St Joseph's College

## Years 12-13 Enrichment

Maths Challenges

Senior Maths Challenge Course - Free enrolment - Click here

UKMT Senior Maths Challenge Papers - Click here

Canadian Maths Challenges - Click here (Grade 11 is equivalent to Year 12 in the UK)

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Reading Material

This is a list of my recommended books to read - Click here

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Challenging Questions involving A-Level Maths

Ritangle is a team competition, no knowledge of mathematics beyond A level Mathematics required - Twenty five questions providing information needed for the final challenge - Click here

Martin Greenhow at Brunel University has set up a suite of mathematics tests for "bored" GCSE & A-Level students at: Click here

These papers use the C1 to C4 content but are designed to challenge the most able - Click here

Hungarian Maths Problems - The different sets were entrance examinations for different majors, such as engineering or mathematics - Click here

Some interesting questions only requiring AS Maths but more thought provoking can be found here with the Oxford Physics Aptitude Tests Maths section - Click here

MEI Produce regular problems for those who like a challenge - Click here

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Beyond the Curriculum

# Beautiful Mathematical Documentaries to Make you fall in Love with Mathematics

These are very “interesting” because they present either actual mathematics, mathematicians, or mathematics history and are presented in order to attract undergraduates toward mathematics - Click here

Mandelbrot Set Exploration - 6 superbly structured tasks - Click here

Numberphile: A superb range of videos - Click here

Over 2500 lectures on mathematics organized by topic and level of mathematics - Click here

Sixty Symbols Videos: Cool videos about physics and astronomy - Click here

Objectivity Videos: Videos about Cool objects - Click here

Great document of suggestions for sources and ideas beyond the curriculum - Click here

Enrichment Activities and Ideas - Click here

Interesting article entitled "What did Ada Lovelace's Program Actually Do?" - Click here

Fancy something very substantial? Think about one of our challenging open investigations. These will take time and you will need to pose your own questions and make your own mathematical discoveries - Click here

Conway's Game of Life - A beginners Guide - Click here and a summary of some of the key facts - Click here

Explore Eulcid's Elements - Digitised with interactive coloured diagrams - Click here

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Free Online Courses

abakus has researched the many online maths courses that are available and curated 49 utterly helpful and engaging online math courses for total beginners to expert students. There is an online course here for all interested students - Click here

Free Learning from the Open University - Click here

Free online Learning from Future Learn - Click here

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Maths Videos of the Week

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This is a series of over 50 links to an interesting or thought-provoking video relating to the world of mathematics.

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Maths video of the week #1

We see the numbers “million” and “billion” a lot, both in maths and in everyday life, but it can be difficult to visualise their relative sizes. In this week’s video, Tom Scott demonstrates the difference between a million dollars and a billion dollars.

You may want to watch the first 2-3 minutes then skip ahead to a couple of minutes from the end, as there is a lot of driving in the middle!

A Million Dollars vs A Billion Dollars, Visualised: A Road Trip - www.youtube.com/watch?v=8YUWDrLazCg

Maths video of the week #2

Fermat’s Last Theorem is one of the most well-known theorems in mathematics. It was first stated by Pierre de Fermat in 1637 but was not proven until 1994 by English mathematician Andrew Wiles, who has studied and worked at Oxford, Cambridge and Princeton.

In this week’s video Simon Singh introduces the theorem, and explains a little of the history of how it was finally proven. Singh has written a number of popular science and mathematics books (including one on Fermat’s Last Theorem) which are very engaging and recommended if you are looking for some super-curricular reading for mathematics. This could be just for your own interest, or to mention in a UCAS personal statement or interview.

Fermat's Last Theorem - www.youtube.com/watch?v=qiNcEguuFSA

Maths video of the week #3

The theory of surfaces (sometimes called two-dimensional manifolds) is part of an area of mathematics called topology. Students generally first encounter topology at university as part of a mathematics degree.

In this week’s video, Matt Parker explores the question of how many ‘holes’ a surface can have, mathematically speaking. The answer may not be as simple as you first think!

For further reading, the mathematical term for the number of holes in a surface is called its “genus”. It can also be described by a number known as the Euler characteristic. More information here: https://en.wikipedia.org/wiki/Genus_(mathematics)

If you like Matt Parker’s style of communicating high level maths in an accessible way, he has written a number of books including “Things to Make and Do in the Fourth Dimension” www.waterstones.com/book/things-to-make-and-do-in-the-fourth-dimension/matt-parker/9780141975863

Why does this balloon have -1 holes? - www.youtube.com/watch?v=ymF1bp-qrjU

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Maths video of the week #4

The Prisoner’s Dilemma is a classic problem in an area of mathematics called Game Theory. Further Maths students will study a section of Game Theory known as “zero sum games” as part of their course; however, Game Theory is a wide-reaching field which has applications in subjects as diverse as Economics, Marketing, Psychology and Computer Science.

This week’s video outlines the key points of the Prisoner’s Dilemma problem and considers its potential solutions.

