Time Travel and the Speed of Light

This is one of my favourite videos from the legendary Carl Sagan. He explains the consequences of near to speed of light travel.

This topic fits quite well into a number of mathematical topics – from graphing, to real life uses of equations, to standard form and unit conversions. It also challenges our notion of time as we usually experience it and therefore leads onto some interesting questions about the nature of reality. Below we can see the time dilation graph:

which clearly shows that for low speeds there is very little time dilation, but when we start getting to within 90% of the speed of light, that there is a very significant time dilation effect. For more accuracy we can work out the exact dilation using the formula given – where v is the speed traveled, c is the speed of light, t is the time experienced in the observer’s own frame of reference (say, by looking at his watch) and t’ is the time experienced in a different, stationary time frame (say on Earth) . Putting some numbers in for real life examples:

1) A long working air steward spends a cumulative total of 5 years in the air – flying at an average speed of 900km/h. How much longer will he live (from a stationary viewpoint) compared to if he had been a bus driver?

2) Voyager 1, launched in 1977 and now currently about 1.8×10^10 km away from Earth is traveling at around 17km/s. How far does this craft travel in 1 hour? What would the time dilation be for someone onboard since 1977?

3) I built a spacecraft capable of traveling at 95% the speed of light. I said goodbye to my twin sister and hopped aboard, flew for a while before returning to Earth. If I experienced 10 years on the space craft, how much younger will I be than my twin?

Scroll to the bottom for the answers

Marcus De Sautoy also presents an interesting Horizon documentary on the speed of light, its history and the CERN experiments last year that suggested that some particles may have traveled faster than light:

There is a lot of scope for extra content on this topic – for example, looking at the distance of some stars visible in the night sky. For example, red super-giant star Belelgeuse is around 600 light years from Earth. (How many kilometres is that?) When we look at Betelgeuse we are actually looking 600 years “back in time” – so does it make sense to use time as a frame of reference for existence?

Answers

1) Convert 900km/h into km/s = 0.25km/s. Now substitute this value into the equation, along with the speed of light at 300,000km/s….and even using Google’s computer calculator we get a difference so negligible that the denominator rounds to 1.

2) With units already in km/s we substitute the values in – and using a powerful calculator find that denominator is 0.99999999839. Therefore someone traveling on the ship for what their watch recorded as 35 years would actually have been recorded as leaving Earth 35.0000000562 years ago. Which is about 1.78seconds! So still not much effect.

3) This time we get a denominator of 0.3122498999 and so the time experienced by my twin will be 32 years. In effect my sister will have aged 22 years more than me on my return. Amazing!

If you enjoyed this topic you might also like:

Michio Kaku – Universe in a Nutshell

Champagne Supernovas and the Birth of the Universe - some amazing pictures from space.

Simulations -Traffic Jams and Asteroid Impacts

This is a really good online Java app which has been designed by a German mathematician to study the mathematics behind traffic flow.  Why do traffic jams form?  How does the speed limit or traffic lights or the number of lorries on the road affect road conditions?   You can run a number of different simulations – looking at ring road traffic, lane closures and how robust the system is by applying an unexpected perturbation (like an erratic driver).

There is a lot of scope for investigation – with some prompts on the site.  For example, just looking at one variable – the speed limit – what happens in the lane closure model?  Interestingly, with a homogenous speed of 80 km/h there is no traffic congestion – but if the speed is increased to 140km/h then large congestion builds up quickly as cars are unable to change lanes.   This is why reduced speed limits  are applied on motorways during lane closures.

Another investigation is looking at how the style of driving affects the models.  You can change the politeness of the drivers – do they change lanes recklessly?  How many perturbations (erratic incidents) do you need to add to the simulation to cause a traffic jam?

This is a really good example of mathematics used in a real life context – and also provides some good opportunities for a computer based investigation looking at the altering one parameter at a time to note the consequences.

Another good simulation is on the Impact: Earth page.  This allows you to investigate the consequences of various asteroid impacts on Earth – choosing from different parameters such as diameter, velocity, density and angle of impact.  It then shows a detailed breakdown of thee consequences – such as crater size and energy released.   You can also model some famous impacts from history and see their effects.   Lots of scope for mathematical modelling – and also for links with physics.  Also possible discussion re the logarithmic Richter scale – why is this useful?

