Der Beitrag The Nobel Prize for Mathematics erschien zuerst auf Heidelberg Laureate Forum.

]]>However, the origins of the HLF are inseparably linked to the existence of the Lindau Nobel Laureate Meetings, which took place from June 30 – July 5, and celebrates the laureates of a much more famous and highly-regarded set of prizes – the Nobel Prizes. Known throughout the world as the pinnacle of scientific achievement, Nobel prizes are awarded in Chemistry, Physics, Medicine and Economics (as well as Literature and Peace). But no specific prize exists for maths, or for computer science – which meant mathematicians and computer scientists couldn’t make the journey to Lindau to celebrate with their peers. The HLF was created to remedy this, by recognising equivalent levels of achievement through the other awards.

It’s not clear why the Nobel prize categories specifically exclude maths and computer science – untrue rumours circulate about the originator of the prizes, Alfred Nobel, refusing to include maths as a prize category due to a grudge against a prominent mathematician, who his wife supposedly had an affair with. This is easily called into question, given that Nobel himself never married. Several sources surmise that the real reason is more likely to be either that Nobel himself didn’t rate mathematics enough to award a prize for it, or that he was aware of existing prizes in maths being given out by the King of Sweden at that time, so didn’t think it was worth including.

But even though the Nobel prize isn’t awarded for mathematics, that doesn’t mean it’s never been awarded to a mathematician. Here are some examples of mathematicians who’ve managed to bend the rules (isn’t that what maths is all about really?) and, either individually or jointly with someone else, be awarded one of Alfred’s coveted gold medals.

Given that there is a Nobel in economics – formally, the Nobel Memorial Prize in Economic Sciences – which as a discipline makes heavy use of mathematics, you’d imagine there might be some mathematicians who have managed to win this. One of the most famous Economics Nobel winners who’s broadly considered to be a mathematician is **John Forbes Nash**, who won in 1994 for his work in the 1950s on the emerging field of game theory. He remains the only person to have won both a Nobel prize in Economics and an Abel Prize.

Game theory is the study of the mathematical structures behind situations – the way incentives motivate people to make decisions, and how rational people might choose to behave given particular conditions. The most famous example of the type of reasoning involved is the prisoners’ dilemma – a setup in which two people have to choose whether to cooperate or be selfish, with different rewards or punishments for each. It’s often used as a model, not just for simple game-based situations, but for studying behaviour in economics, and even international politics.

Nash was jointly awarded the Nobel for his work on the concept of Nash Equilibrium: studying particular states within a game where each player, in full knowledge of the other players’ strategies and motivations, has no incentive to change their strategy – the system has reached equilibrium. While such equilibria can be described by considering simple N-player games, the resulting conclusions about behaviour can be applied to larger, more complex systems, and be used along with probability – each player’s strategy might mean they would take different actions with given probabilities.

In addition to his work in economics, Nash did work in other areas of maths, including geometry and differential equations – but he’s best known for his study of game theory, as it has such broad applications. And he isn’t the only game theorist to have been given a Nobel in Economics for their work – **Robert J. Aumann** was jointly awarded a Nobel for Economics in 2005 for his work on conflict and cooperation through game-theory analysis.

Other mathematicians have also been awarded Economics Nobels: **Kenneth Arrow**, who was jointly awarded the prize in 1972 for contributions to economic equilibrium theory, is also well-known for a result called *Arrow’s Impossibility Theorem*.

This is a mathematical statement about voting systems, which says that if you have three or more parties to choose between, and would like to choose a voting system which meets a set of standard fairness criteria (including things like, adding in additional options to the poll shouldn’t change which of two existing options is in the lead), then no possible system can satisfy all these criteria. Arrow first published the theorem in his doctoral thesis, and it’s often quoted in discussions about voting theory.

A colleague of Arrow, **Gerard Debreu**, won an Economics Nobel in 1983 for ‘having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium’ – using fixed-point theorems from mathematics to make statements about economic equilibria. Another Nobel Economist from 1975, **Leonid Kantorovich**, applied mathematical techniques to the theory of optimal allocation of resources. Kantorovic widely is regarded as the founder of linear programming – the field of studying systems of linear relationships to find optimal values.

