This month is Women in History Month, and 8th March was International Women’s Day. In celebration of all the brilliant women who’ve made mathematics what it is, here’s a celebration of mathematical firsts – pioneers …

Der Beitrag Women Pioneers of Mathematics erschien zuerst auf Heidelberg Laureate Forum.

]]>This month is Women in History Month, and 8th March was International Women’s Day. In celebration of all the brilliant women who’ve made mathematics what it is, here’s a celebration of mathematical firsts – pioneers who changed history and paved the way for others to follow.

Using the links in this article, you can read much more about their lives and work on Wikipedia – and I include these links with special thanks to Jess Wade for her sterling work in improving coverage of female scientists and mathematicians on Wikipedia, and to anyone who has helped her in that goal.

Historical accounts are patchy, and obviously people did a lot of mathematics without leaving a record of it – but one of the oldest well-documented examples of a female mathematician is Hypatia of Alexandria, who lived in Alexandria from around 350CE. Being from a mathematical family, Hypatia was also a philosopher and astronomer, and was known to walk around the city in philosopher’s robes, giving impromptu public lectures.

She was considered to be a great teacher, and much of her work in mathematics was in the form of preserving existing maths texts and contributing commentary on the work of others, as was common for mathematicians at the time. She developed an improved method for long division for use in astronomical calculations, and showed people how to construct astronomical calculation devices like astrolabes and hydrometers.

She is also known to have written a commentary on Diophantus’ Arithmetica – a book of algebraic problems and their numerical solutions (you may remember Diophantine equations from our post around this time last year) – verifying the solutions and adding problems of her own. The number system in use at the time was sexagesimal – base 60 – and Hypatia used her own algorithm for division, which has allowed historians to determine which parts of the text she contributed.

Hypatia became embroiled in politics, acting as an advisor to one of the Roman prefects, and as part of an unfortunate political spat she was brutally murdered by a mob of Christians in 415CE.

Cambridge University has a long and distinguished history of producing excellent mathematicians. The Mathematical Tripos examination – their official undergraduate exam for maths students – is the oldest Tripos examined at the university. The tradition of announcing the results from a balcony in the Senate House continues to this day.

The Tripos has historically been considered a gruelling test of mathematical knowledge – in 1854, the Tripos was made up of 16 papers totaling 211 questions taken over 8 days – a combined 44.5 hours of examination. Only around a quarter of students taking the exam achieved a passing mark, and passing students awarded a first-class degree were referred to as ‘Wranglers’ – with the ‘Senior Wrangler’ being the highest scoring student, ‘Second wrangler’, ‘Third wrangler’ etc below that, with ‘Senior Optime’ being the highest scoring student awarded a second-class degree, and so on.

Women were not officially allowed to take the Tripos (with few exceptions) until 1880, when Charlotte Scott got special permission to take the exam, and placed eighth – but the title of ‘Eighth Wrangler’ went to a male student. However, this opened the doors for more women to take the exam.

The first woman to achieve the top score on the Tripos was Phillippa Fawcett, who did so in 1890 – but since women still weren’t considered officially part of the system, she was ranked ‘Above the Senior Wrangler’. Her achievement gained worldwide media coverage, and created much discussion around women’s rights. The first woman to officially be awarded the position of Senior Wrangler was Ruth Hendry, in 1992.

Sofya Kovalevskaya, born in 1850, was a Russian mathematician notable for her contributions to analysis, mechanics and partial differential equations. She was also a pioneer in several aspects – as well as being the first woman to obtain a doctorate in mathematics (in the modern sense), she was also the first woman to be appointed to a full professorship in Northern Europe, and among the first women to be appointed as an editor of a scientific journal (the prestigious maths journal Acta Mathematica, in 1884).

Also of note is Martha Euphemia Lofton Haynes, the first African-American woman to achieve a PhD, in 1943. Her dissertation title was *The Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences*, and she became a leading figure in maths education, teaching in schools for 47 years, chairing the D.C. board of education and going on to establish the mathematics department at the University of the District of Columbia.

The name Agnesi might already be familiar to mathematicians because of the cubic curve which has become called the ‘Witch of Agnesi’. The curve was studied by, among others, Maria Gaetana Agnesi, an Italian mathematician, philosopher and theologian.

As well as being considered to be the first woman in the Western world to have achieved a reputation in mathematics, Agnesi was the first woman to write a mathematics handbook and the first woman appointed as a mathematics professor at a university (the second woman ever to be appointed a professor in any subject – a physicist, Laura Bassi, being the first).

The Royal Society in London grants fellowships to individuals who have made a ‘substantial contribution to the improvement of knowledge’, and is considered a significant honour – past luminaries include Isaac Newton, Srinivasa Ramanujan and Alan Turing, and the total list of around 8000 Fellows of the Royal Society (FRS) includes over 280 Nobel Laureates.

Around 60 new Fellows have been elected each year since 1663, including many mathematicians, but it took until 1947 before one of those mathematicians was a woman. While she wasn’t the first woman to be an FRS (that happened two years earlier, in 1945, when crystallographer Kathleen Lonsdale was awarded the honour), mathematician Mary Cartwright was the first mathematician.

Having graduated from St. Hugh’s College at Oxford University in 1923 (being the first woman to attain the final degree lectures there, and the first woman to obtain a first) Cartwright has a mathematical theorem named after her (Cartwright’s Theorem) and also published a simplified elementary proof of the irrationality of π.

Incidentally, mathematician and engineer Hertha Ayrton was nominated for fellowship in 1902, on the basis of her work on the electric arc, her application was turned down by the Council of the Royal Society, who decreed that married women were not eligible to be Fellows.

The Royal Statistical Society, the UK’s professional body for statisticians and data scientists, was founded in 1834, and fellows of the society represented a huge section of fields, including politics, the army, law, history, physical science, philosophy, the church, art, journalism, medicine and philanthropy as well as economists and statisticians.

The first woman to be granted the honour of membership was someone you may well have heard of, but possibly not in her capacity as a statistician – founder of modern nursing Florence Nightingale used statistical charts and evidence in her campaigns for social reform, and the RSS proudly claims her as their first female member in 1858. The first female president of the RSS was Stella Cunliffe, who took the role in 1975.

The various mathematical learned societies all have long histories of queues of similar-looking male presidents, but thankfully all of them have at least once been led by a female mathematician. The Institute of Mathematics and its Applications had its first female president in 1978, lattice researcher and education reformer Kathleen Ollerenshaw. Dorothy Lewis Bernstein, who worked in applied maths, statistics and on the Laplace Transform, was the first female president of The Mathematical Association of America, elected in 1979.

The London Mathematical Society and the Mathematical Association (a more teaching-focused organisation) both had their first female presidents in the form of previously mentioned Mary Cartwright – a pioneer in so many things. The American Mathematical Society’s first female president was Julia Bowman Robinson, elected in 1983, and the European Mathematical Society has Marta Sanz-Solé, who along with many other directorships and council memberships throughout the mathematical field, was elected president of the EMS in 2010.

As well as achievements in learning and in being part of mathematical organisations, mathematicians are often honoured for their work by prizes and awards. The National Medal of Science is awarded annually since 1963 by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge.

The National Medal of Science is awarded in many categories, and in the Mathematical, Statistical, and Computer Sciences category, it’s been given to many famous mathematicians including Kurt Gödel, Donald Knuth, Saunders MacLane and Solomon Golomb. The first (and only, as far as I can tell) time it was awarded to a woman was in 1998, when it was given to dynamicist (and second ever female president of the AMS) Cathleen Morawetz.