For further reading on Game Theory, there are dozens of books available – a search for “game theory” on most online book retailers will bring up several easily accessible introductions to the subject. For example, this one on Amazon, which can be ordered in paperback form or downloaded onto a Kindle - www.amazon.co.uk/Introducing-Game-Theory-Graphic-Guide-ebook/dp/B01J4P6L90/ref=tmm_kin_swatch_0?_encoding=UTF8&qid=1633548440&sr=8-4

The Prisoner's Dilemma - www.youtube.com/watch?v=t9Lo2fgxWHw

Maths video of the week #5

This week’s video is a particularly interesting one because it spans two very different areas of mathematics. Steve Mould considers a system of two springs, and asks what would happen if the rope between them were cut. He then applies the somewhat counterintuitive and unexpected result to the question of speeding up the flow of traffic through a road network. This is a topic that is studied by our Further Maths students in the Discrete maths topic of Network Flows.

Steve Mould mainly creates videos on Physics, Chemistry and Biology which frequently overlap with Mathematics, so it is worth checking out his channel if you are interested in these areas too!

The Spring Paradox - www.youtube.com/watch?v=Cg73j3QYRJc

The traffic analogy he mentions is Braess’s paradox, after German mathematician Dietrich Braess. For further reading, see https://en.wikipedia.org/wiki/Braess%27s_paradox

Maths video of the week #6

This week’s link is actually a series of podcasts by +Plus Magazine, which is a maths website run by the University of Cambridge. The podcasts cover a wide range of mathematical topics, such as modelling climate change, making predictions during the COVID-19 pandemic, the dynamics of crowds and the maths of card shuffling.

You can stream the podcasts straight from the website, or download them onto a phone or device to listen to later (click on your chosen episode, then “Listen to the podcast”, then either play it from there or use the three dots to download it)

Some of these podcasts may provide you with a starting point for further reading, if you are looking for topics to discuss in a UCAS personal statement or university interview. Why not choose a few that interest you to listen to over the half term break?

https://plus.maths.org/content/Podcast

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Maths video of the week #7

The US presidential elections, on 3rd November 2020, led to Joe Biden and Kamala Harris becoming the president and vice president of the US.

You may remember at the time there was a lot of discussion in the news and on social media of cheating and electoral fraud. In particular, a mathematical concept known as Benford’s Law made headlines when it was claimed by some Republicans to be proof that the Democrats had fixed some of the votes for Biden.

Benford’s Law is a common pattern in statistics that some sets of data follow, while others do not. In this video, Matt Parker describes what Benford’s Law is and explains why in some areas, votes for Trump followed the rule while votes for Biden did not.

Why do Biden's votes not follow Benford's Law? - www.youtube.com/watch?v=etx0k1nLn78

If you are interested in statistics and the analysis of UK election data, you may like to spend some time looking at this website www.britishelectionstudy.com/ In particular, the “News” section has analysis from the 2019 General Election which compares variables such as age, demographic and which party people voted for, and in the “Data” section you can create your own chart or graph using your chosen variables, under “Data Playground”.

You can read more about the mathematics of Benford’s Law here: https://mathworld.wolfram.com/BenfordsLaw.html

Maths video of the week #8

You may have heard of or seen some examples of fractals before. If not, it is well worth going on Google images and searching “fractals” before you watch this video to get a feel for the beautiful and intriguing patterns that can be created.

This week’s video is an introduction to fractals, or “self similar” shapes, in particular the Sierpinski triangle and the Koch snowflake curve. In future weeks we’ll be sharing further videos on other types of fractal such as the Mandelbrot set and the Dragon curve.

Fun with Fractals - www.youtube.com/watch?v=XwWyTts06tU

Maths video of the week #9

Following on from last week’s introduction to fractals, this week is all about the Mandelbrot set. If you’ve not seen it before, you might like to Google “Mandelbrot set” and look at some images before you watch the video to see the amazing patterns it creates.

The Mandelbrot set is a fractal pattern made by complex numbers. Further maths students will already be familiar with complex numbers; if you haven’t come across them before there is a nice introduction below by Khan Academy which will tell you all you need to know to understand the Mandelbrot set. Don’t worry if you don’t fully understand the complex numbers part; it’s possible to enjoy the Mandelbrot set as a beautiful design without getting too bogged down in the mathematical detail!

The Mandelbrot Set - www.youtube.com/watch?v=NGMRB4O922I

Introduction to complex numbers | Imaginary and complex numbers - www.youtube.com/watch?v=SP-YJe7Vldo

Maths video of the week #10

Howie Hua is a Maths instructor at Fresno State University in California. He makes a series of TikTok videos demonstrating mathematical concepts in different visual and intuitive ways. I particularly like his demonstration of the infinite sum 1/3 + 1/9 + 1/27 + 1/81 +… in a way that is really easy to understand.

This is a collection of some of his TikTok videos, on YouTube.

Howie_Hua's Math TikTok videos (Part 1) - www.youtube.com/watch?v=59g9BuHISxg

Maths video of the week #11

Some of you may already be familiar with the YouTube channel 3Blue1Brown. For those who have not come across it before, it is an excellent channel that explains some complex ideas in mathematics (often degree level and beyond) in a way that is accessible and easy to understand.

This video looks at a scenario called the brachistochrone problem, which poses the question of how to get from one point in space to another in the shortest amount of time. This mathematical question has clear applications in Physics and Engineering.

The brachistochrone problem is one that most mathematics undergraduates first encounter at university, so some of the maths involved in the solution gets a bit heavy towards the end of the video – try not to get bogged down by the details; a general understanding of the solution can be gained without following every step!

