The Museum Lates happen three times a year and are a chance for adults to get into the museum after hours to look at the collections whilst enjoying a cocktail and some live music. But they are much more than that! Each Late is themed in some way, and the theme this month was ‘Vikings’ to accompany the museum’s special exhibit. So guests were encouraged to dress up as Vikings, get their faces painted, touch Viking objects, make Viking paraphernalia (but NOT horned helmets – they aren’t Viking!), listen to Viking stories and eat Viking food. It was the perfect opportunity for us to get in some stealth maths engagement…

(From left) Joshua, Helene and Madeleine all Vikinged up and ready to go.

My co-conspirator in this project was Madeleine Shepherd (who you may remember from lots of previous projects, including the last Alice In Wonderland themed museum late and also the Mathematicians’ Shirts project), and on the night we had two lovely assistants: Helene Frossling (Madeleine’s colleague at ICMS) and Joshua Prettyman (an undergraduate on my maths outreach team). Together we presented the public with…The Valknut Challenge!

In Viking mythology, the Valknut was the special symbol of Odin, king of the gods. It consisted of three interlocking circles, but if you removed any one of the circles then the other two would fall apart and would not be linked together at all. The Valknut Challenge is: can you draw/make the symbol? If you haven’t seen this before, you should have a go before scrolling down to see the picture.

The Valknut symbol is often found on Viking stones where Odin is depicted going into battle or standing over fallen warriors. It’s not hard to see why the symbol would be synonymous with strength in battle: the three interlocking circles symbolise the strength we have in acting together which falls apart when we go it alone. The same symbol has been found in many civilisations – for example, signifying the Trinity in Christianity.

The Stora Hammars stone showing a Valknut above a scene of human sacrifice and next to some warriors. Powerful stuff.

Mathematically the symbol is called the Borromean rings (named for the Italian Borromeo family who used the symbol on their coat of arms). There are actually infinitely many different ways of solving the puzzle, and the Valknut is a special case of a more general puzzle where you have n circles and have to interlink them so that removing one ring makes the rest fall apart. Such a link is called a Brunnian link and they are particularly cool topological objects.

The standard Borromean rings.

A different solution to the Valknut challenge.

And another solution!

Most people have created a Valknut/Brunnian link in the course of their lives without ever realising it. Every knot or link can be drawn as a braid, and the Valknut is actually the standard 3-stranded braid that girls do in their hair all the time. Try making one and pulling out a strand – you’ll find that the other two strands become instantly untangled. This means you can quickly make a braid and harness the power of Odin whenever you need it – handy to remember next time you find yourself in battle.

The next Museum Late will be on 17th May with the theme of ‘Dinosaurs’.  If you missed out this time around then make sure you get tickets early for the next one! Anyone have any ideas for some surreptitious Jurassic mathematics that we can do?

I thought I would go through the numbers and say what they mean and why we chose them. All physical constants are in standard SI units.

1. ℏ × 10³⁴ = 1.05 is the reduced Planck constant; that is, the Planck constant divided by 2π. This number relates the energy of a photon to its angular velocity and is one of the basic units in quantum mechanics. Its existence means that energy only comes in discrete packets, or quanta, which was a massive discovery in 20th century physics.
2. is the sum which turns out to be exactly equal to 2. It’s an example of a geometric series and was first discussed in ancient Greece when Zeno came up with his paradoxes. He claimed that motion was impossible, because before you could get somewhere you’d have to get halfway there, but then you’d have to get halfway to the halfway point and so on. However, despite the infinite number of steps in the sequence it actually has a finite value.
3. c × 10^-8 =2.9979 is the speed of light in a vacuum, measured in metres per second. The theory of relativity says that this is constant, regardless of the observer, and that nothing in the universe can go faster than this. The fact that c is constant leads to lots of crazy consequences, such as time passing slower for moving objects and moving objects shortening in length.
4. is the integral (area under the curve) of the natural logarithm function between and . Logarithms were invented by the Scottish mathematician John Napier and are important in many branches of maths and science. The natural logarithm is related to the mathematical constant e, which arises whenever a quantity is growing at a rate proportional to its current value; for example, compound interest.
5. det(12n₄₈₈) is the determinant of the 488th non-alternating 12-crossing knot. This was a knot which featured in my thesis, as I used new techniques to show that it had infinite order in the knot concordance group. Finding the determinant was the first step in this method. You can find tables of knotty things on the website KnotInfo, including any values which are still unknown.
6. 29iⁱ = 6.03 shows off the surprising fact that an imaginary number (the square root of -1) to the power of an imaginary number is actually a real number. Actually, iⁱ  is equal to .
7. is the golden ratio squared and then squared again. The golden ratio arises if you want to divide a line into two parts so that the ratio of the larger to the smaller part is the same as the ratio of the larger part to the whole line. It is often described as the most aesthetically pleasing ratio, and is also the ‘most irrational’ number in the sense of being most badly approximated by rational numbers.
8. N_A.k_B = 8.314 is the ideal gas constant which relates the energy of a mole of particles to its temperature. It is the product of N_A, which is Avogadro’s constant (the number of particles in one mole of a substance – see 12), and k_B, which is the Boltzmann constant (which relates energy of individual particles to temperature).
9. m_e × 10³¹ = 9.109 is the rest mass of an electron which is one of the fundamental constants of physics and chemistry. It is actually impossible to weigh a stationary electron, but we can use special relativity to correct our measurements of the mass of a moving electron.
10. g = 9.81 is the acceleration of objects in a vacuum at sea level due to Earth’s gravity. This is actually an average value, as the force of gravity varies at different points on the Earth’s surface. Things affecting the apparent or actual strength of Earth’s gravity include latitude (since the Earth is not a perfect sphere) and the local geology (since the Earth does not have uniform density). The force is weakest in Mexico City and strongest in Oslo.
11. = 11.23 is a combination of everyone’s favourite mathematical constants, pi and e. These numbers are not only irrational but are transcendental, meaning that they are not the roots of any integer polynomial equation. It is still unknown if is transcendental, although we know that is.
12. ¹²C = 12 is the isotope of carbon having six protons and six neutrons in its nucleus, giving it a mass of 12 atomic mass units. More importantly for chemists, IUPAC defines the mole  as being the amount of a substance containing the same number of  atoms (or molecules, electron, ions etc) as there are the number of atoms in 12g of carbon-12 (which is Avogadro’s number – see 8).