The higher up the mathematical ladder of achievement you go, the more recently you need to look to see women achieving honours. The Fields Medal, one of the prizes whose laureates are invited to the HLF each year, is awarded to four mathematicians once every four years, at the International Congress of the International Mathematical Union. It is regarded as one of the highest honors a mathematician can receive, and considered to be the mathematician’s equivalent of a Nobel Prize.

In its 84-year history, the Fields medal has been awarded to a woman exactly once – in 2014, when geometer Maryam Mirzakhani was awarded the prize “For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.” Mirzakhani was also the first Iranian to be awarded the prize.

The Abel prize, a similarly prestigious honour and HLF laureate criterion, has been awarded annually since 2003 by the King of Norway, and laureates include the late Michael Atiyah, Fermat’s Last Theorem prover Andrew Wiles and game theory founder John Forbes Nash Jr.

The award was finally given to a woman just last year, in 2019, when it was awarded to American mathematician and founder of the field of geometric analysis Karen Uhlenbeck, as I wrote about last April.

These giants of mathematics have opened the gates to many more women studying and working in the subject over the years. We salute their courage and willingness to pave the way for future generations – and we enjoy the fruits of their mathematical labours whenever we quote their theorems or reference their work, especially knowing that they made their contributions despite the extra challenge of being the first.

Der Beitrag Women Pioneers of Mathematics erschien zuerst auf Heidelberg Laureate Forum.

]]>The 2019-2020 Australian wildfires burned an estimated 186,000 square kilometers (72,000 square miles), ravaging the country and killing up to 1 billion animals. In California, the 2019 wildfire season ravaged over 1,000 square kilometers, often …

Der Beitrag How Data Science Can Help Us Understand (and Fight) Wildfires erschien zuerst auf Heidelberg Laureate Forum.

]]>The 2019-2020 Australian wildfires burned an estimated 186,000 square kilometers (72,000 square miles), ravaging the country and killing up to 1 billion animals. In California, the 2019 wildfire season ravaged over 1,000 square kilometers, often in or close to inhabited places. As climate change continues to mount, wildfires are expected to become a more and more serious problem – and data science can be an important ally in this fight.

Dr. Tom Beer watched in dismay as wildfires ravage through Australia. Beer read the reports and followed the news, dreading the effects of a devastating wildfire season. As anyone keeping an eye on the situation, Beer was devastated; but he wasn’t surprised.

In 1986, Beer was working at CSIRO, an Australian federal government agency responsible for scientific research. He was studying wildfires and was instructed to take a look at how the greenhouse effect was changing wildfires. The study, which was published in 1988, was almost prophetic.

The greenhouse effect, the study documented, would make wildfires more likely and more dangerous. More greenhouse gases meant higher temperatures, which in turn mean less moisture and more dry vegetation — perfect fuel for wildfires. Furthermore, climate change would fuel stronger winds, further exacerbating the spread of wildfires. Rising temperatures make worse wildfires, the paper concluded.

The results were, at best, ignored. Dr. Graeme Pearman, Beer’s supervisor (who had instructed him to look at this topic in the first place) went on to give over 500 presentations on climate change in future years, many of which addressed wildfires. Both Beer and Pearman, however, received criticism for this. The science was “shoddy”, some would say. Large wildfires did not happen for ten years, and climate change acceptance was painstakingly slow.

So their results went largely unnoticed. It took decades before science (and tragic events) vindicated them.

As time passed, our understanding of wildfires changed and improved. Satellite data offered large-scale observation data, and increasing computation power allowed the development of increasingly complex models. Wildfire studies became more quantitative and more detailed – and computer models became an important tool.

Wildfire models use numerical data to simulate the behavior of wildland fires. Models start out with simple inputs, such as the surface geometry and the type of fuel — in most cases, the vegetation which is expected to burn.

Intuitively, this is easy to understand. Dry wood burns more than wet wood, small twigs burn faster than larger ones, and so on. But numerically modeling these parameters is an entirely different problem.

Fire (and wildfire in particular) is a notoriously difficult thing to predict. This is why, in addition to well-established chemical and physical equations, wildfire models also tend to include semi-empirical fire spread equations, which are usually published by the forestry services of several countries. This makes wildfire modeling particularly interesting and particularly challenging.

In addition, models must also consider fire as an active parameter that could actively change its surroundings: a strong fire can produce storms and strong winds, which further affect it, creating a complex feedback loop.

Models are also influenced by weather and moisture, and must, therefore, be coupled with atmospheric simulations — adding another layer of complexity into the mix. It is not surprising then that modern models are already incorporating massive amounts of data and machine learning; some models require supercomputers to be able to run.

Ultimately, a risk index is developed as an output, and several predictions are made about the emergence and evolution of wildfire.

The connection to climate change is, in one regard, pretty obvious. Many areas on Earth are already seeing average temperatures 1°C above preindustrial levels, excess drought, and an increasing frequency of extreme weather events. In places such as Australia and California, drought has become a key parameter affecting wildfires, and conditions created by a heating climate are exacerbating wildfire risk.

In this sense, the devastating wildfires in Australia are not surprising. They are part of a larger trend which we will see in many parts of the world (although in truth, they were worse than anything models had predicted) — a trend which was predicted by numerous models.

But computer models can do more than just help us understand wildfires — they can help us tackle them.

The ultimate goal of computer models isn’t only to gain a better understanding of the process, but also to direct effective policy. For wildfires, in particular, this is essential in the case of a controversial practice: controlled fires.

Wildfires, it should be said, are a natural process. Natural wildfires eliminate excess fuel (such as fallen wood and litter) and keep forests widely spaced. Under normal conditions, natural wildfires can help replenish forests.

But as it so often happens, the natural balance has been heavily tipped by human activity.

In addition to climate warming, human activity has affected forests in more direct ways, through forest management practices. Humans don’t really like wildfires — they threaten our safety and infrastructure — so most preventive measures focus on reducing wildfires as much as possible.

This phenomenon was traditionally counterbalanced through controlled fires. A growing body of evidence (much of it based on models) suggests that carefully prescribed fires can have long-lasting beneficial effects. However, controlled fires can be risky, and they tend to attract negative media coverage, so most policymakers have shifted away from this practice. Disasters such as the one at the Bandelier National Monument in New Mexico in 2000 (when a prescribed fire went out of control and forced the displacement of 400 people) definitely don’t help the case.

This is where wildfire models come in handy. Behavior modeling can predict how the fire will develop and where it will go, finding under what conditions prescribed fires can be safely carried out. Modeling can also estimate ecological and hydrological effects, along with other effects (such as the amount of smoke produced).

At a local and regional scale, data science can help forest planners safely deploy prescribed fires, reducing the risk of catastrophic damage caused by excess fuel in the forest. We can learn to map and classify wildfires and build algorithms to make predictions about fire likelihood.

On a global scale, we can better understand how climate change is increasing the risk of wildfires. Beer and Pearson correctly predicted that climate change will increase the risk of wildfire more than 30 years ago; more refined models can help us understand that risk even better.

The threat of wildfires is very serious, as we’ve seen in recent years – and is likely to become even more prevalent as global temperatures continue to increase. We’d be wise to learn from Australia’s lessons and work to limit future tragedies as much as possible.

Der Beitrag How Data Science Can Help Us Understand (and Fight) Wildfires erschien zuerst auf Heidelberg Laureate Forum.

]]>John Hopcroft is an ACM A.M Turing Award laureate, which he won in 1986 together with Robert E. Tarjan for fundamental achievements in the design and analysis of algorithms and data structures. He distinguished himself …

Der Beitrag John Hopcroft on the teaching of today and tomorrow erschien zuerst auf Heidelberg Laureate Forum.

