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Divisibility tests allow us to calculate whether a number can be divided by another number. For example, can 354 be divided by 3? Can 247,742 be divided by 11? So what are the rules behind divisibility tests, and more interestingly, how can we prove them?

**Divisibility rule for 3**

The most well known divisibility rule is that for dividing by 3. All you need to do is add the digits of the number and if you get a number that is itself a multiple of 3, then the original number is divisible by 3. For example, 354 is divisible by 3 because 3+5+4 = 12 and 12 can be divided by 3.

We can prove this using the modulo function. This allows us to calculate the remainder when any number is divided by another. For example, 21 ≡ 3 (mod 6). This means that the remainder when 21 is divided by 6 is 3.

So we first start with our number that we want to divide by 3:

n = a + 10b + 100c + 1000d + …….

Now, dividing by 3 is the same as working in mod 3, so we can rewrite this n in terms of mod 3:

n = a + 1b + 1c + 1d + ……. (mod 3)

(this is because 10 ≡ 1 mod 3, 100 ≡ 1 mod 3, 1000 ≡ 1 mod 3 etc)

Now, for a number to be divisible by 3 this sum needs to add to a multiple of 3. Therefore if a + b + c + d + …. ≡ 0 (mod 3) then the original number is also divisible by 3.

**Divisibility rule for 11**

This rule is much less well known, but it’s quite a nice one. Basically you take the digits of any number and alternately subtract and add them. If the answer is a multiple of 11 (or 0) then the original number is divisible by 11. For example, 121 is a multiple of 11 because 1-2+1 = 0. 247,742 is because 2-4+7-7+4-2 = 0.

Once again we can prove this using the modulo operator.

n = a + 10b +100c + 1000d + …..

and this time work in mod 11:

n = a + -1b +1c + -1d + ….. (mod 11)

this is because for ease of calculation we can write 10 ≡ -1 (mod 11). This is because -1 ≡ 10 ≡ 21 ≡ 32 ≡ 43 (mod 11). All numbers 11 apart are the same in mod 11. Meanwhile 100 ≡ 1 (mod 11). This alternating pattern will continue.

Therefore if we alternately subtract and add digits, then if the answer is divisible by 11, then the original number will be as well.

**Palindromic Numbers**

Palindromic numbers are numbers which can be read the same forwards as backwards. For example, 247,742 is a palindromic number, as is 123,321. Any palindromic number which is an even number of digits is also divisible by 11. We can see this by considering (for example) the number:

n = a + 10b + 100c + 1000c + 10,000b + 100,000a

Working in mod 11 we will then get the same pattern as previously:

n = a – b +c – c +b – a (mod 11)

so n = 0 (mod 11). Therefore n is divisible by 11. This only works for even palindromic numbers as when the numbers are symmetric they cancel out.

**IB Maths Revision**

I’d strongly recommend starting your revision of topics from Y12 – certainly if you want to target a top grade in Y13. My favourite revision site is Revision Village – which has a huge amount of great resources – questions graded by level, full video solutions, practice tests, and even exam predictions. Standard Level students and Higher Level students have their own revision areas. Have a look!

The Goldbach Conjecture is one of the most famous problems in mathematics. It has remained unsolved for over 250 years – after being proposed by German mathematician Christian Goldbach in 1742. Anyone who could provide a proof would certainly go down in history as one of the true great mathematicians. The conjecture itself is deceptively simple:

*“Every even integer greater than 2 can be written as the sum of 2 prime numbers.”*

It’s easy enough to choose some values and see that it *appears* to be true:

4: 2+2

6: 3+3

8: 3+5

10: 3+7 or 5+5

But unfortunately that’s not enough to *prove* it’s true – after all, how do we know the next number can also be written as 2 primes? The only way to prove the conjecture using this method would be to check every even number. Unfortunately there’s an infinite number of these!

Super-fast computers have now checked all the first 4×10^{17} even numbers 4×10^{17} is a number so mind bogglingly big it would take about 13 billion years to check all these numbers, checking one number every second. (4×10^{17})/(60x60x24x365) = 1.3 x 10^{10}. So far they have found that every single even number greater than 2 can indeed be written as the sum of 2 primes.

So, if this doesn’t constitute a proof, then what might? Well, mathematicians have noticed that the greater the even number, the more likely it will have different prime sums. For example 10 can be written as either 3+7 or 5+5. As the even numbers get larger, they can be written with larger combinations of primes. The graph at the top of the page shows this. The x axis plots the even numbers, and the y axis plots the number of different ways of making those even numbers with primes. As the even numbers get larger, the cone widens – showing ever more possible combinations. That would suggest that the conjecture gets ever more likely to be true as the even numbers get larger.

A similar problem from Number Theory (the study of whole numbers) was proposed by legendary mathematician Fermat in the 1600s. He was interested in the links between numbers and geometry – and noticed some interesting patterns between triangular numbers, square numbers and pentagonal numbers:

*Every integer (whole number) is either a triangular number or a sum of 2 or 3 triangular numbers. Every integer is a square number or a sum of 2, 3 or 4 square numbers. Every integer is a pentagonal number or a sum of 2, 3, 4 or 5 pentagonal numbers.*

There are lots of things to investigate with this. Does this pattern continue with hexagonal numbers? Can you find a formula for triangular numbers or pentagonal numbers? Why does this relationship hold?

If you like this post you might also like:

How Are Prime Numbers Distributed? Twin Primes Conjecture. Discussion on studying prime numbers – in particular the conjecture that there are infinitely many twin primes.

Fermat’s Last Theorem An introduction to one of the greatest popular puzzles in maths history.