The Brachistochrone, with Steven Strogatz - www.youtube.com/watch?v=Cld0p3a43fU

For further reading on this topic, you may find the following links a good place to start:

www.cantorsparadise.com/the-famous-problem-of-the-brachistochrone-8b955d24bdf7

https://en.wikipedia.org/wiki/Brachistochrone_curve

Maths video of the week #12 – Christmas edition!

At this time of year, many people like to get into the Christmas spirit by doing a Secret Santa, where everyone picks a name out of a hat and buys a present for that person. In this week’s video, Hannah Fry considers the mathematical implications of this, some problems it causes and finds a suitable mathematical solution.

www.youtube.com/watch?v=5kC5k5QBqcc

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Maths video of the week #13

Most people have heard of Florence Nightingale, a nurse famous for her work in military hospitals in the Crimean War (1853-1856) However, she was also a statistician and social reformer who pioneered the use of statistics and analysis of data in the field of medicine. She developed several types of statistical diagrams to represent the data she collected, and used these to improve healthcare and patient outcomes. In this week’s video, Professor Mike Merrifield discusses this, and the benefits of displaying data in a circular diagram rather than in an x-y plane. The video was recorded in spring 2021 so the Covid data used is from that period.

Nightingale Diagrams - www.youtube.com/watch?v=VTdVPNvwULM

For further reading: www.sciencemuseum.org.uk/objects-and-stories/florence-nightingale-pioneer-statistician

Maths video of the week #14

You may have read the book Jurassic Park by Michael Crichton, or seen the 1993 film and its sequels (if not, I can highly recommend reading any of Michael Crichton’s books – they are excellent sci-fi thrillers based on a whole range of topics in science and technology, all of which are very well researched by the author!)

In this week’s first video, Rob Eastaway sheds some light on an interesting fractal pattern called the Dragon Curve that features at various stages throughout the book. The second video (Unfolding the Dragon) is a really nice computer simulation showing repeated iterations of the pattern as it ‘unfolds’.

Dragon Curve by Rob Eastaway - www.youtube.com/watch?v=wCyC-K_PnRY

Unfolding The Dragon - www.youtube.com/watch?v=UBuPWdSbyf8

Rob Eastaway has written a number of popular mathematics books, including “Maths on the Back of an Envelope”, “Why Do Buses Come in Threes?” and “The Hidden Maths of Sport”. For more information, see his website: https://robeastaway.com/

Maths video of the week #15

There are many links between Cartography (the science and art of maps) and Mathematics.

For example, maps can be used as a type of chart to display data, using colours or other labels. A choropleth map uses different shades of colour to show regional variations in a chosen measurable variable. A cartogram is a type of graph where the area of each region is distorted to represent something other than its land area (for example, its population or GDP). The World Mapper website is an excellent and fascinating source of cartograms on a wide range of data sets and is well worth checking out (link in the comments)

There is a branch of mathematics called projective geometry which mathematics undergraduates are often given the opportunity to study at university, which can be used to find different ways of transferring the location of places on a spherical globe onto the flat, rectangular plane of a sheet of paper (representing the Earth as a flat map) Some of these projections are better than others in terms of preserving the areas or distances between places.

In this week’s video, Hank Green summarises 42 interesting maps (just a warning, he talks very quickly!)

42 Amazing Maps - www.youtube.com/watch?v=dldHalRY-hY

Links for further information:

World Mapper: https://worldmapper.org/

Choropleth maps: https://en.wikipedia.org/wiki/Choropleth_map

Ten different map projections: https://futuremaps.com/blogs/news/top-10-world-map-projections

Projective geometry: https://brilliant.org/wiki/projective-geometry/

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Maths video of the week #16

In this week’s video by 3Blue1Brown, Grant Sanderson ponders the relationship between the area of a circle and the surface area of a sphere. The mathematical reasoning he uses links to the ideas touched upon in last week’s topic of projective geometry, projecting a sphere onto a flat plane like the Earth’s surface onto a map.

The maths used is quite complicated in parts, but the visuals are very nice. Remember, videos on this channel are often on advanced topics beyond the scope of A Level, so don’t be put off if you don’t understand everything!

But why is a sphere's surface area four times its shadow? - www.youtube.com/watch?v=GNcFjFmqEc8

Maths video of the week #17

If you are interested in computer science and internet security you may already be aware that encryption of data on the internet requires the use of some advanced level mathematics. The first video (by Khan Academy) summarises the basics of encryption and ciphers. The second video (by Computerphile) goes into more detail of how large prime numbers are used in RSA encryption on the internet.

Encryption and public keys | Internet 101 | Computer Science - www.youtube.com/watch?v=6-JjHa-qLPk

Prime Numbers & RSA Encryption Algorithm - www.youtube.com/watch?v=JD72Ry60eP4

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Maths video of the week #18

Mathologer is a YouTube channel that creates videos on a wide range of mathematical topics – it is worth checking out their other videos if you haven’t seen it before.

This video looks at the link between plotting times tables on a circle and the Mandelbrot set, the complex fractal pattern that we saw in video #9 earlier this year. The idea of modular arithmetic (placing integers around a circle and counting around) is studied by our further maths students in the Discrete part of the course.