]]>John Hopcroft is an ACM A.M Turing Award laureate, which he won in 1986 together with Robert E. Tarjan for fundamental achievements in the design and analysis of algorithms and data structures. He distinguished himself by his excellent and inspiring teaching. I talked with him about it during last year’s 7th Heidelberg Laureate Forum.

*Mr. Hopcroft. People who are enthusiastic about teaching generally have a role model from their own school days. Is that also the case for you and if so, who was it?*

I was very lucky because I had a lot of inspiring teachers. Not only in primary and secondary school, but also at the university. For example, I had a very good primary school teacher who taught me algebra at the time. But he was also a football coach and he was good in both subjects. He left his mark in my life, which led me to want to have such an influence on other people. I think what he did was to show the students how important their success was to him. And that hit home! It is important that students realize that the teacher really cares about their future.

*How should a student feel in class?*

It doesn’t matter whether the lecturer has a lot of experience and whether he knows a lot. How he designs the class or the lecture is not the crux of the matter. While this is important, it is far more important that he passes on his passion to the students. That is what makes a good lecturer. Because if you have that in mind, then the first two properties are automatically added. If students do not understand a topic, then the experience of the teacher or the design of the class will not help. But that the lecturer cares about his students does.

*How can you make students feel comfortable in class?*

It has to come from yourself. It must be a matter of course for you to want to achieve this. If a student has a problem with homework, this is not the problem, but that the student did not understand the subject matter. Then I try to explain this topic to him from the start and take the appropriate time. You have to do that in every situation. You have to ask yourself how you can help this person. And if you have a lot of students in a course, you can’t really help everyone personally. That’s a problem. But then you have the opportunity to make the whole team of lecturers aware of the need to take special care of the students.

*Do you change a lecture when you notice that student feedback is not good?*

Yes. I actually always prepare my lectures and I always know what I’m going to say and if after 15 minutes I notice that it doesn’t work, I have to change it. There is no way out. That is another reason why you should not use presentations or similar stuff in class. It has a negative impact on the lecture because you cannot adapt to the circumstances. You are stiff. There is nothing better for teaching than using a blackboard.

* What do you think about unorthodox, newer teaching methods? Do you use them?*

I don’t use them. But that doesn’t mean that they’re not valuable. Some say that you shouldn’t actually teach, but rather take care of captivating the students. Every tool is valuable for this. But you also have to be aware of the fact that you have to invest a lot of time in such methods. It must also be clear beforehand what the goal of university-level education actually is so that these methods act towards this goal. Unfortunately, this is not always clear.

*What do you mean by that?*

Young people do not study to get a job, but to shape their entire future life. They shouldn’t take a job for salary, because the job is a fundamental foundation of every life. And you want to be a winner. You shouldn’t have the feeling that you go to work every day, but that you have fun every day. Pick what you enjoy. At the university, you not only learn the subject matter but also other things, such as interacting with other fellow students. The greatest lessons are found outside of lectures, not during them. In this sense, it is particularly important as a lecturer to pass on knowledge and wisdom to the students even outside of the subject matter. That should be important to every lecturer. For example, if there are electives, students have to pay particular attention to what is expected of the course. You also have to project into the future and be aware that many decisions made during your time at university affect your whole life. You shouldn’t pay attention to the demands you get from the curriculum, but your own demands on life.

*You gave one of the first computer science courses. What do you have to do if you want to teach a new subject and have to design a curriculum for it?*

It depends a lot on the area. As an example, I would like to use deep learning, which is particularly popular today. But the main results are based on experimental approaches. And how do you teach that? There is no mathematical theory that justifies why deep learning works (yet). So if you teach something new like this, you have to put together everything you have available and over time you expand it and include new topics. But whenever you teach something new, you have to develop a theory about it. That is the most important thing for me.

*You are introducing measures and policies in the academic field in China to improve teaching there. But what can you do on a smaller scale to improve teaching?*

If a system doesn’t work, you have to ask the honest question why it doesn’t work. What we did in China was to make it attractive for universities to improve their teaching work. They should have motivation, especially in the financial area, to focus on it. The important thing here is that support for this venture came from above. This is the only way to initiate such changes. I have taught in many countries and have supported everyone who is committed to such changes. However, neither I nor the individual had an actual influence because the attempt was nipped in the bud. The government must support such an idea, have an honest interest in it, and set an example. This is the only way to change something in the long term.

*How do you think teaching will change in the future?*

That is difficult to say. But we can do it differently. It is said nowadays that we are in the middle of an information revolution. So if we look at other major changes in the past, it may help us. The first really fundamental change in human history was probably the agricultural revolution. This led to societies developing because in order for agriculture to thrive, you need a hard-working force. Until then, however, education was not as important because it is not as important in the country as other skills. Then came the industrial revolution, which abolished physical labor in factories and changed everything on a large scale. Nowadays you need at least a school leaving certificate because no matter where you work, you have to be able to deal with numbers and above all, you have to be able to communicate properly with others. That changed the way of education. Until a few decades ago, in the heyday of the industrial age, energy sources and raw materials helped powerful states achieve their status. It is believed that this will change again in the information revolution because it will then be available to every country. Countries will then stand out for their ability to develop talent, which can be achieved primarily through education. So it is the countries that will adequately improve their educational policies that will take the leading positions in the world in the future.

One of the other issues I work on in this context is early childhood. I talked to a lot of scientists who study and analyze them and they all agree that if you invest in education at the earliest possible time, you will get the greatest possible payoff. By “earliest possible time” I mean the first two years in life. That is amazing. My question was whether there really is research to back this up and indeed there have been an incredible number of studies in the last 25 years that show that a peaceful and stable environment during growth is important for the optimal development of an infant’s brain. Children develop more and more skills until they are around 20 and are fully mature, but in the first two years children learn how to learn and that is the important thing here. You have to tackle this successfully so that the children can excel in elementary school later on. But it goes one step further. If you really invest in the earliest possible upbringing, it is also proven to be one of the best investments that can be made in the financial area. But no politician does that, because that would be long-term measures and they only think about the next 5 years.

*How can you make society aware of the importance of talent?*

I often think about strategic opportunities to make society aware of this. People need to be aware that we only live once and that we have to enjoy it. You have to search for what is really fun in life and then invest time in it. I’ve had many PhD students in my life, and only one of them never got his PhD. He came to me once to explain that he wanted to be a ski instructor. And of course it was the right decision. If he had stayed in the academic field, he would certainly not enjoy his work as much as he does today and that is certainly particularly important for his students. It will have a positive impact on their lives and there is nothing better than that. That’s why I try to get people to do what makes them happy. If you do a job that you really enjoy and you have a colleague who may be more brilliant and talented but doesn’t enjoy it as much as you do, you will surely do the better job.

Der Beitrag John Hopcroft on the teaching of today and tomorrow erschien zuerst auf Heidelberg Laureate Forum.

]]>In a post last April, I described some of the mathematics behind combinatorics – the branch of mathematics concerned with counting all the different ways to do something. Often, this kind of maths appears in …

Der Beitrag Picking and choosing erschien zuerst auf Heidelberg Laureate Forum.

]]>In a post last April, I described some of the mathematics behind combinatorics – the branch of mathematics concerned with counting all the different ways to do something. Often, this kind of maths appears in daily life – if you’re trying to work out how many combinations of something are possible, or how many different orders something could occur in.