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

**The Gambler’s Fallacy**

The above video is an excellent introduction to the gambler’s fallacy. This is the misconception that prior outcomes will have an effect on subsequent independent events. The classic example for this is the gambler who watches a run of 9 blacks on a roulette wheel with only red and black, and rushes to place all his money on red. He is sure that red must come up – after all the probability of a run of 10 blacks in a row is 1/1024. However, because the prior outcomes have no influence on the next spin actually the probability remains at 1/2.

Maths is integral to all forms of gambling – the bookmakers and casino owners work out the Expected Value (EV) for every bet that a gambler makes. In a purely fair game where both outcome was equally likely (like tossing a coin) the EV would be 0. If you were betting on the toss of a coin, the over the long run you would expect to win nothing and lose nothing. On a game like roulette with 18 red, 18 black and 2 green, we can work out the EV as follows:

$1 x 18/38 represents our expected winnings

-$1 x 20/38 represents our expected losses.

Therefore the strategy of always betting $1 on red has an EV of -2/38. This means that on average we would expect to lose about 5% of our money every stake.

Expected value can be used by gamblers to work out which games are most balanced in their favour – and in games of skill like poker, top players will have positive EV from every hand. Blackjack players can achieve positive EV by counting cards (not allowed in casinos) – and so casino bosses will actually monitor the long term fortunes of players to see who may be using this technique.

Understanding expected value also helps maximise winnings. Say 2 people both enter the lottery – one chooses 1,2,3,4,5,6 and the other a randomly chosen combination. Both tickets have exactly the same probability of winning (about 1 in 14 million in the UK) – but both have very different EV. The randomly chosen combination will likely be the only such combination chosen – whereas a staggering 10,000 people choose 1,2,3,4,5,6 each week. So whilst both tickets are equally likely to win, the random combination still has an EV 10,000 times higher than the consecutive numbers.

Incidentally it’s worth watching Derren Brown (above). Filmed under controlled conditions with no camera trickery he is still able to toss a coin 10 times and get heads each time. The question is, how is this possible? The answer – that this short clip was taken from 9 hours of solid filming is quite illuminating about our susceptibility to be manipulated with probability and statistics. This particular technique is called data mining (where multiple trials are conducted and then only a small portion of those trials are honed in on to show patterns) and is an easy statistical manipulation of scientific and medical investigations.

If you liked this post you might also like:

Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour

Premier League Finances – Debt and Wages. An investigation into the finances of Premier League clubs.

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

**Cracking RSA Code – The World’s Most Important Code? **

RSA code is the basis of all important data transfer. Encrypted data that needs to be sent between two parties, such as banking data or secure communications relies on the techniques of RSA code. RSA code was invented in 1978 by three mathematicians (Rivest, Shamir and Adleman). Cryptography relies on numerous mathematical techniques from Number Theory – which until the 1950s was thought to be a largely theoretical pursuit with few practical applications. Today RSA code is absolutely essential to keeping digital communications safe.

**To encode a message using the RSA code follow the steps below:**

1) Choose 2 prime numbers p and q (let’s say p=7 and q=5)

2) Multiply these 2 numbers together (5×7 = 35). This is the public key (m) – which you can let everyone know. So m = 35.

3) Now we need to use an encryption key (e). Let’s say that e = 5. e is also made public. (There are restrictions as to what values e can take – e must actually be relatively prime to (p-1)(q-1) )

4) Now we are ready to encode something. First we can assign 00 = A, 01 = B, 02 = C, 03 = D, 04 = E etc. all the way to 25 = Z. So the word CODE is converted into: 02, 14, 03, 04.

5) We now use the formula: C = y^{e} (mod m) where y is the letter we want to encode. So for the letters CODE we get: C = 02^{5 }= 32 (mod 35). C = 14^{5 } = 537824 which is equivalent to 14 (mod 35). C = 03^{5 } = 33 (mod 35). C = 04^{5 } = 1024 which is equivalent to 09 (mod 35). (Mod 35 simply mean we look at the remainder when we divide by 35). Make use of an online modulus calculator! So our coded word becomes: 32 14 33 09.

This form of public key encryption forms the backbone of the internet and the digital transfer of information. It is so powerful because it is very quick and easy for computers to decode if they know the original prime numbers used, and exceptionally difficult to crack if you try and guess the prime numbers. Because of the value of using very large primes there is a big financial reward on offer for finding them. The world’s current largest prime number is over 17 million digits long and was found in February 2013. Anyone who can find a prime 100 million digits long will win $100,000.

**To decode the message 11 49 41 we need to do the following:**

1) In RSA encryption we are given both m and e. These are public keys. For example we are given that m = 55 and e = 27. We need to find the two prime numbers that multiply to give 55. These are p = 5 and q = 11.

2) Calculate (p-1)(q-1). In this case this is (5-1)(11-1) = 40. Call this number theta.

3) Calculate a value d such that de = 1 (mod theta). We already know that e is 27. Therefore we want 27d = 1 (mod 40). When d = 3 we have 27×3 = 81 which is 1 (mod 40). So d = 3.

4) Now we can decipher using the formula: y = C^d (mod m), where C is the codeword. So for the cipher text 11 49 41: y = 11^{3 } = 11 (mod 55). y = 49^{3 } = 04 (mod 55). y = 41^{3 } = 6 (mod 55).

5) We then convert these numbers back to letters using A = 00, B = 01 etc. This gives the decoded word as: LEG.

If you enjoyed this post you might also like:

How Are Prime Numbers Distributed? Twin Primes Conjecture. Discussion on studying prime numbers.