Which one was your favourite number? If you were designing your own geek clock, what numbers would you pick? Leave your answers in the comments!

This was the verdict delivered yesterday by a group of Dungeons & Dragons fans who had come to ICMS for Doors Open Day, after being treated to some maths busking by me. I also think they went away convinced that I was a geomancer instead of a geometer – I really must work on my enunciation…

Spatulamancy: the art of using a humble spatula to predict the future?

[An interesting aside, geomancy is apparently one of the seven "forbidden arts," along with necromancy, hydromancy, aeromancy, pyromancy, chiromancy (palmistry), and spatulamancy. Ah, I love Wikipedia.]

It’s been a stressful week for me, but culminated in a totally wonderful day of maths communication yesterday. In the morning I gave the first Edinburgh masterclass of the season to a group of 82 enthusiastic 13-year-olds, and some equally enthusiastic student helpers. When I commiserated with them on having to get up early on a Saturday morning, the response was “We’d always get up early for lectures if they were as interesting as this!”. Which is lovely and flattering for me, but really makes me sad that we aren’t doing enough in university to bring our subject alive. Of course not every lecture can be as fun as a masterclass, but there are far too many researchers for whom lecturing is a chore and who never make an effort to bring enthusiasm or interest to their subject.

I digress, but there was an interesting blog post on a related theme by Peter Rowlett this week. He asked whether it was possible to pursue a career in university teaching and lecturing whilst not being a researcher – a question I have full sympathy with as someone in exactly that position. For me the story has a happy ending: after a year and a half of trying to persuade the university that a full time outreach/teaching position was a Good Thing, I have finally got my contract extended to 3 years. It is great to know that the department and university value the things I do, but I would despair of being able to find a similar position were I ever to change universities. While good teaching and public engagement are listed as promotion criteria in many places, in practice they are rarely rewarded when compared with research output.

Another side of the story is that there are many people who do public engagement in their spare time who are not recognised for it. A job title such as mine (Mathematics Engagement Officer) can count for a lot, as my friend and collaborator Madeleine Shepherd has found many times. Although we’ve worked on many projects together, with her often the brains behind the ideas, emails proposing new engagement opportunities are often sent to me and rarely to her.

It was wonderful to see ICMS, where Madeleine works, being open to the public yesterday for Doors Open Day. The building, on South College Street, is a converted church and still has an original stained glass window, among other interesting features.

Doors Open Day at ICMS, featuring Penrose tiles, chaotic pendulum and magnets, Tantrix, and me busking to three D&D fans. Click photo for more ICMS images.

This was the first year it had opened as part of Doors Open Day and we had no idea how many visitors would turn up. In the end I think the count was at 229, most of whom were lured in by the promise of maths puzzles rather than an interest in the building itself. I was only able to attend in the afternoon (due to the masterclass in the morning) and had a huge amount of fun showing people my favourite topological tricks, card tricks and mathematical puzzles. Even those of the public who proclaimed they were bad at maths went away enthused by what they had learnt and wanting to share their new knowledge with friends and family. I hope that we can run such events more frequently instead of waiting for Doors Open Day every year!