Student Handout

Asteroid Impact – Why is this important?
Comets and asteroids impact with Earth all the time – but most are so small that we don’t even notice. On a cosmic scale however, the Earth has seen some massive impacts – which were they to happen again today could wipe out civilisation as we know it.

The website Impact Earth allows us to model what would happen if a comet or asteroid hit us again. Jay Melosh professor of Physics and Earth Science says that we can expect “fairly large” impact events about every century. The last major one was in Tunguska Siberia in 1908 – which flattened an estimated 80 million trees over an area of 2000 square km. The force unleashed has been compared to around 1000 Hiroshima nuclear bombs. Luckily this impact was in one of the remotest places on Earth – had the impact been near a large city the effects could be catastrophic.

Jay says that, ”The biggest threat in our near future is the asteroid Apophis, which has a small chance of striking the Earth in 2036. It is about one-third of a mile in diameter.”

Task 1: Watch the above video on a large asteroid impact – make some notes.

Task 2:Research about Apophis – including the dimensions and likely speed of the asteroid and probability of collision. Use this data to enter into the Impact Earth simulation and predict the damage that this asteroid could do.

Task 3: Investigate the Tunguska Event. When did it happen? What was its diameter? Likely speed? Use the data to model this collision on the Impact Earth Simulation. Additional: What are the possible theories about Tunguska? Was it a comet? Asteroid? Death Ray?

Task 4: Conduct your own investigation on the Impact Earth Website into what factors affect the size of craters left by impacts. To do this you need to change one variable and keep all the the other variables constant.  The most interesting one to explore is the angle of impact.  Keep everything else the same and see what happens to the crater size as the angle changes from 10 degrees to 90 degrees.  What angle would you expect to cause the most damage?  Were you correct?  Plot the results as a graph.

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Champagne Supernovas and the Birth of the Universe - some amazing photos from space.

Fractals, Mandelbrot and the Koch Snowflake - using maths to model infinite patterns.

NASA, Aliens and Binary Codes from the Star

The Drake Equation was intended by astronomer Frank Drake to spark a dialogue about the odds of intelligent life on other planets. He was one of the founding members of SETI – the Search for Extra Terrestrial Intelligence – which has spent the past 50 years scanning the stars looking for signals that could be messages from other civilisations.

In the following video, Carl Sagan explains about the Drake Equation:

The Drake equation is:

where:

N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);
R* = the average number of star formation per year in our galaxy
fp = the fraction of those stars that have planets
ne = the average number of planets that can potentially support life per star that has planets
fl = the fraction of planets that could support life that actually develop life at some point
fi = the fraction of planets with life that actually go on to develop intelligent life (civilizations)
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space

The desire to encode and decode messages is a very important branch of mathematics – with direct application to all digital communications – from mobile phones to TVs and the internet.

All data content can be encoded using binary strings. A very simple code could be to have 1 signify “black” and 0 to signify “white” – and then this could then be used to send a picture. Data strings can be sent which are the product of 2 primes – so that the recipient can know the dimensions of the rectangle in which to fill in the colours.

If this sounds complicated, an example from the excellent Maths Illuminated handout on codes:

If this mystery message was received from space, how could we interpret it? Well, we would start by noticing that it is 77 digits long – which is the product of 2 prime numbers, 7 and 11. Prime numbers are universal and so we would expect any advanced civilisation to know about their properties. This gives us either a 7×11 or 11×7 rectangular grid to fill in. By trying both possibilities we see that an 11×7 grid gives the message below.

More examples can be downloaded from the Maths Illuminated section on Primes (go to the facilitator pdf).

A puzzle to try:

“If the following message was received from outer space, what would we conjecture that the aliens sending it looked like?”

0011000 0011000 1111111 1011001 0011001 0111100 0100100 0100100 0100100 1100110

Hint: also 77 digits long.

This is an excellent example of the universality of mathematics in communicating across all languages and indeed species. Prime strings and binary represent an excellent means of communicating data that all advanced civilisations would easily understand.