Cracking ISBN and Credit Card Codes. The mathematics behind ISBN codes and credit card codes

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

**Maths IA – Maths Exploration Topics:**

Over 300 examples of maths IA exploration topics and ideas for IB mathematics. This list is for SL and HL students (exam 2020) and for Applications and Analysis students (exam 2021). If you are doing a Maths Studies IA (exam 2019) or an SL Applications IA (exam 2020) then you may be best going to this page instead.

**NEW**

You can download a **60-page pdf guide** to the entire IA coursework process for the new syllabus (first exam 2021) to help you get excellent marks in your maths exploration. Written by an experienced IB teacher this guide talks you through:

- An introduction to the essentials about the investigation,
- The new marking criteria,
- How to choose a topic,
- Examples of around 70 topics that could be investigated,
- Useful websites for use in the exploration,
- A student checklist for completing a good investigation,
- Common mistakes that students make and how to avoid them,
- A worked example for the maths behind correlation investigations,
- A selection of some interesting exploration topics explored in more depth,
- Teacher advice for marking,
- Templates for draft submissions,
- Some examples of beautiful maths using Geogebra and Desmos.

**You can download this here.**

**IB Revision with** Revision Village

There’s a really great website been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course. You choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s.

The Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year. I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

**Maths IA – Maths Exploration Topics**

Here is a list of over 200 ideas with links to further reading for your maths exploration!

**Algebra and number**

1) Modular arithmetic – This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3.

2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics.

3) Probabilistic number theory

4) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.

5) Diophantine equations: These are polynomials which have integer solutions. Fermat’s Last Theorem is one of the most famous such equations.

6) Continued fractions: These are fractions which continue to infinity. The great Indian mathematician Ramanujan discovered some amazing examples of these.

7) Patterns in Pascal’s triangle: There are a large number of patterns to discover – including the Fibonacci sequence.

8) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. There is a $1 million prize for solving the Riemann Hypothesis and $250,000 available for anyone who discovers a new, really big prime number.

9) Random numbers

10) Pythagorean triples: A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle).

11) Mersenne primes: These are primes that can be written as 2^n -1.

12) Magic squares and cubes: Investigate magic tricks that use mathematics. Why do magic squares work?

13) Loci and complex numbers

14) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 2/3 could be written as 1/6 + 1/2. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?

15) Complex numbers and transformations

16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics.

17) Chinese remainder theorem. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. It involves understanding the modulo operation.

18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved.

19) Natural logarithms of complex numbers

20) Twin primes problem: The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. The twin prime conjecture states that there are infinitely many consecutive primes ( eg. 5 and 7 are consecutive primes). There has been a recent breakthrough in this problem.

21) Hypercomplex numbers

22) Diophantine application: Cole numbers

23) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). ie 6 is a perfect number because 1 + 2 + 3 = 6.

24) Euclidean algorithm for GCF

25) Palindrome numbers: Palindrome numbers are the same backwards as forwards.

26) Fermat’s little theorem: If p is a prime number then a^p – a is a multiple of p.

27) Prime number sieves

28) Recurrence expressions for phi (golden ratio): Phi appears with remarkable consistency in nature and appears to shape our understanding of beauty and symmetry.

29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth $1million to anyone who solves it (not for the faint hearted!)

30) Time travel to the future: Investigate how traveling close to the speed of light allows people to travel “forward” in time relative to someone on Earth. Why does the twin paradox work?

31) Graham’s Number – a number so big that thinking about it could literally collapse your brain into a black hole.

32) RSA code – the most important code in the world? How all our digital communications are kept safe through the properties of primes.

33) The Chinese Remainder Theorem: This is a method developed by a Chinese mathematician Sun Zi over 1500 years ago to solve a numerical puzzle. An interesting insight into the mathematical field of Number Theory.

34) Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2?. A post which looks at the maths behind this particularly troublesome series.

35) Fermat’s Theorem on the sum of 2 squares – An example of how to use mathematical proof to solve problems in number theory.

36) Can we prove that 1 + 2 + 3 + 4 …. = -1/12 ? How strange things happen when we start to manipulate divergent series.

37) Mathematical proof and paradox – a good opportunity to explore some methods of proof and to show how logical errors occur.

38) Friendly numbers, Solitary numbers, perfect numbers. Investigate what makes a number happy or sad, or sociable! Can you find the loop of infinite sadness?

39) Zeno’s Paradox – Achilles and the Tortoise – A look at the classic paradox from ancient Greece – the philosopher “proved” a runner could never catch a tortoise – no matter how fast he ran.

40) Stellar Numbers – This is an excellent example of a pattern sequence investigation. Choose your own pattern investigation for the exploration.

41) Arithmetic number puzzle – It could be interesting to do an exploration where you solve number problems – like this one.

42) Normal Numbers – and random number generators – what is a normal number – and how are they connected to random number generators?

43) Narcissistic Numbers – what makes a number narcissistic – and how can we find them all?

44) Modelling Chaos – how we can use grahical software to understand the behavior of sequences

45) The Mordell Equation. What is the Mordell equation and how does it help us solve mathematical problems in number theory?

46) Ramanujan’s Taxi Cab and the Sum of 2 Cubes. Explore this famous number theory puzzle.

**Geometry**

1a) Non-Euclidean geometries: This allows us to “break” the rules of conventional geometry – for example, angles in a triangle no longer add up to 180 degrees. In some geometries triangles add up to more than 180 degrees, in others less than 180 degrees.

1b) The shape of the universe – non-Euclidean Geometry is at the heart of Einstein’s theories on General Relativity and essential to understanding the shape and behavior of the universe.

2) Hexaflexagons: These are origami style shapes that through folding can reveal extra faces.