This question also seems to occur to marketing executives on a regular basis – when they realise that their product can be sold in many different combinations to the consumer. It seems very exciting to marketing teams, in a way that it’s not necessarily that exciting to anyone else, that the number of possible combinations might be very large.

The problem comes when marketing teams try to calculate the number of possibilities – and even though the maths involved is fairly straightforward, fail to check their answer is actually correct (presumably most companies employ at least one mathematician that they could run it by?) This happens surprisingly often, leading to wildly inaccurate or confusing claims.

A few years ago, I experienced this first hand – when I checked into a hotel, to find they had a sign explaining that their breakfast buffet offered ‘Endless possibilities (well, 35 trillion to be exact)’. My initial curiosity (‘Why would you claim something is endless when it’s finite?’ ‘Is the number exactly 35 trillion?’ ‘How would you calculate that?’) turned to fury when I realised that the answers were, respectively, ‘Because we don’t know how maths works’, ‘Probably not’ and ‘We’re not telling you’.

On a breakfast buffet, you could consider each item as something you either have, or don’t have, on your breakfast. This means the number of possible combinations would be two to the power of the number of things on your buffet – for each, you choose one of ‘yes’ or ‘no’. For the number to be in the region of 35 trillion, a suitable number of breakfast buffet options should be 45 – since 2^{45} = 35,184,372,088,832. This is, regardless of your level of precision, not in any way exactly 35 trillion (it’s out by a little over 184 billion), but this could explain where that number has come from.

In the wake of this realisation, I looked carefully at the buffet to see if there were exactly 45 different foods to choose from, but things quickly became unclear (Do you count different flavours of jam as different options? And is two sausages a different item to one sausage?). I also surmised that since a sizeable proportion of the possible combinations would be objectively horrible (with several other suggestions for horrible combinations contributed by my Twitter followers), this impressive number of possibilities doesn’t necessarily mean an impressive number of enjoyable breakfast combinations.

Part of the problem with these kinds of adverts is that it’s not always clear what counts as a combination. Last year the team at The Aperiodical wrote about a promotional Superbowl advert run by the makers of Pringles, in which they introduce the concept of stacking three different flavours of Pringles together and eating them all in one go – clearly a delicious thing to do – and claimed excitedly that there were 318,000 possibilities.

Several people on Twitter tried to work out where this oddly specific number came from, and realised there are a lot of considerations. How many flavours are you choosing from? How many Pringles go in a stack – does it have to be three, like in the advert, or can it be another number? And does it matter what order the flavours go in? (Presumably, you get a stronger hit of whichever one touches your tongue first, so it would be a different experience if you put a BBQ one underneath a Sour Cream & Onion).

One person took the step of directly contacting Pringles about this, and their social media team stepped up. Their reply, which answered all these questions, was clearly informed by someone having done some actual maths. With 25 Pringles flavours, in a stack of two, three, or four, and counting all the different orderings, you do get a total of 318,000. You need to first calculate the number of ways to pick 2 things from 25 – which is 25 choices for the first thing, times 24 for the second – then add this to the number of ways to pick 3 things (25 × 24 × 23) and the number of ways to pick 4 things (25 × 24 × 23 × 22).

Strictly, the wording of their tweet, in which they say the calculation “assumes the same flavors are not stacked in a different order”, could imply that the ordering is not important (so we’d need to divide each term by the number of ways to reorder, two, three or four things respectively), but if you bear in mind the ambiguity of written English and the difficulty of communicating knotty maths concepts like this, I’m happy to give them the benefit of the doubt.

It turns out this kind of maths is difficult – not necessarily because of the numbers, but because you have to know exactly what question you’re answering. In 2002, McDonald’s launched the McChoice menu in the UK – offering eight different items and claiming 40,312 different possible combinations. It’s impressive, but also somewhat unclear where this number comes from (and Twitter wouldn’t exist for another four years, so you couldn’t ask their social media team).

With 8 items, each of which you can either have or not have, there are only 2^{8} = 256 different combinations, which is not nearly as many as they claim. With a little thought, it becomes apparent that they’ve calculated the number of different orderings you can put 8 items of food in (8! = 40,320), but then conscientiously excluded the 8 meals consisting of exactly one item, subtracting 8 to give 40,312, which is patently nonsense.

If this makes you angry, you might be pleased to hear that 154 brave souls complained to the Advertising Standards Agency and McDonald’s were taken to court to fight the claim – which McDonalds won after an appeal, by claiming that’s not what they meant, changing their story and convincing the ASA that they just meant there were a lot of options (and if you include different flavours of milkshake and other variants, there are actually a lot more combinations than 40,312). But this mixing of methods did lead to at least one new mathematical discovery – the McCombination numbers (of the form N! – N), which have yet to find an actual practical use.

While it’s nice to see maths being used, these claims are often actually a bit meaningless. Even if you have exactly 45 items on your breakfast buffet, the number of breakfasts you can have won’t be 2^{45} – since you could have more than one of some items, and eat them in a different order (which might make for a different meal) – so any attempt to calculate the actual total is futile. You certainly can’t claim ‘to be exact’!

Maybe it’s best to follow the lead of one of my favourite combination miscalculations in advertising – the Rubik’s cube. In the early days, it was advertised as having ‘Over 3 billion combinations’ – a very impressive number, but in truth slightly shy of the actual 43,252,003,274,489,856,000 (43 trillion) combinations possible on a 3x3x3 Rubik’s cube.

Before mathematicians worked out the maths behind the cube’s complicated twists and turns, the exact number might not have been known, even to the engineer Rubik who designed the cube. But even though 3 billion is a massive underestimate, in combination with the word ‘Over’, the statement is strictly mathematically accurate! That’s what I like to see.

Der Beitrag Picking and choosing erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag Nothing erschien zuerst auf Heidelberg Laureate Forum.

]]>I try to find a varied range of topics to write about for this blog, but sometimes you run out of inspiration and find you have nothing to write about. Luckily, this has happened to me today – since nothing, also known as the number zero, is a crucial concept in mathematics and a vital ingredient of many mathematical theories and systems. Here are some of the ways in which nothing is really something.

While most of us usually start counting from one (since until you’ve got one of something, there’s no real need to count it), mathematicians and computer scientists often start counting from zero. Sometimes called zero-indexing, doing this makes counting and indexing easier in computer programs, and is a more natural way to think about many mathematical series and patterns.

For example, the zeroth power of a number is always 1, and when thinking about number bases it’s natural to start counting with ones – in decimal, we have ones, tens, hundreds and thousands, and in binary it’s ones, twos, fours, eights and so on. The zeroth derivative of a function is just the function itself, and any sensible list of derivatives would normally include it as the first entry.

It’s most often used in numbering lists of things – you wouldn’t say that the set {a,b,c} contains 2 elements, because you started counting at zero! It’s got three elements, but a is the 0th element, b is the 1st and c is the 2nd.

Numbering the first element zero is sometimes also employed when an extra, more important and fundamental idea is discovered after the list is already established – like the zeroth law of thermodynamics, which was formulated in 1935, decades after the others.

So if you’re counting things and a mathematician or computer scientist is involved, make sure to check whether they’re zero-indexing (since, as the famous quote goes, “There are two hard problems in computer science: cache invalidation, naming things, and off-by-one errors.”) Or, use it as a handy excuse if you yourself have mis-counted something!

When writing numbers, we use a place-value number system – that is, the position of a digit within the number determines how much that digit is worth. For example, the digit 4 in the number 24 represents 4 ones, but in the number 496 it’s worth 400. In a system like this, zeroes are crucial, as if we didn’t have a symbol to indicate ‘no tens’, the number 205 would be indistinguishable from 25.