You’d probably also imagine that the Nobel in physics might have some overlap with the world of mathematics – **John Bardeen**, who has shared two Physics Nobels in 1956 and 1972, and **Max Born** and **Walther Bothe**, who shared one in 1954, are all considered to be mathematicians as well as physicists; Bardeen studied on the graduate program in mathematics at Princeton University, while Born had a PhD in mathematics, having studied at Göttingen with Felix Klein, David Hilbert and Hermann Minkowski.

Nobel-prize winning chemists can also be mathematicians – **John Pople** (1998) and **Herbert Hauptman **(1985) both did maths degrees before winning a Nobel in chemistry – and Hauptman’s prize was awarded for a mathematical approach to determining molecular structures.

Even outside of the scientific disciplines, mathematicians have still managed to win Nobels. Mathematician and philosopher **Bertrand Russell** was awarded the Nobel Prize for Literature in 1950 “in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought”. Russell was famously anti-war, and was sent to prison during World War I for has pacifism.

Within maths, Russell is well known for his work on logic and the fundamental principles of mathematics – *Principia Mathematica*, which Russell wrote with Alfred Whitehead and published in 1910, is an attempt to write out the basic axioms underlying mathematics in order to understand them fully. It famously contains a proof that 1+1 = 2, spanning several hundred pages (it was very thorough). *Russell’s paradox*, a classical mathematical paradox, considers a way of defining sets of things – and in particular, asks the question: if you define a set as containing all the sets that do not contain themselves – does that set then need to contain itself?

The message you’re hopefully getting here is that while mathematics can be considered a discipline in itself, and having awards for incredible achievements in mathematics is important, it’s actually a subject which crosses over with many other types of research and thinking.

Maths is a language that helps us understand and interpret what we see in the world around us – so people working in science need to at least get along with it, if not embrace it as part of their work. We hope the Lindau laureates had a great week at their celebration, and that they can continue to use mathematical ideas to make the world a better place.

Der Beitrag The Nobel Prize for Mathematics erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag The Lynx and the Hare erschien zuerst auf Heidelberg Laureate Forum.

]]>Given a situation in the real world, you might measure and make observations about what happens – counting the number of animals in a population at given time intervals, measuring the temperature of a chemical reaction, or collecting data about any variable which changes with time or given different initial conditions. But while measurements are useful to tell you about what’s happening now, they don’t tell you much about what’s going to happen in the future.

Mathematical modelling allows you to construct a model – a virtual version of the system, which you can use to predict how it’s going to evolve. You could, for instance, study your observations and notice a correlation between two things – when one of them increases, so does the other. If you’re studying the population of an animal (p), you might notice that at times when the amount of food available (f) is high, the population of the animal also tends to be higher.

Looking at the data you’ve collected would allow you to determine the coefficient, c, which describes the correlation – if the quantity of food goes up, how much does the population go up by? This coefficient would be part of your model, and allow you to predict what the effect of a given amount of food being added would be, on the size of the population.

You can use a correlation in a model even if the two things turn out not to be directly related. For example, it’s been observed that cows which are given individual names tend to produce more milk – but that doesn’t mean it’s the name that’s causing Daisy’s milk production to increase. It’s the presence of a third factor: a farmer that cares enough about their cows to name them, and who also looks after them well. (If you’re a fan of spurious correlations, with or without a connecting reason, there are some who make a hobby of collecting them). But you can still use an indirect relationship to make predictions – if you know the two things will increase in tandem, regardless of why, the model is still useful.

Of course, the model given above is very simple. There will almost certainly be other factors affecting the size of a population, and there might be a limit to how many animals can live in a given area – you can’t keep increasing the amount of food indefinitely and expect the population to always keep up, so your model might only work for certain ranges of values.

