (the Advisory Committee on Mathematics Education, a committee of the Royal Society). Of the four phase-specific contact groups, the Early Years and Primary group (chaired by Sue Gifford) are currently focusing on spatial reasoning. The Mathematical Futures Programme has been set up by ACME to look at the mathematical competencies that will be needed in the future and what education systems will be needed to develop these competencies. The group is advocating for cross party agreement on future policy decisions and there will shortly be a discussion paper identifying key issues. One issue Lynne mentioned, for example, is around the high proportion of students who currently fail their GCSE mathematics re-sit. Many of us in the room were genuinely shocked by this high proportion. Ultimately the vision includes a policy unit with paid researchers who will research and collate evidence to support recommendations to policy makers around (for example) teacher recruitment, early career mathematicians, or funding for mathematics research. 8 Mathematical Angles | October 2023 | issue 1
Role of the primary mathematics leader in England’s re-modelled school landscape. Cath Gripton gave an overview of this project, funded by the Wellcome Trust. The changes to the school system mean that current structures, e.g. academies, maths hubs, teaching schools etc. all have different geographical ‘footprints’. Cath et. al. have looked at the impact this has had on CPD for Primary maths teachers: • Equity issue – how fair is this distribution? • How has this impacted on quality? The three localities in the study were purposefully chosen to be different from each other, and London was deliberately not included. The schools were sampled to try to get a mix that roughly reflects the proportional make-up in each locality. Class teachers, maths leads and Head Teachers were interviewed in each school. System leaders in each locality (i.e. from Multi Academy Trusts (MATs), DfE, teaching schools, etc.) were also interviewed. (see Gripton, C., Hudson, G., Greany, T., Noyes, A., & Cowhitt, T. (2022). Teacher learning in a shifting school landscape: The implications of academisation for professional development in primary mathematics. FORUM, 64(3), 32-41. ) KS1 and KS2 tests. Pip Huyton led a discussion and reflection on the recent 2023 KS2 NC tests. Questions for these tests are developed by NFER and ACER (Australian Council for Educational Research). Pip had spent time analysing questions and she shared a few slides in which she drew attention to several questions and raised some interesting and thought-provoking points: • Questions in the papers are not arranged in a graduating level of difficulty now, which may have implications for teaching. • The contexts of some of the questions were queried. • There seemed to be a deliberate choice of difficult numbers. Members suggested that children would benefit from a ‘warm up’ at the start of the arithmetic paper. A point was made that if children come out of the arithmetic paper feeling shaky, then this may impact on the first reasoning paper, which is timetabled on the same day. The group commented that there seemed to be a lack of opportunities for lower attaining students to reason. There are also implications for next year when KS1 SATs are no longer statutory. Royal institution (Ri) Masterclasses. Alison Eves gave us a flavour of the Ri masterclasses. Ri masterclasses generally focus on topics outside the curriculum. These are hands-on sessions led by enthusiasts for children who have a love of maths and/or are interesting thinkers. The children are drawn from a number of schools and the classes offer a great opportunity to meet like-minded people. Some elements run at the Ri, and groups are also run around the country. Linda Glithro spoke about the Masterclasses she ran pre-Covid in Caversham, with lots of support via the Ri for the first year. She invited all schools in Caversham and found the experience very rewarding. Rebecca Turvill described the Masterclasses she has run with Alison Eves through her MAT, all online as the schools are geographically far apart. Rebecca used some 6th-form students to lead sessions, which helps to bring primary and secondary together. The 6th-form students are always amazed at how primary children think. Rebecca Thomas also talked about her experiences; she started as a speaker last year with Alison Eves during her Masters year. She said that it was a very positive experience. https:// www.rigb.org/learning/activitiesand-resources?type=30 An introduction to the fluency research project between England and Jersey schools. Ruth Trundley, Andy