3) Minimal surfaces and soap bubbles: Soap bubbles assume the minimum possible surface area to contain a given volume.

4) Tesseract – a 4D cube: How we can use maths to imagine higher dimensions.

5) Stacking cannon balls: An investigation into the patterns formed from stacking canon balls in different ways.

6) Mandelbrot set and fractal shapes: Explore the world of infinitely generated pictures and fractional dimensions.

7) Sierpinksi triangle: a fractal design that continues forever.

8) Squaring the circle: This is a puzzle from ancient times – which was to find out whether a square could be created that had the same area as a given circle. It is now used as a saying to represent something impossible.

9) Polyominoes: These are shapes made from squares. The challenge is to see how many different shapes can be made with a given number of squares – and how can they fit together?

10) Tangrams: Investigate how many different ways different size shapes can be fitted together.

11) Understanding the fourth dimension: How we can use mathematics to imagine (and test for) extra dimensions.

12) The Riemann Sphere – an exploration of some non-Euclidean geometry. Straight lines are not straight, parallel lines meet and angles in a triangle don’t add up to 180 degrees.

13) Graphically understanding complex roots – have you ever wondered what the complex root of a quadratic actually means graphically? Find out!

14) Circular inversion – what does it mean to reflect in a circle? A great introduction to some of the ideas behind non-euclidean geometry.

15) Julia Sets and Mandelbrot Sets – We can use complex numbers to create beautiful patterns of infinitely repeating fractals. Find out how!

16) Graphing polygons investigation. Can we find a function that plots a square? Are there functions which plot any polygons? Use computer graphing to investigate.

17) Graphing Stewie from Family Guy. How to use graphic software to make art from equations.

18) Hyperbolic geometry – how we can map the infinite hyperbolic plane onto the unit circle, and how this inspired the art of Escher.

19) Elliptical Curves– how this class of curves have importance in solving Fermat’s Last Theorem and in cryptography.

20) The Coastline Paradox – how we can measure the lengths of coastlines, and uses the idea of fractals to arrive at fractional dimensions.

21) Projective geometry – the development of geometric proofs based on points at infinity.

22) The Folium of Descartes. This is a nice way to link some maths history with studying an interesting function.

23) Measuring the Distance to the Stars. Maths is closely connected with astronomy – see how we can work out the distance to the stars.

24) A geometric proof for the arithmetic and geometric mean. Proof doesn’t always have to be algebraic. Here is a geometric proof.

25) Euler’s 9 Point Circle. This is a lovely construction using just compasses and a ruler.

26) Plotting the Mandelbrot Set – using Geogebra to graphically generate the Mandelbrot Set.

27) Volume optimization of a cuboid – how to use calculus and graphical solutions to optimize the volume of a cuboid.

28) Ford Circles– how to generate Ford circles and their links with fractions.

29) Classical Geometry Puzzle: Finding the Radius. This is a nice geometry puzzle solved using a variety of methods.

30) Can you solve Oxford University’s Interview Question?. Try to plot the locus of a sliding ladder.

31) The Shoelace Algorithm to find areas of polygons. How can we find the area of any polygon?

32) Soap Bubbles, Wormholes and Catenoids. What is the geometric shape of soap bubbles?

**Calculus/analysis and functions**

1) The harmonic series: Investigate the relationship between fractions and music, or investigate whether this series converges.

2) Torus – solid of revolution: A torus is a donut shape which introduces some interesting topological ideas.

3) Projectile motion: Studying the motion of projectiles like cannon balls is an essential part of the mathematics of war. You can also model everything from Angry Birds to stunt bike jumping. A good use of your calculus skills.

4) Why e is base of natural logarithm function: A chance to investigate the amazing number e.

5) Fourier Transforms – the most important tool in mathematics? Fourier transforms have an essential part to play in modern life – and are one of the keys to understanding the world around us. This mathematical equation has been described as the most important in all of physics. Find out more! (This topic is only suitable for IB HL students).

6) Batman and Superman maths – how to use Wolfram Alpha to plot graphs of the Batman and Superman logo

7) Explore the Si(x) function – a special function in calculus that can’t be integrated into an elementary function.

8) The Remarkable Dirac Delta Function. This is a function which is used in Quantum mechanics – it describes a peak of zero width but with area 1.

9) Optimization of area – an investigation. This is an nice example of how you can investigation optimization of the area of different polygons.

10) Envelope of projectile motion. This investigates a generalized version of projectile motion – discover what shape is created.

**Statistics and modelling**

1) Traffic flow: How maths can model traffic on the roads.

2) Logistic function and constrained growth

3) Benford’s Law – using statistics to catch criminals by making use of a surprising distribution.

4) Bad maths in court – how a misuse of statistics in the courtroom can lead to devastating miscarriages of justice.

5) The mathematics of cons – how con artists use pyramid schemes to get rich quick.

6) Impact Earth – what would happen if an asteroid or meteorite hit the Earth?

7) Black Swan events – how usefully can mathematics predict small probability high impact events?

8) Modelling happiness – how understanding utility value can make you happier.

9) Does finger length predict mathematical ability? Investigate the surprising correlation between finger ratios and all sorts of abilities and traits.

10) Modelling epidemics/spread of a virus

11) The Monty Hall problem – this video will show why statistics often lead you to unintuitive results.

12) Monte Carlo simulations

13) Lotteries

14) Bayes’ theorem: How understanding probability is essential to our legal system.

15) Birthday paradox: The birthday paradox shows how intuitive ideas on probability can often be wrong. How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday? Find out!