This hope is not a forlorn one, as I have big plans brewing… I am currently recruiting undergraduates and postgraduates to be on my new Maths Outreach Team (with unfortunate acronym MOT), and hope to have a team of 10 people trained up and ready to engage by the middle of October. Once they are unleashed on the unsuspecting city of Edinburgh, there will be no end to the school workshops, festival exhibitions, website articles and puzzles, public lectures and impromptu maths busking. At least, that is the plan. If you know of any maths undergrads who would be interested in this, please spread the word!

On that note, it is time for me to head off and hatch more nefarious outreach plans. Please do leave a comment if you were at Doors Open Day, my masterclass, or if you have comments on the difficulties of being rewarded for good outreach and lecturing. Until next time…

Here I am at one of the beamlines at Petra synchrotron, at DESY, Hamburg. The tube behind me is where the beam comes from… scary!

Albert here! Some of you may recognise me from Haggis’ Twitter feed and from Haggis’ 2011 New Year’s post (along with the rest of our family!). Last week I was in Hamburg at PETRA III, a synchrotron at DESY. After some successful measurements there, I made the short hop across the Baltic Sea to the lovely city of Stockholm, for the 4^th International School on Crystal Topology.

First I should say a little about what I do. I’m interested in chemistry, especially materials called Metal-Organic Frameworks (MOFs).

An example of one of the first MOFs, MOF-5. Chemists use rigid organic struts (top left) to link clusters of metal atoms (in this case four zinc atoms, bottom left) to build open framework-like materials (right).

These are a new type of material made from clusters of metal and oxygen atoms which are linked together by long rigid linkers – think of it kind of like a climbing frame. These materials are interesting as they might help to combat climate change by sieving out CO₂ in a process called Carbon Capture and Storage (CCS made it into the Oxford English Dictionary recently!).

But what does this have to do with topology? Chemists simplify the structures of MOFs down to a series of rods (edges) and nodes where these rods meet (vertices) – the simplified structures are mathematical graphs. We can then see how the structure is connected together as a network, without unnecessary molecular clutter. As chemists we want a way to classify the networks of our materials for two reasons. Firstly, so we can see if similar networks have been made before by other researchers, and secondly to help us design new materials. We might, for example, find that a certain network is really good at storing CO₂; using a linker molecule which holds onto CO₂ really well and the right topology to form our target network, we could make a new material which is even better at capturing CO₂. To classify our networks we need to use graph theory.

Charlotte Bonneau (left), Michael O’Keeffe (middle left), the person I hitched a lift to Stockholm with (middle right), Xiaodong Zou (right)

However chemists are not normally trained in graph theory, so this was the aim of the Stockholm school. The school was taught by Prof. Michael O’Keeffe (emeritus Regents’ Professor at Arizona State University), who taught us about the mathematical ideas necessary to deconstruct a crystalline network, and Dr Charlotte Bonneau (currently a full time mother to the adorable Leonie), who focussed more on the use of software to analyse crystal structures, such as systre and Topos.

During Mike’s lectures we were told about the graph isomorphism problem of determining whether two finite graphs have the same connectivity. This is of importance to chemists, as we want to be able to compare our networks to see if they have been reported before! Graph isomorphism is also a specific example of one of the million dollar maths problems, P versus NP, which asks whether every problem for which a solution can be quickly checked, may also be quickly solved by a computer. One of Mike’s collaborators, Dr Olaf Delgado-Friedrichs, has attempted to address the graph isomorphism problem in the program systre. systre uses a barycentric method to raise the symmetry of a collection of atoms in a graph to the highest symmetry representation. The barycentric representation is effectively like replacing all the edges in the graph with springs and these pulling the vertices to their weighted average positions. Although systre is able to classify most graphs, it is unable to deal with graphs where applying the barycentric approach causes two nodes to collapse into one another (a so-called collision – see picture). So unfortunately, it’s not a complete solution to P versus NP.

A graph showing a collision. When you put this into a baricentric representation, the two red nodes collapse into one another. Back to the drawing board for a solution to the graph isomorphism problem then…

The rest of the course was full of lots of useful information which will help in making new materials and further classifying old ones. The course as a whole was a lot of fun and it was great to meet such a friendly bunch of people! That’s it from me for the minute, but look out for more photos of me on Twitter at exciting scientific/mathematical locations – Albert out.

The theme for this month’s Museum Late was “A Night in Wonderland”, so there were lots of top hats, white rabbits and red queens! (See lots of photos of the event on the Museum’s Flickr page.) Knowing that Lewis Carroll (real name Charles Dodgson) was a mathematician and logician as well as nonsense-poem writer, it seemed wrong for there not to be a mathematical component to the evening so I got together with Madeleine Shepherd (from ICMS) to brainstorm some ideas…

Our first idea was to get the public to make some Fortunatus’ purses. A Fortunatus’ purse appears in the novel Sylvie and Bruno by Lewis Carroll and is based on the old tale of Fortunatus, who has a purse which replenishes itself with money as often as coins are drawn from it. If you read the book you’ll find instructions for making such a purse by sewing together the edges of 3 handkerchiefs in an unexpected way.