Parkinson, Alison Borthwick gave an overview of this project, which started in 2022 at the Easter joint maths conference. Andy participated in a session that Alison and Ruth were leading. Jersey schools currently don’t do the MTC (Multiplication Tables Check), but Andy wondered whether they should. Alison and Ruth’s immediate response was ‘no!’, but they realised that there wasn’t any hard evidence to support their view. This led to the project! Their current research question is: Do children in England perform better/ make use of their understanding of multiplication more than children in Jersey because of the focus on the MTC? They enlisted the support of 17 primary schools in Jersey, 10 in Norfolk, 10 in Devon and Cambridge giving a total of 3213 children across Y4 and Y5. The project was inspired by Anghileri et al’s (2002) research and built on Alison’s previous calculations research They also drew on the Position Statement of this group about teaching of multiplication bonds. They wrote 10 multiplication and division questions, per year group, using a variety of ways to present these. They also wrote questionnaires for teachers and pupils. They wrote 10 multiplication and division questions, per year group, using a variety of ways to present these. They also wrote questionnaires for teachers and pupils. 9 Mathematical Angles | October 2023 | issue 1
Preliminary findings: • Children knew the multiplicative relationship but not necessarily the associated division fact. • They were generally not visualising arrays. • Children who got the abstract calculation correct did not necessarily get the corresponding word problem correct. • The team conjectures that children in England are not encouraged to develop their fluency. Instead it is about rote, with no mathematical structure drawn out. • The team has various working conjectures. They aim to write and publish paper/s on this interesting research in the future. The group look forward to them sharing more in in due course. Future meeting dates and venues Saturday 30 September 2023: online Saturday 27 January 2024: online Friday 7 June 2024: face-to-face (London) Sue Lowndes Teaching Committee News On the morning of Saturday 15th July 2023, Teaching Committee met virtually. This meeting was reschedule from mid-June at the request of our guest speaker, Steve Wren HMI, to coincide with the release of the Ofsted mathematics subject report. We are grateful to Steve for giving up part of his weekend to brief Teaching Committee on this important document. Members of the Joint MA/ATM Primary Group have expressed some reservations about the new National Professional Qualification in Leading Primary Mathematics and it is possible a paper outlining their concerns will emerge from their discussions. A focus for Teaching Committee in the Autumn Term will be the production of a position paper about the withdrawal of a number of mathematics degree courses. More recently our 11-18 subcommittee met on Saturday 16th September, Teaching Committee again met on Saturday 23rd September and the Joint MA/ATM Primary Group met on Saturday 7th October. All three meetings were held virtually. Outcomes from these meetings will be reported in the February edition of this publication. We are planning for a face-to-face meeting of Teaching Committee at our Loughborough Headquarters in January and also intend to hold an Open Meeting of Teaching Committee at the upcoming Joint Mathematical Subject Association Conference ‘Shape Up’ in April 2024. If you think you might be interested in getting involved with the work of Teaching Committee, our 11-18 subcommittee or the Joint MA/ATM Primary Group, please send an email to tc-chair@ma.org.uk . David Miles Chair of Teaching Committee Branches Committee There were a couple of Branch events in June and a few Branch events are already planned for the Autumn term. It is hoped that, now schools have returned, more Branches will start to produce events for their members. We continue to ask Branches to let either myself or MA Headquarters know about any events at their Branch so that they can be advertised on the various MA outlets. Liverpool Branch has advertised a fairly full list of events for this academic year and will have had their first popular lecture on Wednesday October 18th 2023 – ‘Fantastic Numbers and where to find them’ by Tony Padilla Professor of Physics at Nottingham University, at The University of Liverpool, before the date of publication of this magazine. More details of Liverpool Branch events follow under Forthcoming Branch Events. Yorkshire Branch has announced the date of their annual Christmas Quiz on Wednesday 6th December 2023 at The University of Leeds. The next Branches committee meeting is at 9.30 am on Saturday 28th October 2023 via Zoom. Information will be emailed to representatives before this event. It would be great to see every Branch represented! Cindy Hamill Chair of Branches Outcomes from these meetings will be reported in the February edition of this publication. 10 Mathematical Angles | October 2023 | issue 1