Also sometimes called positional notation, this system is the one we’re all used to working with, and it’s a great way to write down numbers. It allows for easy multiplication and addition of numbers (column-by-column), and combined with a decimal point we can represent any number, whole or fractional, using some combination of digits.

Prior to the invention and widespread use of positional notation, numbers were represented in less intuitive ways. For example, Roman numerals use symbols like I to represent 1, V for 5 and C for 100, and these letters will always have these values wherever they appear (except in the case a symbol is placed before a larger number, in which case its value is negative: IV is 5 – 1 = 4). Computing sums and products using this notation is not particularly intuitive – if you’ve ever tried it, you mostly find it’s easier to convert the numbers to decimal first! It’s also not necessary to have a symbol for zero in this kind of system, and the Ancient Greeks didn’t have a symbol for zero at all.

But this doesn’t mean zero wasn’t used or known about in ancient times – place-value systems have been around in some form for a very long time, and in any positional system, you need a zero, or at least some way of indicating it. The Ancient Babylonians used base 60 numbers (sexagesimal), Mesoamericans used base 20 (vigesimal), and early mathematics from many other cultures used a variety of number systems including base 10, and sometimes even binary (base 2). It was common to leave a gap between numbers to indicate a zero, and when specific symbols for zero were introduced they included a pair of slashes, a large dot, and later a circle shape, which became the basis of the symbol 0.

Groups are a fundamental concept in maths, and group structures crop up everywhere from arithmetic to symmetry to matrices. Briefly, groups are sets of objects on which there’s a well-defined combining operation – if you take two objects from the set, and combine them, you get another object from the set. If you read my post from last year on groups (or you know about them anyway) you might recall that every group has an identity element – one which when combined with any other object doesn’t do anything: it leaves the object the same.

One group which you’ve almost certainly already worked with extensively (maybe without realising it!) is the group of whole numbers under addition. It definitely has a group structure – if you add together two whole numbers, unless you’re doing it really wrong, you always get another whole number. In this group, the identity element – the unique important special one that you need for the group to make sense and work properly – is zero. Adding zero to something doesn’t change it!

This works in finite groups as well – if you think of the numbers 1-12 in clock arithmetic (so if you count past 12 you go back to the beginning), the zero here is the number 12, and the set is often considered to be the numbers 0-11. If you add 0 hours to the time (or 12 hours), you don’t change anything.

While the zero might not seem like the most interesting element of a group, the fact that we have it is crucial to the way addition works, and to the structure of the group.

If you ask a mathematician to tell you their favourite maths equation, quite often the answer will be Euler’s identity. Sometimes also called Euler’s formula, it’s an equation which relates some of the most interesting concepts in mathematics, and it’s most commonly seen in this form:

This simple relation connects five things mathematicians really like:

- π, the circle constant
- e, the base of natural logarithms
- i, the imaginary number
- 0, the identity for addition
- 1, the identity for multiplication

Written this way, it also includes all of the important mathematical operations – addition, multiplication (in there between the i and the π) and exponentiation (raising to a power). The formula is fundamental to trigonometry, and shows how all of these mathematical ideas are connected. Isn’t it cool? And zero is the answer to the equation, which cements its place as definitely one of the best concepts in maths.

Binary, which is the basis of all computer systems and many other types of thinking, couldn’t happen without zeroes. For all the reasons outlined in item 1 above, it’s needed to define the absence of a power of two in the number’s expansion – but in the case of binary, it’s even more fundamental. The intrinsic beauty of stripping information down to fundamental ones and zeroes is meaningless without zeroes – they’re literally half of the story.

1 and 0 are also used in logic – as true and false in truth tables, representing the truth value of a statement, and encapsulating the yin and yang, each meaningless without the other. And statements with the truth value 0 are increasingly popular these days! 🙂

One final argument in favour of nothing: mathematicians are obsessed with finding zeroes. In particular, zeroes, also called roots, of functions. For example, if I have a function f(x), I might ask ‘for which values of x does this function equal zero?’

This might seem like a strange question, but if you want to study a function and understand what it looks like and how it behaves, the zeroes are crucial. If the function is continuous and can be plotted as a line on the page, the zeroes will tell you where the function crosses the axis, and how it’s positioned in space. If you’re prepared to do a little calculus and differentiate the function, the zeroes of its first derivative will tell you where it changes direction and where its behaviour changes. Mapping out the zeroes gives a picture of what’s interesting about the function.

In fact, looking for zeroes could potentially be a lucrative hobby – if you can say for sure where all the zeroes of the Riemann Zeta function are, it’ll net you a cool million dollars! As I wrote previously, the Riemann Hypothesis is one of the greatest outstanding problems in mathematics. Despite many decades of mathematicians working on the problem, we still don’t have a definite answer to the question of exactly where all the zeroes are. If you ask anyone working on it, they’ll certainly tell you how important zero is to their work in mathematics!

So, if you’re considering how important zero is, you might think nothing of it – but for mathematicians, it’s second to none.

Der Beitrag Nothing erschien zuerst auf Heidelberg Laureate Forum.

]]>Parallel to the Heidelberg Laureate Forum, the Heidelberg Laureate Forum Foundation organizes an exhibition related to mathematics or computer science, as part of the accompanying program of the HLF. In 2019 it was the mathematics …

Der Beitrag Aldo Spizzichino exhibition at the HLF 2017 erschien zuerst auf Heidelberg Laureate Forum.

]]>Parallel to the Heidelberg Laureate Forum, the Heidelberg Laureate Forum Foundation organizes an exhibition related to mathematics or computer science, as part of the accompanying program of the HLF. In 2019 it was the mathematics of music, in 2018 it was the turn of women in Mathematics. While we wait to know what the exhibition will be in the next edition of the HLF in September 2020, we review the exhibition hosted in the 5th HLF that took place in September 2017, with the work of Italian physicist Aldo Spizzichino on computer-generated art.

For more about Aldo Spizzichino’s work, visit his website,.

Der Beitrag Aldo Spizzichino exhibition at the HLF 2017 erschien zuerst auf Heidelberg Laureate Forum.

]]>I have a strange relationship with my calculator. A lot of people who haven’t met many mathematicians assume when they meet me that my status as a mathematician qualifies me as a super-brain who would …

Der Beitrag Calculating Percentages erschien zuerst auf Heidelberg Laureate Forum.

]]>I have a strange relationship with my calculator. A lot of people who haven’t met many mathematicians assume when they meet me that my status as a mathematician qualifies me as a super-brain who would never resort to working something out using a calculator – once, on being introduced to someone who’d had a few drinks, I was greeted with “Oh, so you’re a mathematician, eh? Then what’s the answer to [mathematical calculation]?” – missing entirely what maths is actually all about – and also thankfully having drunk enough not to be able to come up with a question that was actually difficult. (It turns out, just because numbers are big, it doesn’t mean they’re hard to multiply together, and seven times a million is just seven million).

Being a mathematician is obviously not just all about adding and multiplying numbers. As I hope I’ve been getting across in this column, maths is a rich and beautiful subject full of abstraction and variety, and involves manipulating complicated concepts in your mind to appreciate their beauty. However, it’s unavoidable that mathematicians do sometimes have to do calculations. Occasionally, it’ll be part of work – but most of the time it’s for one of the many mundane reasons that any person has to do a calculation, like working out how much of something you need, or adding up the bill in a restaurant and including a tip. Quite often in these situations, if I want to make sure a small mistake doesn’t result in me being short-changed (or short-changing someone else), I’ll whip out a calculator to save time and effort.