The mathematical object created is one which has no inside or outside – it is called non-orientable, and is (of course) not possible to make in 3 dimensions without part of the purse intersecting itself. Some of you may be thinking that this is a Klein Bottle, but it is actually a different creature called a Projective Plane.

However, whilst doing the practice run for the purse-making, we found that it took quite a long time, was fairly fiddly and would involve giving drunk people sharp needles. Probably not the best idea. (But we might do this in a future maths/craft event!)

So instead we came up with the “Snark Constellation Challenge”, inspired in equal parts by the Lewis Carroll poem The Hunting of the Snark and by a mathematical object in graph theory called a snark. Visitors were invited to play a game which involved colouring the lines between stars in a constellation, and were challenged to colour the lines using only 3 colours.

Can you colour the lines with 3 colours so that at each star, 3 different colours meet?

There were two games the visitors could play: working collaboratively to find a colouring of all the lines, or working competitively to be the last person to draw a valid line. Have a go at the puzzle and see if you can colour the lines before reading on!

Hopefully you had a go at colouring in the picture and didn’t get too frustrated when  you didn’t succeed. The reason you failed is because the picture can’t be coloured with only 3 colours: it is a special type of graph called a snark, whose defining feature is exactly that it can’t be coloured in less than 4 colours.

Snarks were invented back in 1880 by the Scottish mathematician P.G.Tait when he was trying to prove the more famous problem, the Four Colour Theorem. This states that any map can be coloured with at most 4 colours so that no two adjacent regions share the same colour. It was a problem that remained unsolved until the 1970s, and even now remains controversial as we have never found a proof that does not involve using a computer to check cases. Tait’s idea was to change the question about maps into a question about graphs, and showed that the Four Colour Problem was equivalent to showing that no snark was planar; i.e. that no snark could be drawn on paper without at least two lines crossing over each other.

You’d probably be surprised to hear that the first snark was not discovered until 1898 – nearly 20 years after the definition was made! The name of ‘snark’ (made by Martin Gardner many years later) was given because in Lewis Carroll’s poem, the snark is an elusive creature which disappears as soon as anyone sees it. It seemed a fitting description of these strange mathematical pictures which were extremely hard to discover! Even today there aren’t so many known snarks. You can see most of them on this page. And there are still open questions about snarks waiting to be solved, such as the Cycle Double Cover Conjecture.

Going back to our museum event, we wanted our visitors to have some artistic fun as well as learning deep and exciting maths, so we set up a table with a “Snark Hunters’ note book” and invited guests to draw their impressions of what a snark might look like.

Madeleine looking after the beaver making lace and Snark Hunters’ note book.

One artist’s idea of what a snark might look like.

It was a really great evening – we had a lot of fun and met lots of interesting characters! I shall leave you with just one more photo of our intrepid snark hunters. Contact us if you ever need help with your own snark hunting!

Snark hunter helpers Hanyi Chen (left), Julia Collins (middle), Madeleine Shepherd (right) and Jakub Marecek (behind the camera!).

So, here are the answers!

The Monty Hall problem is always a good excuse for a picture of a goat.

1) The Monty Hall Problem: 3 doors; 2 hiding goats and 1 hiding a car. You pick a door, then the gameshow host shows you another door hiding a goat. He asks you if you want to swap to the other remaining door, or stick with the one you first picked. The vast majority of the public, of whatever age, said that there was no difference between sticking or swapping, and most preferred to stick. (People seem to find it psychologically more distressing to lose a prize they had briefly owned than to not win a prize they could potentially have had.)

In actual fact, they would have been twice as likely to win the car by swapping doors than by sticking with their original choice. How can this be? After the host opens a door to reveal a goat, there are two doors left, so surely the probability is 50-50? The reason this is not true is because the host’s choice of door depends on the player’s initial choice of door. If the player chooses a door hiding a goat the first time (which happens 2 times out of 3) then the host has no choice but to reveal the other goat-hiding door, meaning that he definitely leaves the car behind the other door. To repeat, 2 times out of 3, the car has to be behind the other door. In other words, the player only wins the goat by sticking if they pick correctly the first time, which happens 1 in 3 times.

2) Heads or Tails: I flip two coins and tell you that at least one of them is a head. What is the chance that the other coin is also a head? Again, intuition tells most people that it should be 50-50. How can the first coin influence the second? And yet the knowledge that at least one coin is a head doubles the chance of the other coin being a tail! Consider the different possible combinations of two coin tosses: Tail/Tail, Tail/Head, Head/Tail or Head/Head. The knowledge that at least one coin is a head eliminates the first possibility, leaving 3 remaining, of which two give the other coin as being a tail.

What’s the chance of correctly choosing 4 numbers from 10?