Despite all this, population dynamics remains one of the easiest types of system to model – the variables involved tend to be fairly simple, and other minor confounding factors often don’t have a large impact on the final values. Once nice set of equations developed for modelling the interactions between two species – the predator-prey equations, also called the **Lotka-Volterra equations**, give a lovely approximation to the dynamics of two species in competition.

Generally, for modelling an animal population, you need to take into account a number of factors, including the rate of reproduction – how quickly that particular animal creates new animals; and the rate of death, which might depend on the average lifespan, but also other hazards which might cause them to die – in particular, the number of predators in the area, and how many of their prey each predator needs to sustain themselves.

The Lotka-Volterra equations describe the relationship between two populations as follows:

In these equations, x is the number of prey animals in the population, and y is the number of predators. Δ here is a Greek letter which mathematicians use to represent change – so Δx is the change in x. What these equations are saying is that the rate at which the prey population changes will depend on the size of the prey population, but also on the size of the predator population; and vice versa.

The values of a, b, c and d in these equations will depend on the particular animals involved – their rates of reproduction, how frequently they prey on each other, how much food the predators need to live – observing them and recording some values will allow you to establish what these values might be for a given pair of species, and then you can use this model to make predictions of what will happen in the future.

Looking at the equation for Δx, you might notice the term which depends on the number of predators (y) is negative – meaning the more predators you have, the more this will reduce the prey population; in the Δy equation, the term dependent on the number of prey (x) is positive, because the more prey there are around, the more food for predators so the population will go up more quickly.

Since these two values both depend on each other, you can end up with some interesting results – if there are more prey around relative to the number of predators, that will increase the number of predators, which will in turn reduce the number of prey, which will have a knock-on effect on the number of predators, and so on. The graphs produced by these models often involve two similar-looking oscillating curves, one of which follows the other with a delay (as effects take time to manifest in the population numbers).

If the populations are nicely balanced when you start your model, this system can be relatively stable. (There’s obviously one other stable system you can have here, which is when x and y are zero, but that’s boring). For stability, you ideally want

as your starting states. Any changes to the system – maybe a flood which wipes out a chunk of the predator population, or an introduction of more animals from another area – will destabilise the equilibrium, and the equations will let you model what might happen.

One of my favourite things in maths is when something you’ve predicted comes true, and there’s a wonderful example of the patterns predicted by this model being exactly replicated in a real situation. Around the early 1900s, the Hudson Bay Company traded in fur pelts – from lynxes and hares trapped locally. What they might not have realised at the time was that in doing so, they were collecting data about the local hare and lynx populations – which were predator and prey to each other.

In the early 20th century, biologist Charles Gordon Hewitt analysed the data and discovered that the pattern of increase and decrease in the two populations matched the Lotka-Volterra model pretty well. The two populations were in equilibrium, with their sizes varying from year to year in a way that could be quite nicely predicted, and shows the characteristic delayed peaks of predator and prey.

This model is relatively simple, but mathematical modelling in general is a constant balancing act – if you can model the system with a simple set of equations, and not lose too much of the detail, that’s great. If your model is too simple, you’ll be able to compute your predictions quickly, but they’re less likely to be accurate, as you won’t have taken into account all the possible factors that could affect what’s going on. There’s a famous saying in statistical modelling: *“All models are wrong, but some are useful”*. The compromise you’re aiming for is a model which is the least possible amount of wrong, while still being reasonably useful.

Most real-world systems are more complicated than this simple population example, and couldn’t be captured with such a minimal model. Meteorologists spend their time predicting the weather, using many overlapping and detailed models – but a true model to accurately predict the entire weather system would need billions of variables and would probably take so long to compute, it’d be quicker just to wait and look out of the window.

Der Beitrag The Lynx and the Hare erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag The maths behind the Fields Medal erschien zuerst auf Heidelberg Laureate Forum.

]]>The front of the medal contains an image of Archimedes, along with the quotation “TRANSIRE SUUM PECTUS MUNDOQUE POTIRI” (“To transcend one’s human limitations and master the universe”). The back of the medal has another inscription, “CONGREGATI EX TOTO ORBE MATHEMATICI OB SCRIPTA INSIGNIA TRIBUERE”, which translates as “The mathematicians having congregated from the whole world awarded [this medal] because of outstanding writings.”