Answer in white text below (highlight to read)

Arrange the code into a rectangular array – ie a 11 rows by 7 columns rectangle. The first 7 numbers represent the 7 boxes in the first row etc. A 0 represents white and 1 represents black. Filling in the boxes and we end up with an alien with 2 arms and 2 legs – though with one arm longer than the other!
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Cracking Codes Lesson - a double period lesson on using and breaking codes

Cracking ISBN and Credit Card Codes- the mathematics behind ISBN codes and credit card codes

Cracking ISBN and Credit Card Codes

ISBN codes are used on all books published worldwide. It’s a very powerful and useful code, because it has been designed so that if you enter the wrong ISBN code the computer will immediately know – so that you don’t end up with the wrong book. There is lots of information stored in this number. The first numbers tell you which country published it, the next the identity of the publisher, then the book reference.

Here is how it works:

Look at the 10 digit ISBN number. The first digit is 1 so do 1×1. The second digit is 9 so do 2×9. The third digit is 3 so do 3×3. We do this all the way until 10×3. We then add all the totals together. If we have a proper ISBN number then we can divide this final number by 11. If we have made a mistake we can’t. This is a very important branch of coding called error detection and error correction. We can use it to still interpret codes even if there have been errors made.
If we do this for the barcode above we should get 286. 286/11 = 26 so we have a genuine barcode.

Check whether the following are ISBNs

1) 0-13165332-6
2) 0-1392-4191-4
3) 07-028761-4

Challenge (harder!) :The following ISBN code has a number missing, what is it?
1) 0-13-1?9139-9

Answers in white text at the bottom, highlight to reveal!

Credit cards use a different algorithm – but one based on the same principle – that if someone enters a digit incorrectly the computer can immediately know that this credit card does not exist.  This is obviously very important to prevent bank errors.  The method is a little more complicated than for the ISBN code and is given below from computing site Hacktrix:

You can download a worksheet for this method here. Try and use this algorithm to validate which of the following 3 numbers are genuine credit cards:

1) 5184 8204 5526 6425

2) 5184 8204 5526 6427

3) 5184 8204 5526 6424

Answers in white text at the bottom, highlight to reveal!

ISBN:
1) Yes
2) No
3) No
1) 3 – using x as the missing number we end up with 5x + 7 = 0 mod 11. So 5x = 4 mod 11. When x = 3 this is solved.
Credit Card: The second one is genuine

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NASA, Aliens and Binary Codes from the Stars - a discussion about how pictures can be transmitted across millions of miles using binary strings.

Cracking Codes Lesson - an example of 2 double period lessons on code breaking

Cracking Codes Lesson 1

Introduction: 5 minutes – Use a Morse Code Generator to play some (very slowed down) messages for students to decode.  Discuss why this is was a good way to transmit data in the past.

Brainstorm: 5 minutes – Why are codes important?  Who uses them?  Why do mathematicians go into this career?  Look at all data transmission – TVs, internet, mobile phones.  Discuss the picture at the top of the page - this was transmitted from Mars – which is on average 225 million km from Earth (why on average?)  So, how can we transmit data across such a huge distance?

Video: 10 minutes:  Watch Marcus De Satouy video explaining codes (stop around 8.30):

Worksheet:  Between 30 mins and 50 minutes depending on ability and hints - Give out code challenge worksheet – Murder in the Maths Department.  Working in groups of 2-3.  Students will probably need direction – but try to limit this to a minimum to encourage problem solving.  (First students to finish should create their own coded messages for each other).

Lesson 2:

Binary Codes Introduction: 5 minutes -  Can we see the link between binary codes and Morse codes?  Why are binary strings good for sending data?  Link back to Mars picture.  Talk about SETI – what is SETI (Search for Extra-Terrestrial Intelligence), what do they do?  (Scan sky looking for non-random data strings)

Binary Code Worksheet: 25 minutes - Students need  to convert the binary string codes into pictures.

Extension material: 25 minutes – Handout Vigenere Cipher, ISBN codes and Credit Card Codes for top ability students

Lesson Resources:

Large Code Challenge Resource Pack (including Binary code and Vigenere Cipher Worksheets and Murder in the Maths Department) for download on TES.

Additional Resources:

Crypto Corner is the newest and best code making and code breaking website online – it’s got a huge amount of code information and also allows you to generate your own codes.

CIMT Code resources - a fantastic resource with a large number of ready made worksheets and teacher notes on lots of different codes.