11 “Don’t let me influence you!” Abi is showing Jamal a card trick. She explains that he will choose how many cards from the full deck to turn face-up and he will decide where to hide them within the deck. Then he will choose a number of cards from the deck to make a new stack, and she will then make a prediction about which way up the cards are in the two stacks. “Don’t let me influence you” she tells him before they start. STAGE 1: She first deals 13 cards facedown to Jamal and says “We’ll start with the number of cards in each suit, but many people think that is an unlucky number, so let’s increase it – you decide by how much. Choose a number between one and ten.” Jamal might answer “Seven” and she would deal him seven more. She then tells him to turn all his cards face-up and ‘hide’ them in various places anywhere within the facedown deck. STAGE 2a) Starting the new stack: “Now choose another number between one and ten.” “Nine” “Would you like them all from the top, all from the bottom, or a mixture? – don’t let me influence you! “Three from the top and six from the bottom.” She takes 9 cards as instructed and without turning them over puts them down to one side. STAGE 2b) continuing the new stack: “Now choose another number between one and ten” “Eight” “Would you like them all from the top, all from the bottom, or a mixture? – don’t let me influence you!” “Five from the top and three from the bottom.” She takes 8 cards as instructed and puts them onto the new stack. STAGE 2c) continuing the new stack: “Now choose another number between one and ten” “Five” “Would you like them all from the top, all from the bottom, or a mixture? – don’t let me influence you!” “All five from the bottom.” She takes 5 cards as instructed and puts them onto the new stack. STAGE 3: The prediction “Now, you do agree, don’t you – that you chose how many cards were turned face up? . . and that you had a free choice of the cards that were chosen for the new stack? . . . So how could I predict what will happen when I turn the new stack over?” She turns over the new stack and declares: “There are two more face-up cards here than there are in the other stack.” When Jamal checks, he is amazed to find that she is correct!! Of course, many of the choices that Jamal has been offered are completely irrelevant and are nothing more than misdirection. These are the only important facts: The number of face-up cards (f) was originally 20, and Abi decided to stop after Stage 2c) – when the number of cards (n) chosen for the new stack happened to be 9 + 8 + 5 = 22 (which is 2 more than f)! Abi could have gone on with extra stages such as 2d) and 2e) etc. if she had wished, when the difference would have been greater, but still could be calculated by comparing f and n. This result is quite surprising but easily shown to be true with a little bit of logic. Let’s suppose that within the new stack there happen to be x face-up cards. Here is the situation before the new stack is turned over: New stack (n cards) Face down (n – x) cards Face up x cards Remaining stack (________) cards Face down (_____________) cards Face up (f – x) cards When the new stack is turned over the (n – x) cards which were face down become face-up and so the difference between the two sets of face-up cards will be (n – x) – (f – x) = n – f. When n is less than f, there will be more face-up cards in the ‘remaining’ stack – as there would have been if Abi had stopped after Stage 2b) in our example. This trick can be made even more impressive if Jamal is given a final free choice of which stack to turn over, but if he chose the ‘remaining’ stack, Abi would need to do a different calculation to make a prediction about the face-up cards. You will need to fill in the missing expressions in the brackets above to help you to find out what that calculation would be! Erick Gooding p-adic numbers and how to get from Hereford to Peterborough This is an early version of one of the chapters in my YA novel The Irrational Diary of Clara Valentine, in which Clara helps her (very hot, but not so good at maths) classmate Ty find his brother Oli, who’s gone missing (she also has to deal with her parents acting secretive, her little sister morphing into