It was on such an occasion recently that I discovered an interesting quirk of how the calculator I was using (the calculator app, on my Android phone) works. I needed to work out the total of my food, plus a 10% service charge – so I added together my main, side and drink, and then calculated that total with 10% added. Normally, I’d do this by typing:

(Here multiplying by 1.1 gives me 110% of the original value, so I’ve added 10%). But I noticed that my friend nearby, who was undertaking a similar process, was using the % key on the calculator. As someone used to converting in my head between percentages and decimals, I’d never really used that key much – but it turns out, Android interprets it in a very natural way. You can type:

And it’ll give the same calculation. What’s more interesting is, you can do this without the brackets, and it’ll still apply it to the whole thing:

As someone who’s involved in both mathematics and maths teaching, this grated on me a little. I appreciate that this is how the average calculator user might want this to work – there aren’t many situations where you would specifically want the percentage only to be applied to the last entry in the sum, so it makes sense that doing this would work in this way. But in my mind, the operation ‘add 10%’ is interchangeable with ‘multiply by 1.1’ – and under the standard operator precedence rules, the multiplication should be calculated before the addition. Indeed,

gives a different answer. I can force the calculator to work it out this way – for instance,

will do the thing I would expect it to do. But it’s intriguing that the calculator takes a natural, but slightly non-mathematical approach here. Since I can get a rough idea of what the answer should be in my head (the total is going to be around 18-19, so adding a 10% tip should make it around or slightly more than 20, since that’ll be around 1.8-1.9), I can usually tell which of these things the calculator has done, but someone relying solely on their calculator to give them the answer is unlikely to notice.

It hasn’t always been this way – stock apps like the calculator are updated all the time, and with new versions of the operating system they can change their behaviour. Previous versions of Android included a calculator app which would interpret ‘10%’ just to mean 0.1 (which is precisely what it means, and much closer to my mathematical interpretation of the calculation). So you’d get

It’s taken the sum of the three numbers, then added 0.1. Unhelpful! Under this interpretation of what % means, to work out my bill I could just multiply by 110%, which the calculator would treat as multiplying by 1.1 as I’d usually do. But would it make sense to someone who doesn’t think of it that way?

Writing a calculator app can be a big challenge. Phones can perform incredible feats of processing – streaming video, running games and presenting huge complex websites, coordinating multiple input and output devices, all on something that fits in your pocket. So you’d think that to make something that you operate by pressing buttons on a screen, which output the answer to a mathematical calculation – something computers do all the time – would be trivial.

But calculators have always been an interesting challenge for programmers. The way computers store and process numbers often causes hiccups and issues with doing precise calculations. For instance, numbers are stored to a particular degree of precision – a fixed number of total decimal places. Called ‘floating-point arithmetic’, it involves storing a fixed number of digits, as well as recording the position of the decimal point (using a power of the base you’re calculating in, just like the standard scientific form of [number] × 10^{[power]}). So if you’re calculating with very big, or very precise values, sometimes rounding occurs, and can cause errors.

Another issue is with certain numbers which behave differently in binary. Some numbers in decimal can’t be stored using a finite string – for example, 2/3 = 0.666666 (recurring forever) – so it’ll be stored by a calculator with a finite memory as 0.666666667 (for a given size of memory). When computers convert numbers to binary, they can still store non-whole numbers – for example, ½ in binary is 0.1. The columns which are numbered 1,2,4,8… going off to the left are just 2^{0}, 2^{1}, 2^{2}, 2^{3}… – so the columns off to the right are 2^{-1}, 2^{-2}, 2^{-3} and so on, giving ½, 1/2^{2}, 1/2^{3}… In this way, ¼ is 0.01 and ⅛ is 0.001.

This means that certain numbers can’t be stored in binary using a finite string of digits either – and it’s a different set of numbers than it is in decimal. For example, 0.34 has a finite decimal expansion, but in binary it’s 0.0[10101110000101000111], where the section in brackets repeats forever. So this number, along with infinitely many others, can’t be stored exactly in binary, meaning errors can creep in if not handled carefully.

But as well as all these technical challenges, designers of calculator apps also have to consider the way people will use the calculators. Given a concept like the percentage button, there are any number of ways to interpret what the user could mean when they use it. This seems strange, since it’s a specific mathematical concept, and ideas in maths are usually well-defined – but it’s also a concept in general use, so it needs to work the way most normal people (one way to define non-mathematicians) expect it to.

Another popular type of calculator people carry around in their pockets is the iPhone – and the built-in calculator there actually behaves differently. If you type in a calculation like 12.50+2.95+3.5+10%, as soon as you press the percent symbol, it’ll calculate and display the percentage – in this case, 10% of (12.50+2.95+3.5). If you then proceed to press equals, it’ll add this to the total, and show you the correct value.

This is equivalent to what the Android calculator does – it’s still giving you the answer you need so you can hand the server the correct amount of money – but it’s at least acknowledging that there’s an intermediate step, so you can see a bit of what’s going on.

Perhaps this strange percentage handling is a function of calculators designed for general use. Scientific calculators, made for engineers and mathematicians to do precise and complex calculations, handle the percentage key very differently. I’ve been playing with the scientific calculator I used for my school exams, and it gives some very strange answers when I try to add a percentage – multiplying by one is fine, as you would expect, but it turns out that even different calculator manufacturers have implemented wildly differing interpretations of what ‘18.95 + 10%’ means, and it took me a few minutes to work out what my Casio was even doing.

I suppose the real lessons here are, 1. Don’t take your scientific calculator to a restaurant, 2. Make sure you know what you’re asking a calculator to do so you can check it’s doing the right thing, and 3. Always tip your server (the right amount).

Der Beitrag Calculating Percentages erschien zuerst auf Heidelberg Laureate Forum.

]]>A friend of mine recently kindly made me a present, using their 3D printer – a puzzle made up of 21 pieces, three each of seven different shapes. The statement of the puzzle is as …

Der Beitrag Building Blocks erschien zuerst auf Heidelberg Laureate Forum.

]]>A friend of mine recently kindly made me a present, using their 3D printer – a puzzle made up of 21 pieces, three each of seven different shapes. The statement of the puzzle is as follows:

“Can you arrange the pieces into seven groups of three so that for each possible pair of shapes, there is one group containing that pair?”

That is, I need to find a way to arrange the 21 pieces into 7 groups, so that if you name any two shapes, I can point to a group (and only one group) which contains those two pieces. The pieces are mounted on bases shaped like thirds of a circle, so groups of three fit together nicely, and we’ve had a lot of fun coming up with names for the seven shapes.

It’s an interesting puzzle, and it took me a few minutes to work out a solution – mostly by trial and error. I first noticed that it didn’t matter which three shapes I picked to put in the first group, since there are the same number of each shape, so they’re interchangeable. Then I tried some other groupings alongside that, and worked backwards whenever I reached a problem.

If you’d like to have a go at the puzzle, I’ve made a PDF of 21 pieces you can cut up, with three each of seven different shapes – you’ll need access to a 2D printer and a pair of scissors.

One way to extend this puzzle is to try it without paying attention to the colours first, then once you have a solution, try to rearrange the pieces so each group has one piece of each colour in it.

This kind of problem – looking at different ways to combine things so that they satisfy statements like ‘any two shapes occur in exactly one group’ is part of a branch of maths called combinatorics. I’ve mentioned it before in a previous blog post, as it also covers counting and enumerating combinations, but this kind of problem constitutes one particular branch – and solutions to them are called combinatorial block designs.