Times Tables, Mandelbrot and the Heart of Mathematics - www.youtube.com/watch?v=qhbuKbxJsk8

Maths video of the week #19

This week’s link is a series of podcasts called “Taking Maths Further”. They are relatively short (20-25 minutes each) and cover a wide range of applications of maths including forensics, astrophysics, accountancy and links with art and music. They are an excellent super-curricular resource which may give you inspiration for your UCAS personal statement, and may introduce you to different careers in mathematics that you may not have previously thought of.

They can be accessed by either following the link below (you need to click on “Guest Access” then scroll down almost to the bottom, and click on the “podcasts” link) or I have placed a folder containing all 20 as mp3 files in the Files section of this Teams page in Class Materials. You can play them directly or download them on to a device to listen to later. Let me know if you have any problems accessing them!

Taking Maths Further podcast - https://amsp.org.uk/resource/students-fmsp-legacy-resources-archive (click on guest access then scroll down to podcasts)

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Maths video of the week #20

Mathematics is an important part of the programming and modelling process in animated series and films. In this video, Zach Star talks about some of the mathematics used in making Disney Pixar films.

He also discusses some of the considerations made by structural engineers and architects when designing new buildings in order to make them safe, such as the impact of vibrations and resonance.

Math and physics can show up when you least expect - www.youtube.com/watch?v=FrbK2c-UMTY

Maths video of the week #21

You may have seen the display in the Maths corridor in Ash about the 7 Millennium problems, which were set in the year 2000 as important unsolved problems in mathematics, each with a $1,000,000 prize for the person who solves it. So far, only one of these problems has been solved; the Poincaré Conjecture was solved in 2003 by Grigori Perelman (he declined the prize money)

This week’s video explains one of the still unsolved problems, the Riemann Hypothesis.

The Riemann Hypothesis, Explained - www.youtube.com/watch?v=zlm1aajH6gY

For more information about the Millennium Prize Problems: www.claymath.org/millennium-problems

Maths video of the week #22

Dr Tom Crawford is a Mathematics tutor at the University of Oxford; he also does outreach projects with schools and produces a lot of interesting Maths videos and articles on his website Tom Rocks Maths. This week’s video is a short (60 second) summary of his PhD thesis on the mathematical modelling of ocean pollution. There is also a longer video which goes into this in more depth, and a link to his thesis and other articles for further reading.

My Maths PhD on Ocean Pollution Explained in 60 Seconds - www.youtube.com/shorts/gZrJMkUmpgg

Where Does River Water Go? - www.youtube.com/watch?v=5mGh0r3zC6Y

Link to Tom’s Thesis (if you’re interested in reading it!): https://tomrocksmaths.com/read/thesis/

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Maths video of the week #23

Crash Course is a brilliant YouTube channel which produces dozens of series of short videos covering all kind of topics, including History and Politics, Economics, and the Physical and Social Sciences. They are all high quality, engaging and very well researched – it is definitely worth spending some time looking at the topics to find one that takes your interest (I watched all of the World History ones a few years ago and learned all kinds of new things!)

This week’s video is from the series on Statistics; it looks at Science Journalism, and how data is interpreted to create news headlines.

Science Journalism: Crash Course Statistics #11 - www.youtube.com/watch?v=ZwqOoD17_LU

To browse all of the Crash Course topics, look here: www.youtube.com/c/crashcourse/playlists

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Maths video of the week #24

How many people have you met in your life who share the same birthday as you? Assuming you’re not a twin, the number might be quite low.

Now imagine a room full of people; what is the likelihood that two people in the room share the same birthday? This will clearly depend on the number of people in the room. This famous problem is known as the birthday problem, and the probabilities involved may surprise you.

Check your intuition: The birthday problem - David Knuffke - www.youtube.com/watch?v=KtT_cgMzHx8

Maths video of the week #25

You have probably played with a Rubik’s Cube at least once in your life; you may have even been able to solve it. The mathematics behind the Rubik’s Cube is quite complex, and falls into an area of mathematics called Group Theory and Permutations. Solving the cube has become a hobby or interest for millions of people since its invention in 1974. This week’s video gives an introduction for one method of solving the cube.

How to Solve a Rubik's Cube - www.youtube.com/watch?v=R-R0KrXvWbc

More info: https://en.wikipedia.org/wiki/Rubik%27s_Cube

Some further reading on the mathematics of the Rubik’s Cube, from MIT (Massachusetts Institute of Technology): http://web.mit.edu/sp.268/www/rubik.pdf

Maths video of the week #26

This week’s video considers a problem in Physics known as the book stacking problem – the question of how far you can allow a pile of objects such as books to overhang a surface without falling over the edge. This fits in nicely with the year 2 mechanics topic of Moments. Why not watch the video, then try it for yourself (with a pile of non-breakable objects!)?

Book Stacking Problem - Calculating the Overhang - www.youtube.com/watch?v=J9uJi_DmZgc

Maths fact: The amount added on to the overhang each time is related to the harmonic series 1/n, which is the infinite sum 1/1 + ½ + 1/3 + ¼ + 1/5 +…

The solution forms half of the harmonic series, 1/(2n), so the overhang is the sum ½ + ¼ + 1/6 + 1/8 + 1/10 + … and so on.

Maths video of the week #27

There are very close links between Mathematics and Architecture; in order to build a structure an architect needs to ensure the forces involved are stable and balanced out as much as possible. In this week’s video, the University of Oxford’s Dr Tom Crawford looks at the mathematics of building a dome.