a demon child and best friend M going silent on her – not to mention that uni applications are due. It’s a pretty full start to the year!) Oli’s a genius computer hacker, and the pair are trying to get into a laptop that Oli left behind, but it’s been programmed to ask for the answer to maths problems instead of a password. The chapter below changed quite a lot in the final version of the book, and in fact I ended up swapping out the problem about p-adic numbers for one that made more sense for the plot. Even though it was the right thing for the story I was a little sad, because p-adic numbers are my dad’s life’s work and I’ve been hearing about them since I was very little! I hope you enjoy learning a little about them here. Ty typed in the answer we’d worked out together carefully, his long index Your new look content starts here! Mathematical Angles | October 2023 | Issue 1
12 finger pressing down each key with a click. This time we got the black tick showing we’d got it right, and then another question popped up on the screen. Sure, it was disappointing that we were still not inside the computer, but compared to my heart seizure when I thought I’d messed everything up, this was pretty much the best news of the year. Ty didn’t seem to feel the same way. He covered his face with his hands. ‘Are you crying or laughing?’ I asked. ‘I don’t know,’ he said from behind his hands. ‘Both?’ I made a face that I hoped was a calming smile and looked at the screen. Question 73: In the 3-adic world, which number is farthest from 5: 14, 15 or 59? One glance at the question was enough to tell me there was no way I could do it right then and there – I’d never heard of adic anything. ‘I don’t know what any of this means,’ I said to Ty regretfully. I really wanted to see his eyes go all huge and extra eyelash-y when I got the answer again. ‘But I’ll figure it out.’ ‘I know,’ he said quietly. ‘I know you will.’ He let his hands drop from his face and looked at me. ‘Honestly, sometimes I think I’m being crazy and Oli’s just off somewhere for a laugh and he’d be in hysterics if he knew what was going on in my head. Maybe this is all just a big ploy to make me finally give maths a real go.’ He smiled, but he was obviously pretending. I mean, if I’ve managed to convince myself that Oli is on the run from the Russian mafia after he found proof they hacked the US election, I wouldn’t want to know what scenarios are going round Ty’s head. Back home I looked up the term 3-adic, which brought up lots of articles about the more general term of p-adic numbers. I got that the ‘p’ in ‘p-adic’ represented a prime number (3 is prime, and so 3-adic is just one specific example of p-adic), but everything else read like random word association. I clicked on link after link full of stuff I didn’t understand – it seemed wild that people had written all these articles about a topic that I literally had not known existed just a day ago. I learned the term p-adic for the first time today, and Google spits out over five million results for it of people who have worked on them forever. I looked it up and it’s a German mathematician called Kurt Hensel who first described them in 1897. Well – that’s old, but at least it’s not Euclid old. I’m only a century and a bit late (see, this is the problem; even when I am actually working, I still find a way to procrastinate!) I turned back to the last article I’d been reading. One sentence stood out to me, something about the definition of distance in the p-adic world. It stuck with me because I’d never thought about distance needing to be defined. It seems a pretty obvious thing – how far away two things are from each other. Well, actually, it turns out that what I naturally thought of when I read the word distance is just one example of the mathematical concept of distance. The mathematical definition is simple. A distance is a measure between two things, say x and y, which obeys just these three rules: • Rule 1: The distance from x to y is the same as the distance from y to x. • Rule 2: The distance between x and y is always greater than or equal to zero, and the only case where it is equal to zero is when x and y are the same (x = y). • Rule 3: If you stop off at another point on your way from x and y, then you’ll have travelled more distance than if you go straight from x to y. I could definitely see that our usual idea of distance in the world – the one where you measure the straight line between two things – obeys all the rules above. But actually, there are lots of other distances which also obey the three rules. Here’s one: You start with a reference point A. Then the rule is that