3) Lottery: what is the chance of winning the jackpot in a lottery where 4 numbers are chosen from 10? Younger visitors to the museum tended to vastly overestimate the probability, quoting odds of about 1 in 4 or 1 in 5. Older visitors tended to be extremely skeptical, thinking the odds were more like 1 in a million! The actual odds were somewhere in between, being about 1 in 200. Here’s how to work it out.

The first ball chosen has a 4 in 10 chance of matching with one of the numbers you’ve picked. That’s because there are 10 numbers in total and it could match any one of your 4 chosen numbers. Now we want the next ball to match. There are 9 numbers left to choose from, but the chosen ball can match with any 3 of your remaining numbers, so that’s a 3 in 9 chance. Continuing similarly, we get a 2 in 8 chance for the third ball to match and a 1 in 7 chance for the final fourth ball to match.  To get the final probability, multiply all the chances together:

[As an aside, I analysed the lottery tickets of 480 people who played our lottery and found that 0 was by far the least popular number, followed (surprisingly) by 3. The number 2 was (marginally) the most popular.]

4) The coin flip: if 100 people flip a fair coin 5 times each, how many of them would we expect to flip 5 heads? There is a probability of of the coin being a head, and we need this to happen 5 times in a row. Each flip is independent of the last, so the calculation is . So we expect 3 people out of 100 to flip 5 heads. Small children are especially incredulous that such an event could ever happen, claiming that anyone who flipped 5 heads in a row had to be cheating.

Motto: Even unlikely events become probable if the experiment is repeated enough times. There may be a 1 in a million chance of a piece of DNA matching, but this means we expect 60 people in the UK to be potential suspects of a crime. Just because there is a small chance of finding a piece of evidence (such as a DNA sample), this does not mean there is a correspondingly small chance that the suspect is innocent. This argument is known as the prosecutor’s fallacy and I discussed an example of it in my last blog post.

5) The Birthday Problem: How many people need to be in the room for it to become likely that there is a shared birthday? The most common reply to this is about 180 – simply by

If I had a birthday this would be my cake.

dividing the total number of possible birthdays (365) by 2. People are always shocked to be told that the answer is only 23. How is it possible that only 23 people could produce a shared birthday when there are so many possible birth dates? The key to thinking about this is to remember that it could be a shared birthday between any two people, not between any two particular people. As the number of people in the room increases, the number of possible pairings gets exponentially big and the corresponding chance that nobody shares a birthday becomes very low. (For details of the calculations, see the Wikipedia page.) This mathematical result was well demonstrated on the first few days in the museum, when we had our first match at the 17th person (3 days in a row!). The next couple of days we had to wait until about the 30th person, but this was still well within most people’s expectations. And by 100 people we always had a match of 3 people on the same day!

The calculation relies on birthdays being equally distributed throughout the year, which in fact they are not, so the matches become even more likely. A recent article documented the frequency of birthdays on each day of the year, finding that August – October was the most popular time to have a baby. Lots of cuddling up in the winter months it seems!

6) The eyewitness: if an eyewitness to a crime says she saw a person wearing a hat, and if she has a 2/3 chance of correctly spotting hats, and if only 10% of the population wear hats, what is the chance that the suspect was actually wearing a hat? This is a difficult question to get across so we did this by having 30 Playmobil figures of which 3 were wearing hats. The visitors then had to sort the characters into the different categories:

• Wearing a hat and correctly identified: there are (2/3) x 3 = 2 people in this category;
• Wearing a hat and incorrectly identified: there is (1/3) x 3 = 1 person in this category;
• Not wearing a hat and correctly identified: there are (2/3) x 27 = 18 people in this category;
• Not wearing a hat and incorrectly identified: there are (1/3) x 27) = 9 people in this category.

In total then, there are 2+ 9 = 11 people who were thought to be wearing hats, and only 2 of them were actually wearing  a hat. So the chance of the suspect wearing a hat is only 22% – much lower than the 66% most people expect!

Motto: If a situation is particularly rare, then even a test for it which is quite accurate will produce more false positives than true positives. This is a common fallacy in both law and medicine and is one of the reasons we need to have better teaching of Bayesian statistics in these professions.

How did you do on these questions? Did you fall into the common fallacies? Have you heard of them before? Do you disagree with any of the answers? Leave me your comments!

Today’s post is about our science festival fun. We (the School of Maths) teamed up with the School of Chemistry and went for a CSI-themed activity.The premise was that a priceless Egyptian vase had been stolen from the Museum and the visitors had to work out whodunnit. Using chemistry they had to analyse fingerprints and blood samples, and use UV and infrared data to identify substances found at the scene of the crime. After deciding on their prime suspect, they came over to the maths section, which was the courtroom. Here they had to weigh up the probabilities and statistics and then decide on whether their suspect was innocent or guilty.

Just as in real life, we didn’t reveal who actually did it, because we often don’t know for sure. And actually, we hoped that (despite all the evidence) the visitors would vote ‘Innocent’ because the evidence certainly didn’t prove anything beyond all reasonable doubt.