The cosh function mentioned in the video is one of the hyperbolic functions studied by further maths students, and has close links with the trigonometric function, cosine.

How to build a Giant Dome - www.youtube.com/watch?v=WIibcLd_oYU

If you visit Barcelona in northern Spain, you will most likely see the great unfinished basilica (church) La Sagrada Familia, designed by the architect Antoni Gaudí. The towers of this building are based on the same idea of the cosh curve or ‘catenary’, using his models made from string and weights – more information here:

http://dataphys.org/list/gaudis-hanging-chain-models/

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Maths video of the week #28

We often use the word “average” to mean “typical” or, in some sense, “normal”. We can calculate various different averages of a set of data using statistical methods – but what do we really mean when we say something is “average”? In this week’s video, Matt Parker ponders the question of whether the average person really exists.

Does The Average Person Exist? - www.youtube.com/watch?v=NbiveCNBOxk

Maths video of the week #29

This week’s video was sent in by a student – thanks Morgan! If anyone has any interesting videos that they think would make a good video of the week, please do send them in to us! ðŸ˜Š

It focuses on problem called the Collatz Conjecture (in mathematics, a conjecture is a theorem that hasn’t been proven yet) While the problem itself seems straightforward to understand, it turns out that proving it is much more difficult.

The Simplest Math Problem No One Can Solve - www.youtube.com/watch?v=094y1Z2wpJg

Maths video of the week #30

During World War II, a top secret government codebreaking centre was set up at a country house in Bletchley, Milton Keynes, to decipher intercepted messages. The most famous of these codes used the German Enigma and Lorenz ciphers. One of the most well-known code breakers in Bletchley Park at this time was Alan Turing; there is more information about his work and life on the Famous Mathematicians display board in the Ash corridor.

This week’s videos give an insight into how the Enigma machines worked and into the codebreaking that took place at Bletchley Park. The second video is a 360 degree tour of Bletchley Park; if you watch it on a phone you can see the full view of some of the rooms.

158,962,555,217,826,360,000 (Enigma Machine) – Numberphile - www.youtube.com/watch?v=G2_Q9FoD-oQ&t=0s

Bletchley Park 360 tour: How Britain cracked Nazi Enigma - www.youtube.com/watch?v=wlWVpOzgrL4

Bletchley Park is open to the public, and is well worth a visit if you are in the area. https://en.wikipedia.org/wiki/Bletchley_Park

Maths video of the week #31

This week’s video is a fascinating proof by 3blue1brown on why slicing into a cone at an angle results in an ellipse, or a circle stretched in the x- and y-directions. Further Maths students will have encountered ellipses in the conic sections part of the course.

Why slicing a cone gives an ellipse - www.youtube.com/watch?v=pQa_tWZmlGs

Maths video of the week #32

Chaos theory looks at systems of objects that are highly sensitive to initial conditions and can quickly diverge away from what is initially predicted; for example, the weather, which can be almost impossible to predict with accuracy after more than a few days.

This week’s first video looks at different examples of Chaos theory, including the famous Three Body Problem which models three objects such as planets and their movement in relation to each other.

The second video looks at how chaotic systems can be modelled using computer animations.

Chaos: The Science of the Butterfly Effect - www.youtube.com/watch?v=fDek6cYijxI

Chaotic Balls (and other animations) – www.youtube.com/watch?v=6z4qRhpBIyA

Maths video of the week #33

Many people will recall the iconic scene in the ‘90s sitcom Friends where the characters attempt to move a sofa up some stairs whilst shouting “PIVOT!”. You may even have encountered this problem in your own life; trying to move a sofa or large piece of furniture around a corner in a narrow corridor or hallway.

This week’s video looks at this precise scenario; “The Moving Sofa Problem” helps to determine mathematically the best shape of sofa to maximise the area that can be rotated around a corner.

The Moving Sofa Problem - www.youtube.com/watch?v=rXfKWIZQIo4

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Maths video of the week #34

Hi everyone! Each week throughout the year we’ll be sending out a different “Maths video of the week”. This could be something relating to an interesting area of mathematics or a use of mathematics in the real world.

This week we’ll be starting with the classic question of whether Maths is something that was something that always existed and was discovered by people, or whether we invented it.

Is math discovered or invented? - Jeff Dekofsky - www.youtube.com/watch?v=X_xR5Kes4Rs

Maths video of the week #35

In this week’s video, Matt Parker and Steve Mould discuss a problem relating to hanging a picture on the wall. This problem links to an area called knot theory, which mathematics students often get the chance to study at university, as part of a branch of mathematics called Topology.

How to mathematically hang a picture (badly) - www.youtube.com/watch?v=x5h3yTxeCew

More information: www.britannica.com/science/knot-theory

Maths video of the week #36

This week, our video is about a famous mathematical theorem called the Four Colour Theorem, which was proven in 1976. The theorem states that any map can be coloured in using four colours or fewer, in such a way that no two countries sharing a boundary have the same colour. It caused some controversy at the time as it was the first mathematical proof that required the use of a computer to check all of the possible cases.