Secret Codebreaker also has  a lot of information about different codes

Counton website to generate different codes - generate your own codes

Nrich has a nice article about the history of codes and mathematics

NASA codes from the stars - more explanation on binary string codes.

Khan Academy code breaking videos- a large number of short videos looking at both codes through history and more modern code methods.

Numberphile video on public key encryption using prime numbers.

The Mathematics of Cons – Pyramid Selling

Pyramid schemes are a very old con – but whilst illegal, still exist in various forms. Understanding the maths behind them therefore is a good way to avoid losing your savings!

The most basic version of the fraud starts with an individual making the following proposition, “pay me $1000 to join the club, all you then need to do is recruit 6 more people to the club (paying $1000 each) and you will have made a $5000 profit.”

There are lots of variations – and now that most people are aware of pyramid selling, now normally revolve around multi-level-marketing (MLM).  These are often still pyramid schemes, but encourage participants to believe it is a genuine business by actually having a sales product which members have to sell.  However the main focus of the business is still the same – taking money off people who then make their money back after having signed up a set number of new recruits.

The following graphic from Consumer Fraud Reporting is a clear mathematical demonstration why these frauds only end up enriching those at the top of the pyramid:

You can see that if the requirement was to recruit 8 new members, that by the 9th level you would need to have 1 billion people already signed up.  Even with the need to recruit just 4 new members you still have rapid exponential growth which very quickly means you will run out of new potential members.  For pyramid schemes it is only those in the first 3-4 levels (the white cells) that stand any real chance of making money  – and these levels are usually filled by those in on the scam.

Ponzi schemes (like that run by Bernie Madoff) use a similar method.  A conman takes money from investors promising (say) 10% annual returns.  Lots of investors sign up.  The conman then is able to use the lump sum investments to pay the 10% annual returns.  This scam can last for years, with people thinking that they are getting a good rate of return, only to find out eventually that actually their lump sum investment has gone.

This is a good topic to look at with graphs (plotting exponential growth), interest rates, or exponential sequences – and shows why understanding maths is an important financial skill.

If you like this topic you might also like:

Benford’s Law – Using Maths to Catch Fraudsters - the surprising mathematical law that helps catch criminals.

Amanda Knox and Bad Maths in Courts - when misunderstanding mathematics can have huge consequences .

Imagining the 4th Dimension

Imagining extra dimensions is a fantastic ToK topic – it is something which seems counter-intuitively false, something which we have no empirical evidence to support, and yet it is something which seems to fit the latest mathematical models on string theory (which requires 11 dimensions).  Mathematical models have consistently been shown to be accurate in describing reality, but when they predict a reality that is outside our realm of experience then what should we believe?  Our senses?  Our intuition?  Or the mathematical models?

Carl Sagan produced a great introduction to the idea of extra dimensions  based on the Flatland novel.  This imagines reality as experienced by two dimensional beings.

Mobius strips are a good gateway  into the weird world of topology – as they are 2D shapes with only 1 side.  There are some nice activities to do with Mobius strips – first take a pen and demonstrate that you can cover all of the strip without lifting the pen.  Next, cut along the middle of the strip and see the resulting shape.  Next start again with a new strip, but this time start cutting from nearer the edge (around 1/3 in).  In both cases have students predict what they think will happen.

Next we can move onto the Hypercube (or Tesseract).  We can see an Autograph demonstration of what the fourth dimensional cube looks like here.

The page allows you to model 1, then 2, then 3 dimensional traces – each time representing a higher dimensional cube.

It’s also possible to create a 3 dimensional representation of a Tesseract using cocktail sticks – you simply need to make 2 cubes, and then connect one vertex in each cube to the other as in the diagram below:

For a more involved discussion (it gets quite involved!) on imagining extra dimensions, this 10 minute cartoon takes us through how to imagine 10 dimensions.

It might also be worth touching on why mathematicians believe there might be 11 dimensions.  Michio Kaku has a short video (with transcript) here and Brian Greene also has a number of good videos on the subject.

All of which brings us onto empirical testing – if a mathematical theory can not be empirically tested then does it differ from a belief?  Well, interestingly this theory can be tested – by looking for potential violations to the gravitational inverse square law.