16) Are we living in a computer simulation? Look at the Bayesian logic behind the argument that we are living in a computer simulation.

17) Does sacking a football manager affect results? A chance to look at some statistics with surprising results.

18) Which times tables do students find most difficult? A good example of how to conduct a statistical investigation in mathematics.

19) Introduction to Modelling. This is a fantastic 70 page booklet explaining different modelling methods from Moody’s Mega Maths Challenge.

20) Modelling infectious diseases – how we can use mathematics to predict how diseases like measles will spread through a population

21) Using Chi Squared to crack codes – Chi squared can be used to crack Vigenere codes which for hundreds of years were thought to be unbreakable. Unleash your inner spy!

22) Modelling Zombies – How do zombies spread? What is your best way of surviving the zombie apocalypse? Surprisingly maths can help!

23) Modelling music with sine waves – how we can understand different notes by sine waves of different frequencies. Listen to the sounds that different sine waves make.

24) Are you psychic? Use the binomial distribution to test your ESP abilities.

25) Reaction times – are you above or below average? Model your data using a normal distribution.

26) Modelling volcanoes – look at how the Poisson distribution can predict volcanic eruptions, and perhaps explore some more advanced statistical tests.

27) Could Trump win the next election? How the normal distribution is used to predict elections.

28) How to avoid a Troll – an example of a problem solving based investigation

29) The Gini Coefficient – How to model economic inequality

30) Maths of Global Warming – Modeling Climate Change – Using Desmos to model the change in atmospheric Carbon Dioxide.

31) Modelling radioactive decay – the mathematics behind radioactivity decay, used extensively in science.

32) Circular Motion: Modelling a Ferris wheel. Use Tracker software to create a Sine wave.

33) Spotting Asset Bubbles. How to use modeling to predict booms and busts.

34) The Rise of Bitcoin. Is Bitcoin going to keep rising or crash?

35) Fun with Functions!. Some nice examples of using polar coordinates to create interesting designs.

36) Predicting the UK election using linear regression. The use of regression in polling predictions.

37) Modelling a Nuclear War. What would happen to the climate in the event of a nuclear war?

38) Modelling a football season. We can use a Poisson model and some Excel expertise to predict the outcome of sports matches – a technique used by gambling firms.

39)Modeling hours of daylight – using Desmos to plot the changing hours of daylight in different countries.

40) Modelling the spread of Coronavirus (COVID-19). Using the SIR model to understand epidemics.

41) Finding the volume of a rugby ball (or American football). Use modeling and volume of revolutions.

**Games and game theory**

1) The prisoner’s dilemma: The use of game theory in psychology and economics.

2) Sudoku

3) Gambler’s fallacy: A good chance to investigate misconceptions in probability and probabilities in gambling. Why does the house always win?

4) Bluffing in Poker: How probability and game theory can be used to explore the the best strategies for bluffing in poker.

5) Knight’s tour in chess: This chess puzzle asks how many moves a knight must make to visit all squares on a chess board.

6) Billiards and snooker

7) Zero sum games

8) How to “Solve” Noughts and Crossess (Tic Tac Toe) – using game theory. This topics provides a fascinating introduction to both combinatorial Game Theory and Group Theory.

9) Maths and football – Do managerial sackings really lead to an improvement in results? We can analyse the data to find out. Also look at the finances behind Premier league teams

10) Is there a correlation between Premier League wages and league position? Also look at how the Championship compares to the Premier League.

11) The One Time Pad – an uncrackable code? Explore the maths behind code making and breaking.

12) How to win at Rock Paper Scissors. Look at some of the maths (and psychology behind winning this game.

13) The Watson Selection Task – a puzzle which tests logical reasoning. Are maths students better than history students?

**Topology and networks**

1) Knots

2) Steiner problem

3) Chinese postman problem – This is a problem from graph theory – how can a postman deliver letters to every house on his streets in the shortest time possible?

4) Travelling salesman problem

5) Königsberg bridge problem: The use of networks to solve problems. This particular problem was solved by Euler.

6) Handshake problem: With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else?

7) Möbius strip: An amazing shape which is a loop with only 1 side and 1 edge.

8) Klein bottle

9) Logic and sets

10) Codes and ciphers: ISBN codes and credit card codes are just some examples of how codes are essential to modern life. Maths can be used to both make these codes and break them.

11) Zeno’s paradox of Achilles and the tortoise: How can a running Achilles ever catch the tortoise if in the time taken to halve the distance, the tortoise has moved yet further away?

12) Four colour map theorem – a puzzle that requires that a map can be coloured in so that every neighbouring country is in a different colour. What is the minimum number of colours needed for any map?

13) Telephone Numbers – these are numbers with special properties which grow very large very quickly. This topic links to graph theory.

14)The Poincare Conjecture and Grigori Perelman – Learn about the reclusive Russian mathematician who turned down $1 million for solving one of the world’s most difficult maths problems.

**Mathematics and Physics**

1) The Monkey and the Hunter – How to Shoot a Monkey – Using Newtonian mathematics to decide where to aim when shooting a monkey in a tree.

2) How to Design a Parachute – looking at the physics behind parachute design to ensure a safe landing!

3) Galileo: Throwing cannonballs off The Leaning Tower of Pisa – Recreating Galileo’s classic experiment, and using maths to understand the surprising result.

4) Rocket Science and Lagrange Points – how clever mathematics is used to keep satellites in just the right place.

5) Fourier Transforms – the most important tool in mathematics? – An essential component of JPEG, DNA analysis, WIFI signals, MRI scans, guitar amps – find out about the maths behind these essential technologies.

6) Bullet projectile motion experiment – using Tracker software to model the motion of a bullet.