The Four Color Theorem | Coloring a Planar Graph - www.youtube.com/watch?v=42-ws3bkrKM

More information: https://nrich.maths.org/6291

Maths video of the week #37

In the first video this week, “Measuring Coastline”, Steve Mould explores how the length of a coastline (which often includes many small inlets and headlands) can change depending on the accuracy of the tool used to measure it. This leads him into a fascinating area of mathematics called Fractals.

The second video, “Fun with Fractals”, explores the idea of Fractals in more detail. These self-similar shapes often create beautiful patterns – if you haven’t seen them before then a Google image search for “fractals” is highly recommended!

In the final video, Vi Hart uses doodles to explore the Dragon Fractal.

Measuring Coastline - www.youtube.com/watch?v=7dcDuVyzb8Y

Fun with Fractals - www.youtube.com/watch?v=XwWyTts06tU

Doodling in Math Class: DRAGONS - www.youtube.com/watch?v=EdyociU35u8

Maths video of the week #38

Following on from last week’s videos on fractals, this week we are looking at a set of numbers called the Cantor set. This interesting set has the properties of having infinitely many numbers, but zero length and a fractional dimension.

What happens at infinity? - The Cantor set - www.youtube.com/watch?v=eSgogjYj_uw

Maths video of the week #39

This week’s link is actually a series of podcasts by +Plus Magazine, which is a maths website run by the University of Cambridge. The podcasts cover a wide range of mathematical topics, such as modelling climate change, making predictions during the COVID-19 pandemic, the dynamics of crowds and the maths of card shuffling.

You can stream the podcasts straight from the website, or download them onto a phone or device to listen to later (click on your chosen episode, then “Listen to the podcast”, then either play it from there or use the three dots to download it)

Some of these podcasts may provide you with a starting point for further reading, if you are looking for topics to discuss in a UCAS personal statement or university interview. Why not choose a few that interest you to listen to over the half term break?

https://plus.maths.org/content/Podcast

Maths video of the week #40

If you’re not familiar with the YouTube channel 3Blue1Brown you may be interested in checking it out. Many of their videos, which are all mathematical in nature, are quite advanced and cover topics that extend far beyond the A level Maths course. However, they are very well explained and use excellent graphics to explain things in a clear and engaging way that is accessible to all. The channel is highly recommended for anyone with an interest in mathematics. In this week’s video, the channel’s creator Grant Sanderson explores a famous problem called the Basel problem.

Why is pi here? And why is it squared? A geometric answer to the Basel problem - www.youtube.com/watch?v=d-o3eB9sfls

Maths video of the week #41

If you have ever paid attention to the road signs of the UK you may have noticed that there is something not quite right about the sign used to denote a football stadium. Mathematician Matt Parker has very strong feelings about this, which he explains here!

All UK football road signs are wrong! Join the petition for geometric change! - www.youtube.com/watch?v=btPqKAGyajM

Maths video of the week #42

In mathematics we use the constant pi (3.14…) in many different contexts, including trigonometry and geometry. You may be less familiar with the constant tau, which is another Greek letter used to denote the constant 2pi, or 6.28…

In this week’s video, Matt Parker and Steve Mould go head to head in a battle to decide which constant is the best. Watch the video, pick a side, and let us know what your opinion is!

Tau vs Pi Smackdown - www.youtube.com/watch?v=ZPv1UV0rD8U

Maths video of the week #43

We use the number zero all the time, probably without thinking much about it. However, strange things happen when we try to divide by zero, or calculate zero to the power of zero. This week, we have two videos that discuss the concept of zero from different viewpoints; the first by Matt Parker and James Grime at Numberphile, and the second by mathematics teacher Eddie Woo.

Problems with Zero - www.youtube.com/watch?v=BRRolKTlF6Q

Dividing by zero? - www.youtube.com/watch?v=J2z5uzqxJNU

Maths video of the week #44

Mathematical modelling plays a large part in the study of nature and biology. For example, predator-prey models can be used to predict populations of animals over a period of time, taking into account the interactions between different species and external factors such as climate and food availability.

Many mathematics degrees have the option for students to take modules in Mathematical Biology, where these models are studied and refined. In this week’s video, Dr Tom Crawford at the University of Oxford explains how predator-prey models can be set up and solved. The solution of these problems often uses differential equations, which are covered in the second year of A Level Maths and also explored in more depth in A Level Further Maths.

Predators and Prey - www.youtube.com/watch?v=M0nRWcF1WJw

Maths video of the week #45

Everyone is aware of the concept of infinity, or something being “infinitely large”, but did you know that some infinities are bigger than others, while others are the same size?

The mathematician David Hilbert explained this idea using a thought experiment known as Hilbert’s hotel, which is introduced in this week’s video.

The Infinite Hotel Paradox - Jeff Dekofsky - www.youtube.com/watch?v=Uj3_KqkI9Zo

Maths video of the week #46

“Let’s Make a Deal” was a game show on television in the US that was first broadcast in 1963, hosted by presenter Monty Hall. As part of the show, contestants were offered a deal which gave them the chance to win a car – they were given a choice of three doors, and after making their original choice they were given the chance to switch to a different door or stick with their first choice. This became known as the Monty Hall problem, and the probabilities involved sparked a lot of debate over the years. This week’s video explains the Monty Hall problem and the mathematics behind the strategy that leads the contestants to the best chance of winning.