Behind this inscription is pictured an olive branch, and some lines and shapes. This design was chosen as it illustrates one of Archimedes’ favourite mathematical results – he was so proud of it, he asked for this design to be engraved on his tombstone. But what does it mean?

A version of the diagram without an inscription in front of it is shown below, next to a 3D rendering of what it’s trying to represent – it’s a sphere sitting inside a cylinder of the same height and radius.

You might be familiar with the formulae for the circumference and area of a circle – if the radius of a circle is denoted by r, we have that the circumference is 2πr, and the area of a circle is πr^{2}. These formulae are known to all school students everywhere, and form a fundamental part of our understanding of the geometry of circles.

Archimedes went one step (and one dimension) further, and wanted to know the formulae for the surface area, and the volume, of shapes based on circles, like spheres and cylinders, and how they relate to each other. In his work “On the Sphere and Cylinder,” published in 225 BC, Archimedes gave a variety of results, including ones about how to find the surface area and volume of a cylinder and a sphere.

The formulae for a cylinder are easy enough. The outer surface of the cylinder consists of two circles (each with area πr^{2}) and a curved surface whose height is the height of the cylinder, and which wraps around the whole circle – so its area will be the circumference of the circle times the height of the cylinder, or 2πrh (where h is the height of the cylinder). So the total surface area is 2πr^{2} + 2πrh, or 2πr(r + h).

The volume of the cylinder is similarly easy to work out – if you consider it as a circular prism, made by pushing a circle along a line and including all the space the circle passes through as part of the shape, it’s simply the area of the circle (πr^{2}) times the height of the cylinder, so the formula is πr^{2}h. If your cylinder happens to be the same height as it is wide, its height will be 2r, and so this formula would become 2πr^{3}.

For the sphere, things get a bit harder to visualise. Archimedes proved that the surface area of a sphere is four times the area of a circle found by cutting the sphere exactly in half (known as a Great Circle of the sphere). This means if the radius of the sphere is r, the surface area is 4πr^{2}. Such a neat result!

Now the challenge is to convince yourself it’s true. One way to imagine the surface area of a sphere is to imagine a circle that’s pivoted about its diameter, and rotate the circle fully around – the edge of the circle will sweep out the surface of the sphere. (If you’re struggling to imagine this, spin your Fields Medal on the table).

If we wanted to simplify this, we could reduce the rotating circle to a rotating polygon – imagine it’s, say, a 10-sided shape (decagon), sweeping out the same circle. Since polygons with increasing numbers of sides get closer and closer to an actual circle, this is a simpler version of the problem, made up of straight edges.

Rotating the decagon would produce an approximation to a sphere, made up of sections of cones and cylinders, each of which has a known formula for its area. From this you can get a formula for the surface area of the total shape, and if you repeat for other numbers of sides, you eventually reach a formula that depends on the number of sides the shape has – and as you increase the number of sides, this gets closer and closer to a total surface area of 4πr^{2}.

This mirrors a more modern approach to mathematical problem-solving – in the time of Archimedes, calculus was a distant dream, but it’s based on the idea of limits – letting a value increase until it’s effectively infinite, and seeing what the result gets closer and closer to. It’s also possible to derive the volume formula using modern calculus techniques, but Archimedes would presumably have been able to intuit the limit, even if he didn’t have a word for it.

This method is often given as Archimedes’ formal derivation of the proof, but it’s also suggested that he might have originally noticed this relationship by merely comparing the volumes of cylinders and their inscribed spheres, using physical methods.

Archimedes was, after all famous for jumping out of a bathtub (having realised you could measure volume by water displacement), so he must have had some tricks up his toga sleeve for finding the volume of odd shapes. A parchment discovered in the 20th century describes a method of determining volume, attributed to Archimedes, involving balances and centres of mass.