7) Quantum Mechanics – a statistical universe? Look at the inherent probabilistic nature of the universe with some quantum mechanics.

8) Log Graphs to Plot Planetary Patterns. The planets follow a surprising pattern when measuring their distances.

9) Modeling with springs and weights. Some classic physics – which generates some nice mathematical graphs.

10) Is Intergalactic space travel possible? Using the physics of travel near the speed of light to see how we could travel to other stars.

**Maths and computing**

1) The Van Eck Sequence – The Van Eck Sequence is a sequence that we still don’t fully understand – we can use programing to help!

2) Solving maths problems using computers – computers are really useful in solving mathematical problems. Here are some examples solved using Python.

3) Stacking cannonballs – solving maths with code – how to stack cannonballs in different configurations.

4) What’s so special about 277777788888899? – Playing around with multiplicative persistence – can you break the world record?

5) Project Euler: Coding to Solve Maths Problems. A nice starting point for students good at coding – who want to put these skills to the test mathematically.

6) Square Triangular Numbers. Can we use a mixture of pure maths and computing to solve this problem?

7) When do 2 squares equal 2 cubes? Can we use a mixture of pure maths and computing to solve this problem?

8) Hollow Cubes and Hypercubes investigation. More computing led investigations

**Further ideas:**

1) Radiocarbon dating – understanding radioactive decay allows scientists and historians to accurately work out something’s age – whether it be from thousands or even millions of years ago.

2) Gravity, orbits and escape velocity – Escape velocity is the speed required to break free from a body’s gravitational pull. Essential knowledge for future astronauts.

3) Mathematical methods in economics – maths is essential in both business and economics – explore some economics based maths problems.

4) Genetics – Look at the mathematics behind genetic inheritance and natural selection.

5) Elliptical orbits – Planets and comets have elliptical orbits as they are influenced by the gravitational pull of other bodies in space. Investigate some rocket science!

6) Logarithmic scales – Decibel, Richter, etc. are examples of log scales – investigate how these scales are used and what they mean.

7) Fibonacci sequence and spirals in nature – There are lots of examples of the Fibonacci sequence in real life – from pine cones to petals to modelling populations and the stock market.

8) Change in a person’s BMI over time – There are lots of examples of BMI stats investigations online – see if you can think of an interesting twist.

9) Designing bridges – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse.

10) Mathematical card tricks – investigate some maths magic.

11) Flatland by Edwin Abbott – This famous book helps understand how to imagine extra dimension. You can watch a short video on it here

12) Towers of Hanoi puzzle – This famous puzzle requires logic and patience. Can you find the pattern behind it?

13) Different number systems – Learn how to add, subtract, multiply and divide in Binary. Investigate how binary is used – link to codes and computing.

14) Methods for solving differential equations – Differential equations are amazingly powerful at modelling real life – from population growth to to pendulum motion. Investigate how to solve them.

15) Modelling epidemics/spread of a virus – what is the mathematics behind understanding how epidemics occur? Look at how infectious Ebola really is.

16) Hyperbolic functions – These are linked to the normal trigonometric functions but with notable differences. They are useful for modelling more complex shapes.

17) Medical data mining – Explore the use and misuse of statistics in medicine and science.

18)Waging war with maths: Hollow squares. How mathematical formations were used to fight wars.

**The Gorilla in the Room and Other Great Maths Investigations**

These topics are a great way to add interest to statistics and probability lessons at KS3 and KS4 level, and also a good example of investigations that IB students can conduct. They also have a nice link to ToK – how can we believe what we see or what we hear? To what extent should we trust our senses? And it shows the power of statistics and empirical testing in trying to understand what is externally real and what is our own version of reality.

For each one, have the students make a hypothesis (if possible without giving the endings away!), then collect some data as to how the students react. Then look at how the data could be collected in a larger scale experiment (or how the experiment could be modified).

The first one at the top of the page is the “Fa, Ba” test. This is a really curious experiment that shows that what we “hear” is actually often influenced by what we see.

The second one is the amazing colour changing card trick by Richard Wiseman. This is also a great way of showing how we often fail to see what is really in front of us:

The third video is even more impressive – though it doesn’t work on all students. You have to set this one up so that all students are really intently concentrating on the screen – perhaps a prize for the student who gets the answer correct? Also no talking! Students have to count basketball passes:

The last one is a good test of whether students are “right brain” or “left brain” dominant. They have to stare at a rotating woman – some students will see this going clockwise, others anticlockwise. Some will be able to switch between the 2 views. If they can’t (I initially could only see this going in an anti clockwise direction) near the end of the video it shows the woman rotating in a clockwise direction to help. Then rewinding the video to the start – and as if by magic she had changed direction.

If you liked this post you might also like:

Even Pigeons Can Do Maths A discussion about the ability of both chimps and pigeons to count

Finger Ratio Predicts Maths Ability? A post which discusses the correlation between the two.

**Are You Living in a Computer Simulation?**

This idea might be familiar to fans of The Matrix – and at first glance may seem somewhat unbelievable. However, Oxford University Professor Nick Bostrom makes an interesting case using both conditional probability and logic as to why it’s more likely than you might think.

The summary of Bostrom’s Computer Simulation argument is the following:

*At least one of the following propositions is true: (1) the human species is very likely to go extinct before reaching a “posthuman” stage; (2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof); (3) we are almost certainly living in a computer simulation. It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor-simulations is false, unless we are currently living in a simulation. *

The full paper where he makes his argument is available as a pdf here – and is well worth a read. Alternatively Bostrom makes this case in a detailed interview:

Taking the argument step by step, firstly when Bostrom talks about a “posthuman” stage he is referring to an advanced civilisation with significantly greater technological capabilities than we have at present. Such a civilisation would have the ability to run a computer simulation so accurate that it would be indistinguishable from “real life”.