The current theory expects that the extra dimensions are themselves incredibly small – and as such we would only notice their effects on an incredibly small scale.  The inverse square law which governs gravitational attraction between 2 objects would be violated on the microscopic level if there were extra dimensions – as the gravitational force would “leak out” into these other dimensions.  Currently physicists are carrying out these tests – and as yet no violation of the inverse square law has been found, but such a discovery would be one of the greatest scientific discoveries in history.

Other topics with counter-intuitive arguments about reality based on mathematical models are Nick Bostrom’s Computer Simulation Hypothesis, the Hologram Universe Hypothesis and Everett’s Many Worlds quantum mechanics interpretation.  I will blog more on these soon!

If you enjoyed this topic you may also like:

Wolf Goat Cabbage Space – a problem solved by 3d geometry.

Graham’s Number – a number literally big enough to collapse your head into a black hole.

How Are Prime Numbers Distributed? Twin Primes Conjecture

Thanks to a great post on the Teaching Mathematics blog about getting students to conduct an open ended investigation on consecutive numbers, I tried this with my year 10s – with some really interesting results. My favourites were these conjectures:

1) In a set of any 10 consecutive numbers, there will be no more than 5 primes. (And the only set of 5 primes is 2,3,5,7,11)
2) There is only 1 example of 3 consecutive odd numbers all being primes – 3,5,7

(You can prove both in a relatively straightforward manner by considering that a span of 3 consecutive odd numbers will always contain a multiple of 3)

Twin Prime Conjecture

These are particularly interesting because the study of the distribution of prime numbers is very much a live mathematical topic that mathematicians still work on today. Indeed studying the distribution of primes and trying to prove the twin prime conjecture are important areas of research in number theory.

The twin prime conjecture is one of those nice mathematical problems (like Fermat’s Last Theorem) which is very easy to understand and explain:

It is conjectured that there are infinitely many twin primes – ie. pairs of prime numbers which are 2 away from each other. For example 3 and 5 are twin primes, as are 11 and 13. Whilst it is easy to state the problem it is very difficult to prove.

However, this year there has been a major breakthrough in the quest to answer this problem. Chinese mathematician Yitang Zhang has proved that there are infinitely many prime pairs with gap N for some N less than 70,000,000.

This may at first glance not seem very impressive – after all to prove the conjecture we need to prove there are infinitely many prime pairs with gap N = 2. 70,000,000 is a long way away! Nevertheless this mathematical method gives a building block for other mathematicians to tighten this bound. Already that bound has been reduced to N <60,744 and is being reduced almost daily.

Prime Number Distribution

Associated with research into twin primes is also a desire to understand the distribution of prime numbers. Wolfram have a nice demonstration showing the cumulative distribution of prime numbers (x axis shows total integers x100)

Indeed, if you choose at random an integer from the first N numbers, the probability that it is prime is approximately given by 1/ln(N).

We can see other patterns by looking at prime arrays:

This array is for the first 100 integers – counting from top left to right.  Each black square represents a prime number.  The array below shows the first 5000 integers.  We can see that prime numbers start to “thin out” as the numbers get larger.

The desire to understand the distribution of the prime numbers is intimately tied up with the Riemann Hypothesis – which is one of the million dollar maths problems.  Despite being conjectured by Bernhard Riemann over 150 years ago it has still to be proven and so remains one of the most important unanswered questions in pure mathematics.

For more reading on twin primes and Yitang Zhang’s discovery, there is a great (and detailed) article in Wired on this topic.

If you enjoyed this topic, you may also like:

A post on synesthesia about how some people see colours in their numbers.

A discussion about the Million Dollar Maths problems (which includes the Riemann Hypothesis).

Sierpinski Triangles and Spirolateral Investigation Lesson Plan

Leaning Objective:  Students are introduced to some more complex ideas in mathematics (fractals, infinite perimeter, fractional dimensions), students explore the relationship between maths and art, students conduct an open ended invesigation into patterns and sequences. 

10 minutes

Start the lesson with the Mandelbrot Zoom in the background:

Discussion about what this shows (fractal shapes which repeat infinitely).  Discuss the coastline paradox – what is the perimeter of the coastline of the UK?  Does it have one?  What happens when we try and be more accurate with our measurements? 