The term ‘block designs’ refers to the fact that it’s about ways of grouping things, in such a way that specific combinations occur. For example, a group of friends planning to play a series of games together might wish to work out a set of groupings they can arrange themselves in so that they all get to play against everyone else:

6 people want to play in three teams of two, for five different rounds of a game; each round, everyone would like to be on a team with someone they’ve not been on a team with before. How can they arrange themselves?

(One solution to this problem is given at the end of the post – but see if you can find one yourself!).

One very old problem in this area, which was originally stated in 1850 and appeared in several recreational mathematics books at the turn of the century, can be stated as follows:

Fifteen people wish to arrange themselves into groups of three, with a different arrangement each day for a week, so that no two people are in the same group on more than one day.

This problem can be solved (and solutions are given on the Wikipedia page) using a **block design**. This is a way of splitting up a set of things, call it X, into a collection of smaller sets – called blocks – with various constraints which can be applied depending on the desired properties – including:

- Uniformity – all the blocks are the same size (the most interesting and most often studied type)
- Any element of X is contained in a specific number of blocks – which may be one, or more than one if the sets can overlap
- Any set of elements from X of a given size is contained in a specific number of blocks

For example, in the 3D printed puzzle I was given, there are 7 blocks of size 3, any one shape is contained in exactly 3 blocks (as there are three pieces of each shape) and for any set of shapes of size 2, there is exactly one block which contains that set.

In the case of the 19th-century maths problem, there are 15 people who need to be arranged into groups across all seven days – so there are five groups of three each day, and each person is in exactly seven groups, one for each day.

Then the constraint is that no pair of people are in the same group more than once. This problem has the additional constraint that the seven groups each person is in all occur on different days – the groups form a partition of the people each day, but the groups themselves can be partitioned into seven days which contain each person once.

In cases where the design needs to have combinations of things occurring exactly once, such block designs are called **Steiner Systems** and are denoted S(t,k,n) where n is the total number of things, k is the size of each block, and t is the size of combination that has to occur exactly once. My puzzle is an S(2,3,7) system – and so is the diagram given here.

This shape is called the **Fano plane**, and is a visual representation of an S(2,3,7) system. It has seven points, which are arranged into lines – each of which passes through exactly three points, and any two points you can choose lie only on one line.

If you want an easy way to solve the seven shapes puzzle, or you’ve solved it and want to compare your solution to this structure, you could arrange each set of three shapes so it sits on one of the seven points – in such a way that each line has the three pieces of that shape sitting along it.

It’s possible to construct Steiner systems of the form S(2, q+1, q²+q+1) – the Fano plane is for the case q=2. There’s a chance you may also have encountered equivalent structures for other values of q – if you’re a fan of children’s card games.

Dobble (sold in some places as Spot-It!) is a simple picture-spotting game, which comes in a bright yellow and purple tin. It consists of a set of circular cards, each of which is covered in pictures of familiar objects (a car, a key, a turtle, etc.) – eight objects appear on each card, and there are various ways to play, but all involve comparing two cards and finding an object which appears on both cards.

Of course, if this game weren’t mathematically interesting, I wouldn’t be talking about it – and in fact, for any pair of cards in the game, there will always be a symbol, and exactly one symbol, which appears on both cards. In a way which has no impact whatsoever on children’s enjoyment of spotting and matching and shouting out the name of the symbol they’ve found in common, the game has a beautiful underlying mathematical structure.

Since there are 8 symbols on each card, and 57 symbols in total, it’s clear that this is an S(2,8,57) system (S(2, q+1, q²+q+1) where q=7). It also means that each symbol occurs on exactly 8 of the cards, and a quick rifle through the set I have confirms that there are only 8 cards with the turtle on.

There are various versions of the game available, including Beach Dobble – with waterproof cards, designed for use at the beach. Since this was clearly designed to be a portable version of the game, it’s got a smaller deck – but it’s still a Steiner system! Each card has 6 symbols on it, so we must have q=5; and indeed, there are 5²+5+1 = 31 different symbols, with each symbol appearing on 6 cards.

Since the whole thing is symmetrical, it’s also possible to construct 31 cards to go in this set – and in standard Dobble, there are 57 possible cards you could construct using the 57 symbols. However, for reasons unknown to mathematics, and presumably related to printing constraints, the makers of Dobble have decided to only include 55 cards in the game! Beach Dobble, with 31 possible cards, only includes 30.

This means there are two more combinations in standard Dobble, and one in Beach Dobble, which are missing from the set. Perhaps they wanted to give mathematicians the gift of an additional puzzle – to find which cards they are missing. If you’ll excuse me, I’m going to find count how many occurrences of each symbol there are on the cards, determine what the missing ones should be and design and print my own replacements. Hours of fun for all the family!

- Read Christian’s blog post about 3D printing the puzzle, including a link to 3D printing files you can use to print your own.
- One solution to the ‘six people, three pairs five ways’ puzzle from the top of the post: AB CD EF / AC BE DF / AD BF CE / AE BD CF / AF BC DE

Der Beitrag Building Blocks erschien zuerst auf Heidelberg Laureate Forum.

]]>Often at this time of year, people are visiting family, and sometimes traditional games and toys come out to occupy your time. If you’ve got a deck of playing cards handy, you can perform a …

Der Beitrag A mathematical card trick erschien zuerst auf Heidelberg Laureate Forum.

]]>Often at this time of year, people are visiting family, and sometimes traditional games and toys come out to occupy your time. If you’ve got a deck of playing cards handy, you can perform a trick to amuse people – and since it’s a mathematical card trick, you don’t need to be a magician to make it work!

- Count out 27 cards from your deck, and put the rest away.
- Ask a volunteer to shuffle the 27 cards, pick one, look at it, and return it to the stack (in its original position, or anywhere they like) – all without you seeing the card.
- Deal the 27 cards into three face-up piles (placing one card in each pile, then repeating this 9 times). Ask your volunteer to watch for the card they picked – once you’ve finished dealing all the cards, they should tell you which pile it’s in.
- Pick up the three piles, and return them to a stack of 27 cards, but making sure you place the stack your volunteer indicated
**in between the other two**. - Repeat steps 3 and 4 two more times, so you’ve done the process three times in total.
- Count cards off from the top of the deck and turn over the 14th card (using the excuse that 13 is an unlucky number, so 14 must be a lucky number) – this should be your volunteer’s card.

You can try this trick and make sure it works – if it goes wrong, it’s usually a miscount (check you actually have 27 cards!), or a step somewhere that’s gone wrong, so try again or ask a friend to watch and make sure you’re doing it right.

Given that you know this is a mathematical card trick, it should make sense that as long as all the steps are followed correctly, you end up with the same result every time. But this trick has some interesting underlying mathematics, which can be used to perform an even more impressive version of the trick. First we need to understand what’s happened in our 14th-card version of the trick. Imagine your volunteer indicates the card is in the left pile.

When the cards are re-stacked, the pile containing your volunteer’s chosen card is placed in the middle. This means that when you deal the second time, you know that the card you’re looking for isn’t in the first nine cards, or the last nine cards. Since we deal the cards across the three piles, this means your chosen card won’t be in the bottom three or top three cards of any of the three piles. (Think about this a bit more if you’re not convinced.)

Now imagine your volunteer indicates the right-hand pile. This tells you which set of three cards (from the middles of the three piles) the card is in – and this set of three, if you place the pile in the middle again, will spread itself out as the middle card in each of the three piles on the third deal.

If the pile pointed out to you the third time is then placed in the middle, the card will be the middle card in the middle pile of three, or in other words, in the middle of the whole stack – which makes it the 14th card. Magic!