Most £20 notes have traces of cocaine on them.

I’ve had my heart set on doing something like this for a while because I wanted to publicise the great work that our Forensic Statisticians (Colin Aitken and Amy Wilson) are doing right here in Edinburgh. They are analysing the occurence of drugs like cocaine on banknotes to help the police decide when someone is really a drug dealer. Apparently (and don’t quote me on this) most £20 notes (like, over 80%) have got traces of cocaine on them, so the police need help in deciding when the notes have been involved in drugs crime or when they have just accidently been placed next to the ‘dirty’ notes in a shop till.

Colin has also appeared as an expert witness in a few trials and has helped to write books to educate judges and lawyers about statistics. Like the general population, judges and lawyers often have a very bad intuition about probabilities. But unlike the general public, their decisions can really affect people’s lives. The classic example is the Sally Clark case. An expert witness for the prosecution claimed that there was a 1 in 73 million chance that two cot deaths could happen naturally in the same family, and therefore that Sally must have murdered her children. Not only was this statistic wildly wrong (the actual figure is about 1 in 100,000) but the conclusion of guilt is also wrong. Neither side took into account the probability of her innocence: despite the unlikelihood of double cot death, double murder is (statistically) even more unlikely. Such a mistake is called the Prosecutor’s Fallacy. In Sally’s case, it led to her spending 3 years in prison for a crime she never committed and then committing suicide a few years after she was freed.

So anyway, the idea behind our science festival exhibit was to show people how bad they were at judging probabilities and to introduce the idea of Bayesian Statistics (which is behind things like the Prosecutor’s Fallacy). Have a go at these questions and see if you can solve them! Answers will be provided in the next blog post.

The Monty Hall problem assumes you'd rather win a car than a goat. This is not true for everybody.

1) One of the most famous examples of conditional probability is called the Monty Hall problem, or the Car-Goat problem. You are on a gameshow, trying to win a car. You definitely don’t want to win a goat. There are three doors, behind which the host of the show has hidden 2 goats and a car. You choose the door which you think conceals the car. The host then opens a different door to reveal a goat. Finally, you get to choose: should you stick with your original choice of door, or should you swap? Or does it make no difference?

2) On very similar lines is the following queston. I flip two coins and tell you at least one of them is a head. What’s the chance that the other one is also showing a head?

3) In a lottery there are a 10 numbers in a bag and you win the jackpot if you correctly predict which 4 numbers get pulled out. What are the chances of winning the jackpot? What are the chances of predicting 3 out of the 4 numbers?

4) If 100 people each flip a fair coin 5 times, how many of them will we expect to flip 5 heads?

5) On the wall there is a calendar for 2012. Visitors to the museum put their birthday on the chart as they come in. After how many visitors do we expect to see the first shared birthday? (I.e. two people with the same birthday.)

Betty only sees what she thinks 2/3 of the time.

6) An eyewitness, Betty, says she saw a suspect leaving the scene of the crime, and that the suspect was wearing a hat. Betty is shortsighted and only correctly identifies hats 2 out of 3 times. That is, 1 time out of 3 she will think that someone is wearing a hat when they aren’t, and 1 time out of 3 she will think that someone isn’t wearing a hat when they are. If 10% of the Edinburgh population wears a hat, what are the chances that the suspect was really wearing a hat?

Needless to say, most visitors to the exhibit found these questions very difficult, but that was the point. We wanted to teach people not to trust their intuition when it comes to probability, and especially not if they are in a jury on a court case!

Many thanks to all who visited us in the Museum and played all our games with us! I hope you all had a good time and learnt something new.

Would your DNA be long enough to weave into a tea towel?

Yesterday I was in Stirling giving my usual knots masterclass when I got asked the most unusual question ever! I was explaining how DNA is like a very very long piece of string sitting inside each of our cells. (It’s actually about 2-3m in each cell!) One pupil then put their hand up and said “If you took all the DNA from all the cells in my body, would it be long enough…to weave into a tea towel?” I was so taken aback by this that I just laughed, but actually the answer is probably yes! Depending on the size of tea towel that he wanted of course. And it would be so thin that it wouldn’t be very good at drying dishes, but that is beside the point.

It is always a pleasure to be surprised by children. And I was especially impressed at all the questions I was asked in Stirling. It helped that the masterclass was much smaller than usual – 15 pupils compared to about 70 in Edinburgh and 40-50 in Glasgow. It meant that I was able to talk to everyone while they were doing exercises, and it was much less intimidating for people to stop me and ask questions during the presentations. But even given those things, I was genuinely taken aback by the insightful questions I was asked and the interest that they showed in what I had to say.