Show video introducing how to create fractals – looking at how to create Sierpinski Triangles and how the Koch snowflake has simultaneously a finite area and an infinite perimeter:

15 minutes

Students create their own Sierpinski Triangle, which can be generated quite easily (instructions here).  Students need triangular paper which can also be printed from the link. 

5 minutes

Introduction to Spirolaterals – these patterns were first discovered by investigations into fractal designs.  Use the computer generator here to show how these shapes are generated.  (Teacher notes on this investigation here).

40 minutes

Distribute the Spirolateral worksheet (download here).  Students need squared paper and start to investigate different patterns and rules.  Which initial starting rules lead to closed patterns?  Which ones lead to infinitely repeating pattern? For an extension students could investigate turns of 45 degrees rather than 90 degrees. 

If you enjoyed this lesson plan, you might also like:

Lesson Plan on Code Making and Code Breaking

Lesson Plan on Modelling Asteroid Impacts - a modelling lesson which demonstrates the power of mathematics in making real world predictions.

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.

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!

Utility Value – How Maths Can Make You Happier

The use of utility curves to make optimal decisions is something which is never really touched on in mathematics – even though they are a powerful tool for making good choices in life.   “Utility” is used to represent personal benefit – and in this graph, maximum utility value has been set as 1.

For these types of concave utility graphs you have a rapid increase in utility value initially (as spending £2000 on a car brings a large increase in utility value than not having a car), but then it quite quickly levels off (as there is little added utility value in spending £22,000 rather than £20,000).

Obviously everyone’s personal utility curve will be different – perhaps someone who spends several hours every day in their car will really value that extra comfort and luxury that spending a few thousand pounds brings – and so their own curve will flatten out more slowly.  Nevertheless they will always demonstrate the law of diminishing returns – where we ultimately end up spending more and more for ever reduced benefits.

The graph above is generated by the function f(x) = 1 + 2/(x+2).  By adding a slider on Geogebra in the form f(x) = 1 + a/(x+a) you can amend the utility curve to have different levels of steepness – so students can easily generate their own utility curves.

The Khan Academy video on Marginal Utility is a really nice example of how we can use these calculations to optimise our happiness.  In the video they discuss how best to spend $5 when faced with the choice of either fruit or chocolate.

Staying with cars, another factor to take into account is depreciation:

This graph is taken from What Car  which allows you to enter any make and model of a car and see how its cost depreciates with time.  In this particular case (with a Ford Focus), the price depreciates from £17,405 to £9,829 in just 12 months – that’s a stunning 44% decrease in value in one year.  Or to think of it another way, you’ve lost £630 every single month.

So it is clear that buying a new car will lead to a massive yearly loss in your investment.  The only strategies that make sense therefore are either buying a new car (with the peace of mind of long term warranties) and driving it for the next decade (thus averaging out the initial large loss in value), or buying a car that is already a year old – and avoiding the sharp depreciation completely.  The absolute worst thing you can do is buy a new car and after 3-4 years sell it to buy another one – this really is just throwing money away!

So, understanding some simple mathematical concepts can help us make better decisions when it comes to investments and how we spend our money – and by helping to maximise our utility also has the potential to make us happier individuals.

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Maths IA – Exploration Topics: 200 ideas for investigations.

The authors of the latest Pearson Mathematics SL and HL books – (which look really good) have come up with 200 ideas for students doing their maths explorations.  These topics touch a really large cross-section of mathematics.  A lot of these ideas would need the students to go away and research independently to find out about them – which in itself would be a nice activity.  There’s also a lot of potential for integrating some of this content into maths lessons and ToK discussions.  I’ve posted the full list here, and included some links and additional information about some of the topics – so visit the full page for more details.   Here are just an edited selection below:

Algebra and number

1) Modular arithmetic
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
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) Odd 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.

Geometry

1) 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.
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.

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.

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
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.

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) Poker and other card games
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

Topology and networks

1) Knots
2) Steiner problem
3) Chinese postman problem
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
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

If these are not enough have a look at the full 200 topics here.

GCHQ – the British cyber spy agency – have had a rough few months following some staggering revelations from Edward Snowden, so they’re doing some positive PR at the moment to highlight the importance of mathematics and computing skills in code-breaking.  There are 4 codes to solve (the first one posted above) – each answer leading on an internet treasure-hunt to the next clue.  Those who can solve all 4 clues stand a chance of winning a Google Nexus and Raspberry Pi – and possibly could lead to a job opportunity with GCHQ.