This is a twist on the traditional “Brain in a Vat” thought experiment much loved by philosophers when trying to argue whether we be sure that anything exists outside our own subjective experience:

Based on the same logic, we have no way of genuinely knowing whether we are really “here” or whether we are nothing but a computer model designed to give the impression that we really exist. Interestingly, the possibility that our individual life, the world around us and indeed everything we know about the universe may be false means that we can never truly claim to have knowledge of anything.

I think that most optimists would think that civilisation has the potential to develop into a “posthuman” phase of advanced technology. Indeed, you only need to look at the phenomenal growth in computer power (see Moore’s Law) to have confidence that should we stick around long enough, we will have the computational power possible to run such simulations.

So if we optimistically accept that humans will reach a “posthuman” stage, then it’s even easier to accept the second proposition – that if an advanced civilisation is able to run such civilisations they will do. After all human nature is such that we tend to do things just because we can – and in any case running such ancestor simulations would potentially be very beneficial for real world modelling.

If we do accept both these premises, then this therefore leads to the argument that we are almost certainly living in a computer simulation. Why? Well, an advanced civilisation with the computational power to run ancestor simulations would likely run a large number of them – and if there is only one *real* world, then our experience of a world is likely to be one of these simulations.

As a ToK topic this is a fantastic introduction to epistemological questions about the limits of knowledge and questions of existence, and is a really good example of the power of logic and mathematics to reveal possibilities about the world outside our usual bounds of thinking.

If you enjoyed this post you might also like:

Imagining the 4th Dimension – How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.

Is Maths Invented or Discovered? – A discussion about some of the basic philosophical questions that arise in mathematics.

**IB Revision**

**Bridge Building Lesson Plan**

Learning Objectives: Students are introduced to one of the many careers that they can pursue through mathematics.

**5 minutes:**

Brainstorm – why is mathematics useful for engineering? What kinds of jobs do engineers do? (refer to maths careers site – a large number of well paid jobs are in engineering)

**5 minute**

Watch Youtube video interviewing 3 young structural engineers:

**5 minutes:**

Use the bridge building game to discuss strategies.

**5 minutes:**

Discuss the different types of bridge structures – plain bridge, arch bridge, suspension bridge. What are the advantages and disadvantages of each of these? Brief discussion about force dissipation – which shapes do this well?

**5 minutes:**

Set up the challenge – each group must build a bridge to span a 1 metre gap. The bridge must be wide enough to support a weight and the stronger the better! Resources are: straws, newspaper, sellotape. Sellotape and newspaper are free – but straws are 1000B a straw. How cheaply can a design be made?

**10 minutes:**

Students start to plan their bridges. Watch Youtube video about what happens when bridge design goes wrong:

**40 minutes:**

Building the bridge.

**10 minutes:**

Weight testing!

**Black Swans and Civilisation Collapse**

A really interesting branch of mathematics is involved in making future predictions about how civilisation will evolve in the future – and indeed looking at how robust our civilisation is to external shocks. This is one area in which mathematical models do not have a good record as it is incredibly difficult to accurately assign probabilities and form policy recommendations for events in the future.

**Malthusian Catastrophe**

One of the most famous uses of mathematical models in this context was by Thomas Malthus in 1798. He noted that the means of food production were a fundamental limiting factor on population growth – and that if population growth continued beyond the means of food production that there would be (what is now termed) a “Malthusian catastrophe” of a rapid population crash.

As it turns out, agrarian productivity has been able to keep pace with the rapid population growth of the past 200 years.

Looking at the graph we can see that whilst it took approximately 120 years for the population to double from 1 billion to 2 billion, it only took 55 years to double again. It would be a nice exercise to try and see what equation fits this graph – and also look at the rate of change of population (is it now slowing down?) The three lines at the end of the graph are the three different UN predictions – high end, medium and low end estimate. There’s a pretty stark difference between high end and low end estimates by 2100 – between 16 billion and 6 billion! So what does that tell us about the accuracy of such predictions?

**Complex Civilisations**

More recently academics like Joseph Tainter and Jared Diamond have popularised the notion of civilisations as vulnerable to collapse due to ever increasing complexity. In terms of robustness of civilisation one can look at an agrarian subsistence example. Agrarian subsistence is pretty robust against civilisation collapse – small self sufficient units may themselves be rather vulnerable to famines and droughts on an individual level, but as a society they are able to ride out most catastrophes intact.

The next level up from agrarian subsistence is a more organised collective – around a central authority which is able to (say) provide irrigation technology through a system of waterways. Immediately the complexity of society has increased, but the benefits of irrigation allow much more crops to be grown and thus the society can support a larger population. However, this complexity comes at a cost – society now is reliant on those irrigation channels – and any damage to them could be catastrophic to society as a whole.

To fast forward to today, we have now an incredibly complex society, far far removed from our agrarian past – and whilst that means we have an unimaginably better quality of life, it also means society is more vulnerable to collapse than ever before. To take the example of a Coronal Mass Ejection – in which massive solar discharges hit the Earth. The last large one to hit the Earth was in 1859 but did negligible damage as this was prior to the electrical age. Were the same event to happen today, it would cause huge damage – as we are reliant on electricity for everything from lighting to communication to refrigeration to water supplies. A week without electricity for an urban centre would mean no food, no water, no lighting, no communication and pretty much the entire breakdown of society.