The result Archimedes was most pleased with, and the reason it was carved on his tomb, is that of the volume of a sphere. You might have learned by heart that the volume of a sphere is given by 4/3πr^{3} – but if you compare this with the volume of a cylinder which has the same radius and height as the sphere – one it would fit exactly inside – you might notice that these formulae are very similar, and differ only by a factor of ⅔. Even if you didn’t notice this pleasing fact, Archimedes certainly did, and he was correct to be proud of it.

It’s fitting that the Fields Medal, a celebration of the pinnacle of mathematical achievement, contains this beautiful example of a mathematical result so important and groundbreaking for its time. A result so simple it’s now used in high-school geometry classes was, in 225 BC, actually itself worthy of a Fields Medal (if such a thing had existed then). I bet today’s Fields Medalists wish they could win the award by proving a result this simple!

Der Beitrag The maths behind the Fields Medal erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag IMU Abacus Medal erschien zuerst auf Heidelberg Laureate Forum.

]]> This year, the IMU has decided to rename the prize, though it will have the ‘same purpose and scope as the Nevanlinna Prize’. This new award will be called the **IMU Abacus Medal**, and will be awarded at the International Congress of Mathematicians for the first time in 2022, and will continue to include a cash award of €10,000 and a medal.

The IMU’s press release describes their reason for the choice of this new name:

“The name IMU Abacus Medal relates to the abacus, an ancient device that was used for numerical computations, and it underscores the importance of calculations already in early mathematics. The exact place and time of origin of the abacus is unknown, and it can be considered a truly global artifact associated with mathematics and computation.”

The origin of the abacus is indeed unknown, and the word has also been used to describe tablets strewn with sand or sawdust, or covered in wax, used for mathematical calculation. Nowadays, it’s mostly understood to describe some kind of wooden rack of beads which can be used for counting and manipulating numbers. Abacuses have been used for centuries in countries all over the world, but rose to prominence in China and Japan, and were also popular in Europe and Russia.

Babies and small children play with toy abacuses, by sliding beads across to practice counting and learn about numbers – but they are also used by adults as a serious calculation tool, and until fairly recently the use of abacuses was taught in schools – and still commonly is in some places. While it might seem a simple device in this era of electronic calculators and computers, the abacus can perform many of the same functions as a calculator, and a trained operator can perform staggering feats of calculation using a simple rack of beads as a mental aid.

A child’s abacus, like the one pictured above, has ten beads on each bar, which can be slid across to denote the number of hundreds, tens and units in each digit of a number. But some types of calculation abacuses are more streamlined and simple to operate. A **soroban**, a type of abacus originating in Japan, can be used to add, subtract, multiply and divide numbers, and calculate square and cube roots. It consists of vertical rods each containing four beads in a lower section, and one in an upper section.

The soroban was developed from an earlier type of abacus from China, called a **suanpan**, which has two beads above and five below the divider – this still can be used for normal calculations in base ten, by leaving the top and bottom beads in place, but also in hexadecimal, which traditional Chinese units of weight were based on.

The 4+1 system of beads can display any number by an arrangement of beads, with each digit of the number corresponding to one column of beads. The number of beads pushed up to the divider could be 1, 2, 3 or 4; the bead above the divider represents a 5, and if it’s pushed down to the bar it means add 5 to whatever’s below. The digits 0-9 are represented as shown:

Addition is performed digit-by-digit, in the same way you would do it using a pen and paper. For example, if you’re adding together two numbers on paper, you first add the right-hand digits together and write the result in the right hand column; if the result is more than 9, you’ll have a digit to carry to the next column. On the abacus, you’d have a number set already – then, to add another number to it, you could work from the right, adding the number to the digit and increasing the next column along if you need a carry.

Subtraction is more difficult – on paper, subtracting two numbers sometimes requires you to ‘borrow’ some of the value of the next column. One way to achieve subtraction on an abacus is using the method of complements: substituting a digit for its value when you subtract it from 9. For example, 0 can be substituted for 9, 1 for 8, 2 for 7 and so on. This is sometimes called ‘9’s complement’, and is a way to subtract things while only performing addition. Subtracting two numbers can be achieved by taking the complement of the larger number, adding them together, and taking the complement of the result.