So the trick works, and it makes sense. True mathematicians will at this point become curious – we placed the pile in the middle each time, but we didn’t have to do that. It could equally have been placed on the top or bottom, on each of the three deals. So what would happen if we did that?

A little thought, and running through the trick with different combinations, reveals the following observations:

- Placing the chosen pile on the top each of the three times makes the card finish on top of the deck
- Placing the chosen pile on the bottom each time makes the card finish at the bottom.

With a bit more investigation, the following observations become apparent:

- Placing the pile on top after the third deal will always result in it ending up in the top third of the deck; similarly placing it in the middle third means it’s in the middle third of the deck, and the same for the bottom third
- Within each third, the position of the pile after the second deal will determine which third of that third it’s in (top, middle or bottom)
- Within that third-of-a-third, the position of the pile after the first deal determines whether it’s the top, middle or bottom of those three cards.

This information can be summarised in a table – here the ‘final position’ is the number of cards that will be on top of the chosen card at the end, with 0 cards on top of the top card, and 26 cards on top of the bottom card.

First deal | Second deal | Third deal | Final position |

top | top | top | 0 |

middle | top | top | 1 |

bottom | top | top | 2 |

top | middle | top | 3 |

middle | middle | top | 4 |

bottom | middle | top | 5 |

top | bottom | top | 6 |

middle | bottom | top | 7 |

bottom | bottom | top | 8 |

top | top | middle | 9 |

middle | top | middle | 10 |

bottom | top | middle | 11 |

top | middle | middle | 12 |

middle | middle | middle | 13 |

bottom | middle | middle | 14 |

top | bottom | middle | 15 |

middle | bottom | middle | 16 |

bottom | bottom | middle | 17 |

top | top | bottom | 18 |

middle | top | bottom | 19 |

bottom | top | bottom | 20 |

top | middle | bottom | 21 |

middle | middle | bottom | 22 |

bottom | middle | bottom | 23 |

top | bottom | bottom | 24 |

middle | bottom | bottom | 25 |

bottom | bottom | bottom | 26 |

This means we have a pattern we can use for finding the combinations of top, middle and bottom we need to put the card in a specific place at the end of the trick. It might seem a little intimidating to memorise all these combinations, but of course mathematics has the answer.

If you’re familiar with binary numbers, you’ll know that it’s possible to represent any number using only ones and zeroes – base 2, which means every digit is one of two options. But if we wanted three options, that’s also possible – using base 3. Ternary numbers, as they’re known, are a way to write any number as a sum of powers of three, and each digit is either 0, 1 or 2.

The table below shows how to write numbers from 0 to 26 in ternary – and it bears a striking resemblance to our other table above – replace 0 with ‘top’, 1 with ‘middle’ and 2 with ‘bottom’ and you have the same table! The columns represent how many 1s, 3s and 9s (reading from right to left) are added together to make each number – for example, 7 = 3+3+1 so it’s written 021, and 12 = 9+3 so it’s written 110.

1s | 3s | 9s | Total |

0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 |

2 | 0 | 0 | 2 |

0 | 1 | 0 | 3 |

1 | 1 | 0 | 4 |

2 | 1 | 0 | 5 |

0 | 2 | 0 | 6 |

1 | 2 | 0 | 7 |

2 | 2 | 0 | 8 |

0 | 0 | 1 | 9 |

1 | 0 | 1 | 10 |

2 | 0 | 1 | 11 |

0 | 1 | 1 | 12 |

1 | 1 | 1 | 13 |

2 | 1 | 1 | 14 |

0 | 2 | 1 | 15 |

1 | 2 | 1 | 16 |

2 | 2 | 1 | 17 |

0 | 0 | 2 | 18 |

1 | 0 | 2 | 19 |

2 | 0 | 2 | 20 |

0 | 1 | 2 | 21 |

1 | 1 | 2 | 22 |

2 | 1 | 2 | 23 |

0 | 2 | 2 | 24 |

1 | 2 | 2 | 25 |

2 | 2 | 2 | 26 |

We can decide the position we want the card to be in at the end of the trick, look it up in the table, and get a pattern of top, middle and bottom (0, 1 and 2) to put the card in the right position.

As we’ve seen, 111 (middle middle middle) gives 9+3+1=13 cards on top, putting it as the 14th card. 201 (bottom top middle) gives 9+9+1=19 cards on top, putting it as the 20th card. In fact, you could start the trick by asking someone to pick a card, and also casually ask them their age (if you think it’s likely to be less than 27, and not cause offence!) or to pick their favourite number between 1 and 27 – then that would be the number you count to at the end of the trick.

If you put the piles together casually enough, nobody will notice you’re stacking them in a particular order, and the effect can be very impressive. Mathematical magic tricks like this are favourites among professional magicians, as they don’t require any sleight of hand, and always work correctly if you get all the steps right. I personally like this trick because it’s a wonderful application of a nice bit of mathematics!

Der Beitrag A mathematical card trick erschien zuerst auf Heidelberg Laureate Forum.

]]>Here are the solutions to the eight candle puzzles we posted earlier. The candles cost €1.05, and the matches cost €0.05. Every 30 seconds the number of people whose candles are lit doubles, starting with …

Der Beitrag Candle puzzles – solutions erschien zuerst auf Heidelberg Laureate Forum.

]]>Here are the solutions to the eight candle puzzles we posted earlier.

- The candles cost €1.05, and the matches cost €0.05.

- Every 30 seconds the number of people whose candles are lit doubles, starting with 1 person at time 0. After 30 seconds, 2 = 21 candles are lit, after 60 seconds 4 = 22 candles are lit, and so on. After 7 30-second gaps, 27 = 128 candles will be burning, so it will take until 8 lots of 30 seconds (4 minutes) have passed for all the candles to be lit.

- This can be achieved by lighting one of the candles at both ends, and allowing it to burn from both ends at the same time – the flames will meet in the middle after 30 minutes, at which point you can light the second candle and let it burn for an hour. (This does assume that the candles burn at the same rate regardless of which way up they are – in reality, if a candle was held upright and lit at both ends, the bottom end would burn more quickly. If you hold the candle horizontally it might also burn at a different rate than it burns from the top if held upright. But it’s a nice puzzle!)

- If you place three of the candles on each side of the scale, this will indicate which group of three the heavy candle is in: if the scale tips one way or the other, it’s in that group of three, and if it balances exactly, it’s in the three candles you didn’t use for this weighing. Take the group of three that the heavy candle is in, and place one candle on each side of the scale – then the same principle will apply to tell you which is the heavy one.

- After burning 100 candles you’ll have 10 candles’ worth of extra wax; this lets you make 10 more candles, each of which will yield another 1/10 of a candle of wax – so you could make 11 extra candles in total.

- If you blow on each candle once, starting with the one nearest to you and going round one at a time, when you get back to the start you’ll have blown out all the candles! This is because they each get blown on three times (three candles change from lit to unlit or vice versa on each blow), which is an odd number of times, so they’ll go from lit to unlit eventually.

- The number of candles you will have blown out is the sum of all the ages you’ve been – when you’re five you’ll have blown ou 1+2+3+4+5 = 15 candles. These are triangular numbers, and the general pattern is N×(N+1) / 2.

- If you have five 1s and five 0s, the first time you’ll hit a problem is when you need six of the same digit – which will happen for the first time when you try to write the number 111111 = 1+2+4+8+16+32 = 63.

Der Beitrag Candle puzzles – solutions erschien zuerst auf Heidelberg Laureate Forum.

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