That’s not to say that such an event will happen in our lifetimes – but it does raise an interesting question about intelligent life – if advanced civilisations continue to evolve and in the process grow more and more complex then is this a universal limiting factor on progress? Does ever increasing complexity leave civilisations so vulnerable to catastrophic events that their probabilities of surviving through them grow ever smaller?

**Black Swan Events**

One of the great challenges for mathematical modelling is therefore trying to assign probabilities for these “Black Swan” events. The term was coined by economist Nassim Taleb – and used to describe rare, low probability events which have very large consequences. If the probability of a very large scale asteroid impact is (say) estimated as 1-100,000 years – but were it to hit it is estimated to cause $35 trillion of damage (half the global GDP) then what is the rational response to such a threat? Dividing the numbers suggests that we should in such a scenario be spending $3.5billion every year on trying to address such an event – and yet which politician would justify such spending on an event that might not happen for another 100,000 years?

I suppose you would have to conclude therefore that our mathematical models are pretty poor at predicting future events, modelling population growth or dictating future and current policy. Which stands in stark contrast to their abilities in modelling the real world (minus the humans). Will this improve in the future, or are we destined to never really be able to predict the complex outcomes of a complex world?

If you enjoyed this post you might also like:

Asteroid Impact Simulation – which allows you to model the consequences of asteroid impacts on Earth.

Chaos Theory – an Unpredictable Universe? – which discusses the difficulties in mathematical modelling when small changes in initial states can have very large consequences.

**IB Revision**

**The Riemann Hypothesis Explained
**

This is quite a complex topic probably only accessible for high achieving HL IB students, but nevertheless it’s still a fascinating introduction to one of the most important (and valuable) unsolved problems in pure mathematics.

Firstly, the Riemann Hypothesis is concerned with the Riemann zeta function. This function is defined in many ways, but probably the most useful for us is this version:

In other words the Riemann zeta function consists of a sum to infinity multiplied by an external bracket. s is a complex number of the form *s* = *σ* + *it. *This formula is valid for Re(s) > 0 . This means that the real part of the complex number must be positive.

Now, the Riemann Hypothesis is concerned with finding the roots of the Riemann zeta function – ie. what values of complex number s cause the function to be zero. However the equation above is only valid for Re(s) > 0. To check for roots where Re(s) is less than or equal to 0 we can use an alternative representation of the Riemann zeta function:

which shows that the zeta function is zero whenever s = -2,-4,-6…. as for these values the sine term becomes zero. (s = 0 has no solution in this representation because it leaves us with a zeta function of 1 in the far right term – which produces a singularity). These values are called the *trivial zeroes* of the zeta function.

The other, non-trivial zeroes of the Riemann zeta function are more difficult to find – and the search for them leads to the Riemann hypothesis:

**The non-trivial zeroes of the Riemann zeta function have a real part of s equal to 1/2**

In other words, *ζ*(*s*) has non-trivial zeroes only when s is in the form s = 1/2 + *it. *This is probably easier to understand in graphical form. Below we have s plotted in the complex plane:

We can see that when s = 1 the function is not defined. This is because when s =1 in the original equation for the zeta function we get a singularity as this causes the bracket to the left of the summation to reduce to 1/0. All the non-trivial zeroes for the zeta function are known to lie in the grey boxed, “critical strip” – and the Riemann hypothesis is that they all lie on the dotted line where the real value is 1/2.

This hypothesis, made by German mathematician Bernhard Riemann in 1859 is still unsolved over 150 years later – despite some of the greatest mathematical minds of the 2oth century attempting the problem. Indeed it is considered by many mathematicians to be the most important unresolved question in pure mathematics. Mathematician David Hilbert who himself collected 23 great unsolved mathematical problems together in 1900 stated,

*“If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?”*

The problem is today listed as one of the Clay Institute’s Millennium Prize Problems – anyone who can prove it will win $1 million and will quite probably go down in history as one of the greatest mathematicians of all time.

One solution for s which gives a zero of the zeta function is 0.5 + 14.134725142 i. Another one is 0.5 + 21.022039639 i. These both satisfy the Riemann Hypothesis by having a real part of 1/2. Indeed, to date, *10 trillion* (10,000,000,000,000) non trivial solutions have been found – and they *all* have a real part of 1/2. But this is not a proof that it is true for all roots – and so the problem remains unsolved.

So, why is this such an important problem? Well, because there is a connection between the Riemann zeta function and distribution of prime numbers. The function below on the left is another way of representing the Riemann zeta function and the function on the right is an infinite product including all prime numbers:

where:

Understanding the Riemann zeta function will help mathematicians unlock some of the mysteries of the prime numbers – which are the building blocks of number theory (the study of integers). For example looking at the graph below (drawn by a Wolfram Mathlab probject) we can see the function pi(x) plotted against a function which uses both the Riemann zeta function and the distribution of its zeroes. pi(x) is blue graph and shows the number of primes less than or equal to x.

With the number of primes on the y axis, we can see that out of the first 420 numbers there are approximately 80 primes. What is remarkable about the red line is that it so accurately tracks the progress of the prime numbers.

If you are interested in reading more on this the Wikipedia page on the Riemann zeta function goes into a lot more detail. A more lighthearted introduction to the topic is given by the paper, “A Friendly Introduction to the Riemann Hypothesis”

If you enjoyed this post you might also like:

Graham’s Number – a number so large it’ll literally collapse your head into a black hole.

Twin Primes and How Prime Numbers are Distributed – some more discussion on studying prime numbers – in particular the conjecture that there are infinitely many twin primes.

The Million Dollar Maths Problems – some general introductions to the seven million dollar maths problems.

**IB Revision**