My other favourite moment was when another child pondered “I could take a one-dimensional piece of string and weave it into a 2-dimensional object…”. (I assume this was inspired by the previous question about tea towels.) Now, I had not mentioned the word ‘dimension’ once in my whole masterclass. Most people don’t understand the concept of dimension. Even the undergraduates I teach would have trouble understanding why a knot is inherently one dimensional and not three dimensional. And here is a 12-year old child explaining to me how you could use one dimension to fill out 2-dimensional space!

In case you still don’t understand why this bowled me over, you should go and Google “space filling curve”. (Wikipedia is not a bad reference, but is a little technical.) In the mid-19th century, mathematicians had the idea that a curve could be drawn inside a square so that it went through every single point of the square. This is counter-intuitive, as it seems like the square is ‘bigger’ than the curve, so how could the curve fill it all out? Cantor had showed in 1878 that the infinity which is the number of points in a line segment is the same as the infinity which is the number of points in a square, but it was not until 1890 that Peano came up with this geometrical argument that demonstrated it.

How to construct a space-filling curve iteratively. After infinitely many iterations, it will fill out the whole square.

It is still very counterintuitive to mathematicians that this curve is continuous (i.e. can be drawn without taking your pen off the paper) but is nowhere differentiable (i.e. every point is a ‘corner’, so the curve is always changing direction) and is everywhere self-intersecting (every point on the curve touches another point on the curve). Maths is full of these great examples that challenge our assumptions and intuitions and I hope that I can teach this to my undergraduates later in the semester.

So although the masterclass pupil probably didn’t have infinities and deep thoughts in mind when he made the comment about weaving, it’s exactly questions like this which got mathematicians discovering such things over 100 years ago. I hope that his teachers continue to encourage this wonderful imagination and willingness to ask questions, however silly they may seem at first.

The Young Apprentice candidates

These are apparently the best 12 young entrepreneurs of the country, chosen from many thousands to compete for Alan Sugar’s money to set up a new business of their own. Initially they were divided into two teams (boys and girls) and asked to choose names for their teams. Independently, both teams went for science-sounding names: Atomic and Kinetic. This bodes well, I thought. At the very least, it shows that young people view science as exciting and fast-paced; at best, maybe these future entrepreneurs have a deep appreciation for maths and science.

Ha! Wishful thinking.

Despite the Physics-sounding team names, only 2 of the 12 contestants are studying any kind of science, and in both cases it is Biology. 3 of them are doing an A-Level in Maths and a few more in Economics, but we’ll see soon enough that this implies nothing about their basic arithmetical skills.

Episode 1: The contestants have to make and sell their own brand of ice cream. Problems start early on, when they have to work out how many kilograms of ingredients are necessary to make so many litres of ice cream. Not just that, but they have to work out, given how many scoops they can sell in an hour, how many litres to make in the first place, and how much profit will accrue from various costings. The voice-over dubs this ‘A-Level Maths’. Hmm. Is multiplication and division really not taught before the age of 16 any more?

Sainsburys clearly need to employ someone with an A-Level in Maths in order to do their pricing...

But maybe the voice-over man is right. The boys have a team member with A-level maths, who copes fine with the numbers. The girls, on the other hand, have sent both their A-levellists off to do the design work, leaving chaos behind them. The remaining team think a gram is equivalent in weight to a litre. They think 3 x 4 is 28. “The real surprise here is that the team cannot add up, subtract, divide or multiply”, says a bewildered Nick Hewer, who is keeping an eye on proceedings. In the boardroom, the project leader defends herself saying “I only have a GCSE in maths”, to which Lord Sugar responds with “I don’t care if you only have air miles in maths – this is baby stuff!”

Episode 2: The contestants have to design a new product for the mother & baby market. Thankfully it’s more about market research and good pitching than making and selling this week, so no maths to worry about.

Episode 3: The contestants have to set up their own floristry businesses, designing and selling displays to corporate customers, as well as flogging bouquets on the street. Each team is divided into two, with some people staying behind to learn flower arranging and the others going to win deals with companies. Maths makes an early appearance, with one team leader choosing to keep one of the girls in the flower arranging group, despite her being a proven success at pitching to companies. The reason? Because she’s the only one who can do the maths in pricing the displays.

One of the boys, who had got the highest score in GCSE Economics in Northern Ireland, proclaimed “I’m not very good with numbers.” Prompted further, he said “There’s not a lot of numbers in Economics”, to which Lord Sugar replied “We’re not talking about quantum physics here, are we? We’re talking ‘this rose costs 40p, so 10 roses costs £4′.”

In the end though, the most frustrating thing for me about watching the show is that the team who gets it right every week is almost certain to lose the task! The teams who work hard to correctly do their costings, to make a good product and to provide good customer service always fall down because they haven’t overcharged for their product like the other team have. It is depressing to live in a world where being ruthless and greedy gets you further in life than being honest and intelligent, and even more depressing to think that young people will be watching this programme and learning exactly this lesson.