if you want to go from a point x to a point y, you always have to go via A (unless x, y and A are all in a straight line – in that case you can just go from x to y). This means that: • If x, y and A are not all on a straight line: distance between x and y = length of the straight line between x and A + the length of the straight line between A and y The distance between x and y is the length of the blue line + the length of the orange line • If x, y and A are all on a straight line Distance between x and y = length of the straight line between x and y The distance between x and y is the length of the orange line People like to call this the British Rail distance, because when you take the train in this country you always have to go through London. I’ll say that is barely an exaggeration, as I learned when I started planning routes to go visit different unis! Thinking about it that way really helped me understand how there could be a different way to define distance too. After all, if you’re taking the train from Hereford to Peterborough, it doesn’t matter how close they are on the map – all that matters is how long you’ll be sitting in a train for the journey. Thursday, October 12th Home from school today. There was a crisis this morning when my angel sister’s babysitter abruptly quit after ten minutes on the job, citing unreasonable behaviour and demanding compensation for psychological damages. I was the only person free to look after the princess who, you’ll recall, is suspended. (If, of course, by free, you mean willing to give up a day of precious education.) Mathematical Angles | October 2023 | Issue 1
13 The sweet child has actually been pretty easy to wrangle. First, she spent the morning sulking in her room, not needing any attention from me. A bit after noon I made lunch for the two of us and listened to a very long-winded story about her classmates, all of whom seem to be named after fruit. Then she made me listen to Ariana Grande, which I enjoyed more than I let on. After Ariana, she showed me all of her moves to Shape of You, singing blithely about how ‘my bedsheets smell like you’. Cute cute. I left her with a mission to learn a new choreography and perform it for me in 60 minutes, and settled on my bed with a bit of a conundrum. On one hand, Oli might have gotten embroiled with a drug cartel who is about to send us his severed ear in a box. On the other, my homework has been piling up so much since I’ve been spending so much time on Oli’s questions that I got in trouble with two different teachers this week for not having done something. On top of that, Mr Kelly has finally read my personal statement for uni applications, and apparently I use too many filler words. He deleted about 30% of my draft and would not accept my complaint that he was cramping my literary style, so I have to redo it for him to check again. Just then my phone beeped with a message from Ty—clearly a sign. Securing my future will have to wait. I opened my notebook back to Oli’s question. Question 73: In the 3-adic world, which number is farthest from 5: 14, 15 or 59? I thought I was settling in for another long day of doing research, but actually it turned out that learning about distances was the hard part! Now that I understood that there isn’t just one distance, I realised exactly how you create the p-adic world: you start by defining a new type of distance. Here’s what distance means in the p-adic world. To find the distance between two numbers a and b: 1. you take the absolute value of their difference, |a – b| 2. you see how many times you can divide |a – b| by p – say you can divide it n times. Then the distance between a and b is defined as 1/pn (We’re only looking at whole numbers here. The p-adic distance also works on fractions, but Oli only asked about whole numbers and I would like to thank him for that because it let me skip at least two pages of symbols!) The p in p-adic can be any prime number. In Oli’s question, p is 3, so we are in the 3-adic world. I followed the definition carefully to calculate the distances he wanted us to compare. The distance between 5 and 14 The absolute value of their difference is 9, which can be divided once to get 3, and then a second time to get 1. This means n = 2, and the distance between 5 and 14 in the 3-adic world is 1/32, which is 1/9. The distance between 5 and 15 The absolute value of their difference is 10, which can’t be divided by 3 at all. This means that n = 0, and the distance between 5 and 15 in the 3-adic world is 1/30, which is 1. The distance between 5 and 59 The absolute value of their difference is 54, which can be divided once by 3 to