The competition started two days ago (10th September) – and there is a six week deadline to solve all clues.  So, get cracking!

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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 = 02⁵ = 32 (mod 35). C = 14⁵ = 537824 which is equivalent to 14 (mod 35). C = 03⁵ = 33 (mod 35).  C = 04⁵ = 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 = 11³ = 08 (mod 55).  y = 49³ = 04 (mod 55). y = 41³ = 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.

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Western music has its roots in the harmonics discovered by Pythagoras – himself a keen musician – over 2000 years ago.  Pythagoras noticed that certain string ratios would produce sounds that were in harmony with each other.   The simplest example is illustrated above with an electric guitar.  When a string is played, and then that same string pressed half-way along its length (in the guitar’s case the 12th fret), then we get the same note – this is a whole octave.

If you were to then half the distance again you would get another octave (which explains why guitar frets get smaller and smaller near the base of the instrument – the frets mark ratios relative to the whole string).

The ratio 1: 1/2 shows the ratio of an octave.  A full length string: half length string.  We can multiply both sides by 2 to remove the fraction to get, 2: 1.  This is the octave ratio.

All the other harmonies that are the basis of Western music can also be understood through similar ratios.  The chord sequence E, A, B – which is the standard progression for blues and modern music comprises of the base note (in this case E), along with the perfect fourth (A) and the major fifth (B) of the base note.

Looking at the guitar fret we can see that the perfect fourth (A), which occurs on the fifth fret, has the ratio 1: 3/4.  That is 1 whole string: 3/4 of the whole string.  We can simplify this to get 4:3.

The major fifth (B) which occurs on the seventh fret has the ratio 1: 2/3 which simplifies to 3:2.

The other most likely note used in the key of E would then also be either G (the minor third) which has a ratio of 6:5, or G sharp (the major third) which has a ratio of 5:4).

It’s interesting that we find these particular whole number ratios pleasing to listen to – indeed these are the notes that often sound “right” when playing through songs.  It’s also helpful to look at the circle of fifths – which shows all notes which are in the ratio 3:2 with each other.  Moving around the circle again produces music which sounds nice.  For an example of this, starting at C, the progression C,G,D,A,E is the one used by Jimmy Hendrix in the classic song, Hey Joe

There are lots of other areas to explore when looking at the relationship between maths and music – one of which is looking at how we can model the wave frequencies of notes using modified sine/cosine curves.  The IB have included a piece of coursework on this as an example for the new exploration topics.

Another interesting exploration is looking at the strange properties of the Harmonic Sequence – which is the sequence 1, 1/2, 1/3, 1/4… This sequence like many of those found in music is said to be in harmonic progression .  There are some interesting paradoxes related to the harmonic sequence – and a variety of methods of proving that the sum of this sequence (the series) actually diverges to infinity – even though you would intuitively expect it to converge.  The video below provides a taster on this topic:

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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.

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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 4×10¹⁷ even numbers  (4×10¹⁷ is a number so mind bogglingly big it would take about 45 trillion years to write out, writing 1 digit every second).  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?

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War Maths – Projectile Motion

Despite maths having a reputation for being a somewhat bookish subject, it is also an integral part of the seamier side of human nature and has been used by armies to give their side an advantage in wars throughout the ages.  Military officers all need to have a firm grasp of kinematics and projectile motion – so let’s look at some War Maths.

Cannons have been around since the 1200s – and these superseded other siege weapon projectiles such as catapults which fired large rocks and burning tar into walled cities.  Mankind has been finding ever more ingenious ways of firing projectiles for the best part of two thousand years.