Monty Hall Problem - www.youtube.com/watch?v=4Lb-6rxZxx0

Maths video of the week #47

We know that the land area of the UK is around 243,000 square kilometres, but how is this calculated, and does it take into account the lumps and bumps or just assume that the land is completely flat? Matt Parker investigates the mathematics behind the calculations.

Does "land area" assume a country is perfectly flat? - https://www.youtube.com/watch?v=PtKhbbcc1Rc

Maths video of the week #48

In mathematics, we use symbols all the time to express concepts and calculations as efficiently and concisely as possible. But have you ever wondered where the symbols came from? This week’s video explores the history behind some of the most common mathematical symbols.

Where do math symbols come from? - John David Walters - www.youtube.com/watch?v=eVm063xmnow

Maths video of the week #49

While we often use trigonometry in the context of triangles (right-angled or otherwise) we know that the trigonometric functions are closely related to circles. This week’s videos contains some mesmerising visuals demonstrating these ideas in 2 dimensions, before extending to the 3 dimensional case.

Beautiful Trigonometry - www.youtube.com/watch?v=snHKEpCv0Hk

Maths video of the week #50

You may have seen the display in the Maths corridor in Ash about the 7 Millennium problems, which were set in the year 2000 as important unsolved problems in mathematics, each with a $1,000,000 prize for the person who solves it. So far, only one of these problems has been solved; the Poincaré Conjecture was solved in 2003 by Grigori Perelman (he declined the prize money)

In this week’s first video, Dr Tom Crawford gives a brief overview of these problems; in the second video, he goes into much more detail in a lecture on these million dollar mathematical questions. If you have time, the second video is worth a watch!

Millennium Maths Problems Explained in 90 Seconds - www.youtube.com/watch?v=ydKApyzwRBs

The Million Dollar Equations - with Tom Crawford - hwww.youtube.com/watch?v=f251NkeDVB8

For more information about the Millennium Prize Problems: www.claymath.org/millennium-problems

Maths video of the week#51

Cryptography, or the mathematics of codes, has been an important way of communicating messages in secret for centuries. It is now more important that ever in the age of internet security, and has close links with computing. In this week’s video, Zach Star explains some of the main types of encryption.

The Mathematics of Cryptography - www.youtube.com/watch?v=uNzaMrcuTM0

Maths video of the week #52

This week’s link is a series of podcasts called “Taking Maths Further”. They are relatively short (20-25 minutes each) and cover a wide range of applications of maths including forensics, astrophysics, accountancy and links with art and music. They are an excellent super-curricular resource which may give you inspiration for your UCAS personal statement, and may introduce you to different careers in mathematics that you may not have previously thought of.

They can be accessed by either following the link below (you need to click on “Guest Access” then scroll down almost to the bottom, and click on the “podcasts” link) or I have placed a folder containing all 20 as mp3 files in the Files section of this Teams page in Class Materials. You can play them directly or download them on to a device to listen to later. Let me know if you have any problems accessing them!

Taking Maths Further podcast - https://amsp.org.uk/resource/students-fmsp-legacy-resources-archive (click on guest access then scroll down to podcasts)

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Maths video of the week #53

Almost everyone will have played with a fidget spinner since they became a global craze a few years ago; you may well have owned one yourself. But have you ever thought about the Maths and Physics behind them? In this week’s video, Steve Mould explains why exponential decay means that it is difficult to increase the spin time by a significant amount.

If you are interested in the Sciences, Steve Mould’s YouTube channel has some excellent videos on a wide range of phenomena in Physics, Chemistry and Biology.

Exponential decay - why your fidget spinner won't spin for longer - www.youtube.com/watch?v=G7hh42AgBlQ

Maths video of the week #54

Wordle is a popular word game on the internet, where the aim is to guess a daily 5-letter word using coloured clues relating to the letters you have chosen. Many people have a favourite “starting word” or strategy for guessing the letters, but is there an efficient way to solve it using a computer algorithm? This week’s video from 3Blue1Brown suggests how information theory can be used to solve the puzzle as quickly as possible.

Solving Wordle using information theory - www.youtube.com/watch?v=v68zYyaEmEA

Maths video of the week #55

Crash Course is a brilliant YouTube channel which produces dozens of series of short videos covering all kind of topics, including History and Politics, Economics, and the Physical and Social Sciences. They are all high quality, engaging and very well researched – it is definitely worth spending some time looking at the topics to find one that takes your interest.

In A Level Maths we introduce the idea of conditional probability, where the likelihood of an event occurring is affected by some additional information that we have been given. This week’s video is from the Crash Course in Statistics; it introduces Bayes’ Theorem which is an important result used in conditional probability. Students who go on to study Mathematics or Statistics at university will be given the opportunity to study Bayes’ Theorem in more detail.

You Know I’m All About that Bayes: Crash Course Statistics #24 - www.youtube.com/watch?v=9TDjifpGj-k

To browse all of the Crash Course topics, look here: www.youtube.com/c/crashcourse/playlists

Maths video of the week #56

Most of the mathematics that we use in A Level Maths (and everyday life) applies in 2 dimensions (on a flat plane, such as a sheet of paper) or 3 dimensions (the 3 dimensions of space that we see around us) However, some interesting things happen when we extend mathematical results to higher dimensions. This week’s video by Zach Star introduces some shapes that only exist in a higher number of dimensions, such as the Klein bottle.