For example, if you wanted to calculate 832 – 491, you can take the complement of 832 (which is 167), add it to 491 (167 + 491 = 658) and then find the complement of 658 (which is 341, the answer we needed). If you’d like to take a minute to convince yourself why this works, please do.

Similar methods exist to perform multiplication and division digit-by-digit, which can be used with an abacus. It’s also possible to use an abacus to find square and cube roots.

In 1946 a competition was held to determine which was quicker at calculating – a human using a soroban, or one using an electric calculator. Each were challenged to add together 50 large numbers, and perform similar feats of subtraction, multiplication and division. The soroban outperformed the electric calculator on addition, subtraction, division and on problems combining multiple operations – the only category in which the electric calculator prevailed was in multiplication.

Another advantage of the abacus was that early calculators were limited to 8-10 digits, and a soroban can handle as many digits as you have columns of beads. Since the only calculations you can’t really do on an abacus are those of higher complexity, such as trigonometric functions, it was not considered to be superseded by the calculator as a tool until relatively recently.

Many people use abacus calculation as a way to improve their general mental agility – abacists (the term for people who use an abacus) have been shown to have improved memory, concentration, speed of thought and mental capacity. In parts of East Asia, there are soroban competitions, where participants compete to solve calculations faster and faster. Schools in Japan run after-school abacus clubs, where students train and practice for competitions.

The competitions even include mental calculation rounds, in which participants visualise an imaginary abacus and use it to perform calculations – adding together a series of numbers flashed up briefly on a screen, and only writing down the final answer. In 2012, a world record was set for the correct addition of 15 three-digit numbers each shown on a screen for only 1.7 seconds each, beating the previous record of 1.8 seconds.

Abacuses similar in design to a soroban, with a piece of rubber or fabric on the back to stop the beads from sliding down, can be used by visually impaired people to perform calculations by feeling the position of the beads. While talking calculators can be used for more complicated calculations, the abacus performs the same function as a pencil and paper would – allowing people to perform digit-wise calculations and keep track of what they’re doing.

As a symbol of mathematics and computation, the abacus is a simple form of technology, integrated with mathematical ideas, which has revolutionised the way people interact with mathematics. The IMU Abacus Medal will continue to recognise the achievements of those who are creating and working with the modern equivalent of the abacus – using hardware and software, combined with the power mathematical ideas, to change the world.

Der Beitrag IMU Abacus Medal erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag Stairs and generalisation erschien zuerst auf Heidelberg Laureate Forum.

]]>Given a real-world physical system, or a particular instance of a puzzle or question, it’s often easy enough to put all of the values into a calculator and find a solution to the problem directly in front of you. For example, if I wanted to walk up a flight of stairs which has 4 steps, and I’m capable of stepping up one or two steps at a time, how many different possible walks up the stairs could I achieve? (Here, I’m assuming that it doesn’t matter which foot I step up with first, and that two walks up the stairs are considered the same if I step on the same treads).

I could take the first two steps together, then the second two together as well, and I could denote this walk up the stairs by (2,2). I could start with a double step, then two singles – (2,1,1). Or I could do two singles first, then a double step: (1,1,2). If I wanted to mix things up, I could do a single step, then two together and a single after that, giving (1,2,1). Finally, I have the boring grown-up option of taking all four stairs separately, denoted (1,1,1,1).

This puzzle is in itself fairly straightforward, and can be solved simply by checking all the cases. If I start with a double step, there are two options for how to finish (another 2, or two 1s). Then I can check what happens if I start with a single, and cover all the possibilities from there. The answer I get is 5 – five possible ways to walk up the stairs.

Of course, a mathematician faced with this problem is probably already thinking of other ways to approach it. If the four-step case seems daunting at first, what happens if we have only three steps? Or two, or one? If we treat one of these simpler cases first, do they give us an insight into what happens for larger staircases?