The motion of a cannon ball can be modeled as long as we know the initial velocity and angle of elevation. If the initial velocity is V_i and the angle of elevation is θ, then we can split this into vector components in the x and y direction:

V_xi = V_icosθ (V_xi is the horizontal component of the initial velocity V_i)
V_yi = V_isinθ  (V_yi is the vertical component of the initial velocity V_i)

Next we know that gravity will affect the motion of the cannonball in the y direction only – and that gravity can be incorporated using g (around 9.8 m/s² ) which gives gravitational acceleration.  Therefore we can create 2 equations giving the changing velocity in both the x direction (V_x) and y direction (V_y):

V_x = V_icosθ
V_y = V_isinθ – gt

To now find the distance traveled we use our knowledge from kinematics – ie. that when we integrate velocity we get distance.  Therefore we integrate both equations with respect to time:

S_x = x =  (V_icosθ)t
S_y = y =  (V_isinθ)t – 0.5gt²

We now have all the information needed to calculate cannon ball projectile questions.  For example if a cannon aims at an angle of 60 degrees with an initial velocity of 100 m/s, how far will the cannon ball travel?

Step (1) We find out when the cannon ball reaches maximum height:

V_y = V_isinθ – gt = 0
100sin60 – 9.8(t) = 0
t ≈ 8.83 seconds

Step (2) We now use the fact that a parabola is symmetric around the maximum – so that after 2(8.83) ≈ 17.7 seconds it will hit the ground.  Therefore substitute 17.7 seconds into the equation for S_x = (V_icosθ)t.

S_x =  (V_icosθ)t
S_x =  (100cos60).17.7
S_x  ≈ 885 metres

So the range of the cannon ball is just under 1km.  You can use this JAVA app to model the motion of cannon balls under different initial conditions and also factor in air resistance.

There are lots of other uses of projectile motion – the game Angry Birds is based on the same projectile principles as shooting a cannon, as is stunt racing – such as Evel Knieval’s legendary motorbike jumps:

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The Birthday Problem

One version of the birthday problem is as follows:

How many people need to be in a room such that there is a greater than 50% chance that 2 people share the same birthday.

This is an interesting question as it shows that probabilities are often counter-intuitive. The answer is that you only need 23 people before you have a 50% chance that 2 of them share a birthday. So, why do you only need 23 people?

The key to understanding this question is realising that when comparing if any of the (n) people in the room share a birthday, you are not simply making n comparisons – but C(n,2) (n choose 2) comparisons. If there are people A, B, C and D in a room I don’t just make 4 comparisons,  I have to compare AB, AC, AD, BC, BD, CD. This is the same calculation as working out 4 choose 2 = 6 comparisons.

Therefore when there are 23 people in the room you actually need to make C(23,2) comparisons = 253. This goes someway to explaining why the number is much lower than you would expect – but it still doesn’t tell us where the number 23 came from.

If we simplify things so that we don’t have leap years then we can approximate the problem by working out the probability p(no shared birthday). When there are 2 people in the room the probability that person B does not share his birthday with person A is 364/365. When a third person enters the room the probability that C doesn’t share his birthday with A or B is 363/365. Carrying on in this manner, when the 23rd person enters the room, the probability that he doesn’t share a birthday with anyone already there is 343/365.

We then work out p(no shared birthday) = 364/365 x 363/365…x 343/365 = 0.4927
So p(shared birthday) = 1 – 0.49 = 0.51 (2 dp).  Therefore when there are 23 people in the room the probability of a shared birthday is around 51%.

However, the method outlined above is a little unsatisfactory – as we start by using the fact that we know the answer is 23 and then work backwards.  So how can we discover the result independently? One possibility is to use the Poisson approximation, P(λ):

(where X is the number of shared birthdays).  If we want the probability of at least 1 shared birthday then we can find 1 – P(X=0).  When k = 0 the formula reduces to e^-λ .  Therefore P(X>0) = 1 – e^-λ

So, if we want the probability to be greater than 0.5 we can then set up an inequality and solve:

1 – e^-λ  > 0.5
0.5 > e^-λ
ln(0.5) > – λ

Next we use the fact that λ is the expected value – and so will be given by C(n,2)/365.  This is because the number of potential birthday combinations divided by 365 will give us the mean number of shared birthdays.  We also use the formula for C(n,r) which is n!/(k!(n-k)!).  This gives C(n,2) = n(n-1)/2

-ln(0.5) < C(n,2)/365
253 < C(n,2)
253 < n(n-1)/2
0 < n² – n – 506

Which we can solve using the quadratic formula to give n = 23 almost exactly.

You can also extend the problem by looking at the probabilities at least 3 people have the same birthday. Wolfram Alpha have an online generator to allow you to do this.

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