The things you'll find in higher dimensions - www.youtube.com/watch?v=dr2sIoD7eeU

Maths video of the week #57

Game Theory is an area of mathematics that deals with the idea of maximising winnings or minimising losses in a game or competitive situation. Game Theory is a wide-reaching field which has applications in subjects as diverse as Economics, Marketing, Psychology and Computer Science. Further Maths students will study a section of Game Theory known as “zero sum games” as part of their course.

This week’s video looks at how Game Theory can be used to predict human behaviour in a classic problem.

Game theory challenge: Can you predict human behavior? - Lucas Husted - www.youtube.com/watch?v=MknV3t5QbUc

For further reading on Game Theory, there are dozens of books available – a search for “game theory” on most online book retailers will bring up several easily accessible introductions to the subject. For example, this one on Amazon, which can be ordered in paperback form or downloaded onto a Kindle - www.amazon.co.uk/Introducing-Game-Theory-Graphic-Guide-ebook/dp/B01J4P6L90/ref=tmm_kin_swatch_0?_encoding=UTF8&qid=1633548440&sr=8-4

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Maths video of the week #58

This week’s video explores a nice problem involving darts and higher dimensional geometry. The Maths gets quite complicated towards the end, so don’t worry if you don’t understand all of it! If you have access to a dart board then perhaps you could try the game yourself and let us know how you got on.

Darts in Higher Dimensions (with 3blue1brown) - www.youtube.com/watch?v=6_yU9eJ0NxA

Maths video of the week #59

Many people who enjoy running take part in Park Run, a free weekly 5km run that is open to all and takes place in parks all over the country. If you yourself are a park runner, have you heard of “Park Run Bingo”? In this week’s video, Matt Parker speaks to some runners and calculates some of the mathematics behind the so-called coupon collector’s problem.

The Coupon Collector’s Problem - www.youtube.com/watch?v=BstloCx8KDk

Maths video of the week #60

Sometimes when solving a mathematical problem, the solution seems counterintuitive. This week’s video is a classic example of such a problem using probabilities, known as the 100 prisoners riddle. See if you can understand why this strategy works!

The Riddle That Seems Impossible Even If You Know The Answer - www.youtube.com/watch?v=iSNsgj1OCLA

Maths video of the week #61

Logic and Reasoning is an area of mathematics which overlaps with Philosophy. Discussions about whether a statement can be proven to be true or false forms the basis of mathematical logic. Gödel’s Incompleteness Theorem is a theorem concerned with the provability of statements, and states that not all true statements can be proved. This week we have two videos on this theorem, both by mathematician Marcus du Sautoy.

The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy - www.youtube.com/watch?v=I4pQbo5MQOs

Gödel's Incompleteness Theorem - www.youtube.com/watch?v=O4ndIDcDSGc

Maths video of the week #62

How many digits of pi do you know? We are all aware of the constant pi, the ratio between the diameter and circumference of a circle, and people have used supercomputers to calculate the digits, which are currently known to over 60 trillion digits.

You may wonder how these digits were originally calculated before supercomputers – this week’s video explains the process of calculating the digits of pi to a required degree of accuracy.

The Discovery That Transformed Pi - https://www.youtube.com/watch?v=gMlf1ELvRzc

Maths video of the week #63

We know that the square root of 2 is an irrational number, in other words that it cannot be written as a fraction with integers on the numerator and denominator. A proof for this, by contradiction, is covered in the second year of A Level Maths. This week’s video demonstrates how this result can be proven in a number of different ways, including algebraic and geometric proofs.

The 5 Best Proofs that the Square Root of 2 is Irrational - www.youtube.com/watch?v=zEXcsZo4hOQ

Maths video of the week#64

Last week’s video showcased 5 different proofs of the irrationality of the square root of 2, using various algebraic and geometric arguments. However, sometime a ‘proof’ can seem convincing upon first seeing it, but lead to an incorrect result. This week’s video shows how geometric arguments can be used to prove something that is actually false – can you spot the flaw in each ‘proof’?

How to lie using visual proofs - www.youtube.com/watch?v=VYQVlVoWoPY

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Maths video of the week #65

Real numbers can be separated into categories such as integers, fractions and irrational numbers. This week’s video explains what is meant by a transcendental number, which is a special type of irrational number.

Transcendental Numbers - www.youtube.com/watch?v=seUU2bZtfgM

Maths video of the week #66

The Golden Ratio is a number that has held people’s fascination for centuries. It has roots in the Fibonacci sequence, and occurs in nature in the optimal spacing of seed pods and in the spirals of many plants and animals, to name just two examples. It is also considered to be important in art, architecture and anatomy, where the ratio between two lengths can often be seen to be close to this value. This week’s videos explore the Golden Ratio further, delving into its history and how it is calculated.

What is the Golden Ratio? - www.youtube.com/watch?v=6nSfJEDZ_WM

The Golden Ratio (why it is so irrational) - www.youtube.com/watch?v=sj8Sg8qnjOg

Maths video of the week #67

The final video of the week of this academic year is the online lecture series from the University of Oxford mathematics department. These lectures are accessible to all and cover a wide range of topics. If you are doing some extras reading and researching for your UCAS personal statement then this would be an excellent place to start www.maths.ox.ac.uk/events/public-lectures-events

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