get 18, then another time to get 6, and once more to get 2. This means that n = 3, and the distance between 5 and 59 in the 3-adic world is 1/33, which is 1/27. In the end, in the 3-adic world, 59 is the closest number to 5, then 14, and finally 15 is the farthest! You can see that in the p-adic world, a number will be the same distance away from lots of other numbers. For example, in the 3-adic world, 2 is a distance of 1/9 away from 11, -7, 20, 65… Because of this, you can’t show all the integers in the 3-adic world in a neat line, like you can on the real number line (which uses our usual definition of distance). But a very pretty way you can represent them is with a fractal, like this: Mathematical Angles | October 2023 | Issue 1
14 You start with 0, 1 and 2 each in their own yellow circle. Then, you add more numbers inside each of the yellow circles based on how far they are from each other in the 3-adic world: numbers who are the same distance away from each other go together. 0, 3, 6 are all a distance of 1/3 from each other, as are 1, 4 , 7 and 2, 5, 8, which is how you arrange the second diagram. The next step is to add numbers inside each of the green circles based on which ones are a distance of 1/9 from each other. The next one would be to add more numbers to the blue circles based on which ones are a distance of 1/27 from each other – and so on forever! Obviously I was very pleased with how groovy my fractal looked, but I wasn’t exactly clear why some people would choose to spend their whole life exploring the p-adic world, when it’s so different from the normal numbers world. It turns out that the p-adic distance actually has some stronger properties than our usual distance (it’s called an ultrametric, whereas our usual one is just a metric) which means that you can solve some problems in the p-adic world more easily than in our normal world – and when you’re lucky, these results have applications back in our normal world too! I barely had time to message Ty to tell him I had the answer before Emma knocked on my door in a full glitter outfit and makeup fit for Rupaul, ready for her performance. Please send a prayer for me as she has picked an Olivia Rodrigo song, which has been playing on repeat for the past hour. Coralie Colmez Introducing Charlie Stripp - MA President 2024 - 2025 Upon receiving the upsetting news of the passing of Vicky Neale - due to become the next President of the Mathematical Association - and following a period of grieving and reflection, the Mathematical Association can now confirm that Charlie Stripp has accepted its invitation to be its President for 2024-2025. Charlie is a passionate advocate of mathematics education at all levels, from primary school to university. As Chief Executive of the maths education charity, Mathematics in Education and Industry (MEI), Charlie leads an organisation which plays a national role in mathematics curriculum development and teacher professional development. MEI plays a key role in the development and leadership of major government-funded programmes to improve maths education: the Advanced Mathematics Support Programme (AMSP), the National Centre for Excellence in the Teaching of Mathematics (NCETM) and the Maths Hubs Programme. Charlie joined MEI in 2000 to lead the initial development of what is now the Advanced Mathematics Support Programme (AMSP), supporting teachers and students of AS/A level Maths, AS/A level Further Maths and Core Maths throughout England. The AMSP and its predecessor programmes have been funded nationally by successive governments since 2005. In 2010, Charlie became MEI’s Chief Executive and since 2013 he has also been Director of the National Centre for Excellence in the Teaching of Mathematics (NCETM), whilst continuing as MEI’s Chief Executive. His curriculum development work includes working on the reforms that led to the current GCSE and AS/A level Mathematics and Further Mathematics qualifications, and pioneering the development of the Core Maths qualifications. This summer he chaired the Department for Education’s Expert Advisory Group on maths to 18. Before joining MEI in 2000, he taught maths in the state sector for 10 years, both in secondary schools and in a large Further Education College. Since first training to be a teacher, Charlie has been a member of the Mathematical Association, serving as Chair of Teaching Committee and a member of the MA’s Council in the late 1990s and early 2000s. He was awarded an MBE for services to education in 2015. Mathematical Angles | October 2023 | Issue 1