Balloons at MathsJam 2010

In a nutshell, MathsJam is a place for people to meet to share mathematical puzzles, games, toys, ideas, stories and tricks. It was originally the brainchild of Colin Wright, who organised the first ever MathsJam conference last year, bringing together geeky enthusiasts from all over the country for a weekend of mathematical fun. We had Rubiks cubes of all shapes and sizes, mathematically folded balloons, mirrors, Klein bottles, magic tricks, soma cubes, post-it note dodecahedra, vortex cannons… And that was just the list of physical toys! I also learnt facts like:

• Round pegs fit into square holes better than square pegs fit into round holes…until you get to 9 dimensions!
• Almost every integer contains a 3.
• That given 5 numbers, you can always find 3 of them which add up to a multiple of 3. (But what is the generalisation?)
• That there is only one number whose spelling is in alphabetical order. (Can you find it?)
• That a blindfolded person given an even number of coins, placed on a table so that half are facing heads up and half are facing tails up, can separate them into two piles so that the number of heads in each pile is the same.
• That you can work out the distance to the moon using only a pendulum.

The weekend was such a success that people started asking “Can’t we have a MathsJam every month?”. Pretty soon there were ‘Jams in Manchester, Nottingham and London with Edinburgh, Glasgow, Reading, Liverpool, Newcastle, Dorset, Leeds, Bath, Dublin and Belfast following on their tails.

Attempting topology at the August Edinburgh MathsJam

Everyone meets on the second to last Tuesday of the month and we have a shared Twitter account, @MathsJam, so that everyone can see the puzzles being worked on around the country. The Edinburgh ‘Jam was set up by myself and Ewan Leeming, and we meet at Spoon Café Bistro on Nicholson Street. Further details are on the MathsJam website, together with an email address and Facebook page, and also contact details for all the other ‘Jams around the UK (and indeed, the world!).

This weekend I travelled down to somewhere near Crewe for the second annual MathsJam conference, together with my buddies Albert, Julia and Michael. I was very excited about all the toys and games I’d get to play with, but at the same time incredulous that the weekend could possibly be better than the first MathsJam weekend. Well, I shouldn’t have had any such thoughts.

Albert wearing his ring-on-a-chain

One thing I loved about this year’s conference was the chance to purchase goody bags with exciting toys to take home and show friends. Last year I shot some videos and got photos, but nothing compares to being able to go home and show your friends in person the amazing things you’ve seen. My favourite was the ring-on-a-chain trick (pictured left) where a ring is dropped from a chain with unexpected consequences. Next favourites the falling rings and James Grime’s amazing non-transitive dice.Maths and science is much more cool than sleight-of-hand magic.

Here are some pencil and paper questions you might like to get your teeth stuck into (metaphorically speaking):

• A consecutive sum is a sum of consecutive digits. Are there any numbers which are not consecutive sums? How many ways can a number be written as a consecutive sum?
• Why is 100/81 equal to 1.2345678…?
• How can you cut any shape out of a piece of paper using only one cut?
• Does a running sand timer weigh more, less or the same as a finished sand timer?
• How do you make 2 paperclips link together using a strip of paper?
• Given that we can make a regular pentagon by tying a knot into a strip of paper, is it possible to make a dodecahedron by folding 12 knots into a piece of paper and then folding it up?
• How is it possible to randomly play two games, each of which would individually lose you money, and make an overall gain? (This is called Parrondo’s Paradox.)
• Split the numbers 1,..,16 into two sets X and Y so that the sum of the elements in X equals the sum of the elements in Y; the sum of the squares of X equals the sum of the squares of Y; the sum of the cubes of X equals the sum of the cubes of Y. (I am currently working on a generalisation!)

Plus I learnt  that a 9999-sided polygon is called a nonanonacontanonactanonaliagon. (This seems to be the most popular thing I have ever posted on Twitter.) I encountered Pat Ashforth, one of the founders of Woolly Thoughts, who showed me her dragon-curve blankets and crocheted hexaflexagons. I also saw a magic square that worked upside down and some Platonic solid maps of the world.

Dragon curves and other mathematical knitting by Pat Ashforth

A magic square cushion which works both ways up

Julia found herself on the panel for the Math/Maths podcast, which you can listen to here,with contributions also from Matt Parker, James Grime and Katie Steckles. The laughter on the podcast is a really good reflection of the fun that everyone had at the MathsJam, and once again I have to extend a huge thank you to Colin and all the other people who helped to organise the event this year. There’s no other conference in the world which is this enjoyable and it is wonderful to see so many people enjoying the fun and beauty of mathematics.

If you’ve never been to a MathsJam, I hope this article persuades you to go along to the next one on 22nd November! They are all over the country now so there’s bound to be one nearby. And if there isn’t, start one up yourself! All you need is a pub and a couple of people willing to come sit with you on a Tuesday evening. I look forward to seeing more people MathsJamming in Edinburgh in a week’s time!