You might be thinking about the interesting fact that if you work out how many ways there are to walk up a three-step staircase, you can then add a single step at the start or end of each of these and get some of the four-step solutions. This tells you that if you increase the number of stairs by one, the number of ways to walk up the stairs will be at least as many as it was previously – but not necessarily twice as many, since adding a single one to the start and end of the sequence (1,1,1) gives you the same result.

You might even now be wondering if there’s a rule for which 3-step sequences will give different 4-step sequences if you add a single step at the start or end, or in the various points in the middle of the sequence – and for a given sequence, how can you work out how many ways there are to add a single step in the middle? And what about increasing the length by switching a single step for a double?

Someone who’s really thinking mathematically, might, despite me not having even suggested such a thing, be trying to work out what the answer to this question is in general, for a staircase of N stairs. Part of the game of thinking like a mathematician is making these kinds of mental jumps – from a specific case to a general case. This process is called **generalisation**, and is used to turn a single instance of a problem into a whole family of problems – by changing the value of a number, or some other property, in the original setup.

Modelling real-world problems is rarely as neat and whole-number-answer as this – but it’s often still possible to generalise a problem. If this bridge can take a certain amount of weight, what happens if I increase its span, or the thickness of the metal bars it’s made of? What would happen to this animal population if I changed the initial numbers of predators and prey? Sometimes curiosity is enough to motivate generalisation, but sometimes it’s useful to produce a model that can be deployed again when circumstances change.

While the staircase problem seems simple for small numbers, it might get very complicated if the numbers get much bigger, and making sure you’ve counted all the possibilities might get messy. If you can work out a solid way to get from the number of ways to walk up the N-step staircase to the number of ways for the N+1-step staircase, you can use mathematical induction – work out the answer for a small number of steps (e.g. one step, for which there’s exactly one way to do it) and use that to find the number of ways for two, then put that answer in to find the answer for three steps, and so on.

Maybe you’ve had a brilliant intuition about how the problem works, and found a guaranteed way to count all the possible paths up the stairs for a given number of steps. Then you probably don’t even need to try counting all the ways to walk up four steps, and when presented with the original problem, you immediately come up with a general formula and put in the value 4 to see what happens.

But not everybody can make that kind of intuitive leap. One more way to attack this kind of problem and get to a general result is to try the first few cases, and see what happens – down where the numbers are countable, for 1, 2, 3 and 4 steps, and maybe even 5 if you have a little patience. Once you have a few values, you can see if there’s any kind of sensible pattern in those numbers and maybe recognise something you vaguely remember having seen before.

Even if it’s not eerily familiar, maybe you can see a pattern in these numbers which might continue, that you can use to make a prediction about what will happen next. Then, if you can be bothered to count all the possibilities for 6, scribble them all down on a nearby bit of paper, and see if it matches your prediction (but of course, like a good thinker, don’t stop looking for new combinations once you’ve found as many as you predicted would exist; only stop when you’re completely sure you’ve checked all the possibilities, and don’t fall into the trap that so many have fallen into before of assuming your prediction is correct before you’ve completed the process of checking it).

Even though the mathematics involved here is fairly straightforward, and could probably be understood and worked out by an inquisitive child with a particular obsession with finding new ways to walk up the stairs (for the record, having an actual set of stairs to try this out on is unnecessary, but does add in an exciting physical element to the puzzle) – if you’ve made it this far and found yourself doing any of the things I’ve attributed to a mathematical thinker, then congratulations. You’re a mathematician, and there’s nothing you can do about it.

Der Beitrag Stairs and generalisation erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag Combinatorics Puzzles erschien zuerst auf Heidelberg Laureate Forum.

]]>Of course, once you reach higher levels of study, mathematics is revealed to be a rich, varied subject with great subtlety. Combinatorics is counting, but a bit harder.

One example of a problem in combinatorics is one we’ve already seen – in my post last November, on permutations.

Puzzle:if ten people are attending a party and want to stand in line for a photo, how many different orders could they stand in?

If you’d like to take a minute now to think about this, and carry on reading once you have an answer, you should do so now. The problem more generally can be stated as, if we have N objects, and want to put them in order in a line, how do we count the number of different ways we could do this?