1 Mathematical Pie | Autumn 2023 | Issue 220 Autumn 2023 No. 220 Time for a Break - Elevenses? 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Leave one of the numbers out and put the other nine in the grid so that rows, columns, main diagonals are divisible by 11. Do you know the test for divisibility by 11? Add the numbers in the odd positions, add the numbers in the even positions, subtract, and if the answer is divisible by 11 the number is divisible by 11. If not then it isn't. For example: 21439, 15 - 4 = 11, so yes 320165, 9 - 8 = 1, so no 7226560, 14 - 14 = 0, so yes. G.C.S. I Have a Good Friend in Maumee I have a good friend in Maumee Whose age's last digit is 3 The square of the first Is her whole age reversed, So what must this lady's age be? From the website: www.mathstunners.org Geometry Can you do this, from an old Geometry exam set to boys aged around 13? Draw a quadrilateral ABCD and construct a triangle of equal area having A for vertex and its base along DC. You draw something like this: And then you have to find something like this, where the blue triangle has the same area as ABCD: W.R. How Many Zeros? The number 11 × 22 × 33 × 44 × 55 × 66 × 77 × 88 × 99 is written out in full. How many zeros are there at the end? C.J.S. Summing Unusual Some numbers are an exact multiple of the sum of their digits - for example the number 133. The sum of the digits is 7; 133 = 7 × 19. It is more unusual if a number and its reverse are both multiples of their digit-sum. The smallest example is the pair 12 and 21. The digit sum (for both, of course!) is 3; 12 = 3 × 4 and 21 = 3 × 7. You can see that 133 is not a number of this type since 331 is not a multiple of 7. Including 12, and 21, there are fourteen two-digit numbers of this kind. Can you find them all? (By the way, let's agree that we will not consider any number that ends in a zero, since on reversing they would then start with a zero, and we don't normally write numbers in that way.) You may have noticed that one particular set of these pairs all have the same factor. If we continue our search into three-digit numbers, would this observation help you? If I tell you that there are eight pairs of such numbers with one of the pair below 199, how quickly can you find them all? (This time we will rule out palindromes such as 171 since reversing them does not give a new number.) Finally - if you like a challenge - can you find the smallest number of this kind that has only prime factors for both the numbers of the pair? E.G
2 Mathematical Pie | Autumn 2023 | Issue 220 π Not Included The design for a teddy bear's head is based on a circle of diameter 10 cm. Its ears are parts of semicircles constructed as shown. What is the area of teddy's ears? H.K.M. Twelve Squares Here are three shapes arranged to make an outline with one line of symmetry. Can you rearrange them to make an outline with four lines of symmetry? E.G. The L-shape How many of the L-shapes could you fit on to this 4x4 grid? You can rotate them but they can't overlap each other. This came from Chris's weekly newsletter (chris@piwire.co. uk) and the following week he said "Since there are just 16 squares to fill and each of our L-shapes covers three squares, the best we could hope to squeeze on to the shape would be five. This is possible! [and he shows a solution] And of course you could ask 'Can every square be left as the only uncovered one?"' Two questions occurred to me: a) Since an L-shape covers three small squares you'd think that three of them could fit in a 3x3 square. But this can't be done. Is that obvious or do you just have to try all possibilities? b) Can you fit eight of the L-shapes into a 5x5 square? Zigzag What fraction of the square is shaded? Catriona Agg Even Numbers Remove six circles to leave an even number of circles in each row and column. (This is an adapted version of The Dyer's Puzzle by Henry Ernest Dudeney in The Canterbury Puzzles.) From the website Transum.org Half Time Oxford beat Ipswich 3-2 at football. How many different half-time scores could there have been? How many could there have been if the final score was 4-3? What is the general link between final score numbers and the number of halftime possibilities? R.F.H
3 Mathematical Pie | Autumn 2023 | Issue 220 Two Squares and a Triangle Two squares with the areas shown beneath a triangle ABC. The angle ABC is 90° and BD is a straight line. What is the area of triangle ABC? E.G. Beyond Pythagoras 2 The picture illustrates the fact that 52 + (2 × 5 + 1) = 62 . So 52 differs from 62 by 2 × 5 + 1. In general, if we want to find two square numbers that differ by 37 say, work out 37 - 1 and halve it, to get 18. 182 + (2 × 18 + 1) = 192 182 + 37 = 192, and rearranging, 37 = 192 − 182. Now back to forming Triple Square equations starting with any odd square and any even square (as in Beyond Pythagoras 1, Autumn 2022): for example start with 12 and 62 and notice that 12 + 62 = 37. So l2 + 62 = 192 − 182, and rearranging, 12 + 62 + 182 = 192. See if you can complete these equations: 52 + 82 + (?)2 = (?)2 and 62 + 92 + (?)2 = (?)2. G.C.S. Adding Up Ivy picked a number, removed one digit, and added the number she was left with to the original number, getting 155667. What was her original number? C.J.S. Any Triangle The medians of triangle PQR intersect at M. Can you explain why triangles PMQ, QMR and RMP must have the same area? H.K.M. Place The Tiles You have six tiles and you must place them to make a six-digit number so that • The number formed from the first two digits is divisible by 2 • The number formed from the second and third digits is divisible by 3 • The number formed from the third and fourth digits is divisible by 4 • The number formed from the fourth and fifth digits is divisible by 5 • The number formed from the fifth and sixth digits is divisible by 6 For example you might try 123645, but this fails because although 12 is divisible by 2, 23 is not divisible by 3, nor 64 by 5, nor 45 by 6. W.R. A Third Start with a square piece of paper. You may only fold it- no instruments or even a pencil are allowed. Can you find a length which is one third of the length of the side of the square? W. Maas, Euclides 79 (2004) The Cube Displacement Puzzle An open cube with side 24cm is half full of water. A solid steel cube with side length 12cm is dropped in the water. How many centimetres will the level of the water rise when the steel cube has been completely submerged? Andrew Sharpe (from the website Puzzle of the Week: www.puzzleoftheweek.com)
4 Mathematical Pie | Autumn 2023 | Issue 220 Squaring the Circle? Here are a circle and a square, tucked into an equilateral triangle. If the area of the circle is 37π m2, what is the area of the square? E.G. Octagon Numbers Place the numbers from 1 to 9 in the octagons so that the products of the numbers in the four octagons that touch each square give the total in the square. W.R. Autumn 2023 NOTES FOR... No. 220 Time for a Break – Elevenses? First things first – I would ask pupils to try to explain why the method shown for testing multiples of 11 works! The number in the central square has to be the middle digit of four threedigit multiples of 11. So, for example, it cannot be 2 since the only possibilities are 429, 528 and 627. It turns out that the only contenders for the central square are 0 and 9. Patience and use of symmetry are needed to show that no solution exists for 9. Can you find a way to fill the grid with nine of the digits to make the sums of the digits in every row, column or diagonal a multiple of 11? I Have a Good Friend in Maumee I think it is easy to see That her age, when reversed, starts with 3 There is only one square That has got a 3 there 63 is what her age must be. Geometry By drawing a line through B parallel to AC, we can find a point X on DC produced that makes triangle ACX equal in area to triangle ACB (same ‘base’ AC and common ‘height’). So triangle ADX and quadrilateral ABCD are equal. Summing Unusual The 14 numbers (or 7 pairs) are: 12 = 3x4, 21 = 3x7 18 = 9x2, 81 = 9x9 24 = 6x4, 42 = 6x7 27 = 9x3, 72 = 9x8 (36 repeated to emphasise patterns) 36 = 9x4, 63 = 9x7 [36 = 9x4, 63 = 9x7] 48 = 12x4, 84 =12x7 45 = 9x5, 54 = 9x6 The sum of the digits is closely connected to the well-known test for multiples of 9, 6 or 3. This means that any multiple of 9 could be one of the numbers we seek: going up to199 for the smaller of the pair, we find these seven pairs (defined by the smaller of the pair): 108, 117, 126, 135, 144, 153, 162. We rule out the palindrome 171 and the multiple of ten 180. Given that we are looking for eight pairs in all, there is just one pair left to find and fortunately its smaller value is a low number within the range 100 to 199: 102 = 3 x 34 (and 201 = 3 x 67). The challenge: this is a bit mean since unfortunately none of the remaining twenty pairs less than 1000 have prime factors in both cases. The answer has four digits: 1011 = 3 x 337 (and 1101 = 3 x 367). How Many Zeros? Each factor of 5 in the given number contributes just one zero when it is multiplied by one of the even factors. So there will be five zeros at the end. π Not Included This neat little theorem is attributed to Hippocrates in c. 460 B.C. I first saw it in PiE No. 36 (May 1962). The area above the diameter can be expressed as either “Two ears plus a
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