Having once asked an audience to think about the number of different ways to shuffle a pack of cards, and being told there a several – a riffle shuffle, an overhand shuffle, a faro shuffle, the Mexican spiral shuffle, and so on, I’m always careful in how I word the question. But hopefully you understand that I mean, ‘How many different orderings could I place the objects in?’.

The answer, as disclosed in the previous blog post, can be found by thinking about how many options I have for each of the items in order – when choosing which item to put first, I have N choices from my N objects; then once I’ve chosen which one goes first, I have N-1 choices for the second, and then N-2 choices for the third, and so on. So the total number of possibilities is N × (N-1) × (N-2) × … × 1, also known as ‘N factorial’ and written N!. For 10 people, 10! = 3,628,800.

This can be applied in any situation where objects can be placed in different orders – but it also comes in to other problems, as we will see.

Puzzle:if the same ten people are at a party and everyone wishes to shake hands with everyone else in the room, how many handshakes will take place?

Again, take a moment to convince yourself of an answer to this. The question we’re asking here is, how many pairs of people are there in a room of 10 people (or, generally, how many pairs of objects are there in a set of N objects)?

One way to think about this is to realise that each of the 10 people in the room will shake hands with each of the other people in the room – which is 9 people. So, you might think the number of handshakes would be 10 × 9 = 90. But this is too many.

In fact, we’ve counted every handshake twice – once from each of the two people involved. So we can divide this number by 2 and get 45, which is the number of handshakes. In general, for N people, we’d have (N×(N-1))/2 handshakes.

Another way to approach it is to imagine that at the party, everyone is shaking hands immediately having stood in line for the photo. It doesn’t matter what order they’re in, but whoever’s at one end of the line could walk along the line and shake hands with the other 9, then go and sit down. Then the next person, who’s already shaken hands with the one person who’s sitting down, merely has to shake hands with the remaining 8 people in the line; then 7, then 6 and so on. So our total will be 9 + 8 + 7 + 6 + … + 1 = 45.

The fact that the number of pairs in N things matches with the sum of all the numbers up to N-1 is true in general, and a useful coincidence. There’s a story (possibly slightly exaggerated) about the young mathematician Gauss, who was annoying a teacher by being too good at mathematics and interrupting the class with clever ideas. The teacher angrily set Gauss the task of adding up all the numbers from 1 to 100, to see if that would give them some respite – but Gauss worked out the general formula, and realised that it was simply (101×100)/2 = 5,050 and answered immediately. Grr!

Puzzle:Now imagine the 10 people at a party would like to go home, and a taxi arrives with space for exactly four people. How many different sets of four people could be chosen to get in the taxi?

This problem is slightly more complicated than the previous examples – and in some ways combines both of them. Like our ordering problem, we can still approach it by thinking about how many options we have at each stage; with 10 people, we could choose from 10 to put in the first seat, then pick one from the remaining 9 to go in the second seat, then 8, then 7; so for the four seats, we’d have 10 × 9 × 8 × 7 = 5,040 ways to choose four people.

This isn’t strictly a factorial – it does count down and multiply the numbers together, but it stops before going all the way down to 1. However, this is a product of things multiplied together, so it can be thought of 10! ÷ 6!. If you were to write both factorials out in full in a fraction, with 10! on top and 6! below, you could see that the last 6 terms would cancel top and bottom, leaving behind the result we want.

So, to choose 4 (= 10 – 6) people from 10, we could calculate 10!/(10-4)!, or in general, to choose K people from N, N!/(N-K)!. But if this doesn’t match up with your answer, because you carefully wrote out and counted all the possibilities, that’s because we’ve missed one thing.

If we were to choose four people (say, A, B, C and D) to go in the taxi, this would be because person A was picked first from the full set of 10, then B picked from the set of 9, and so on. However, if by chance we had picked B from the full set of 10, then A from the set of 9, we’d have B, A, C and D going in the taxi – which is exactly the same outcome as A, B, C and D.

In this case, given the four people are getting in a taxi and it doesn’t matter what order they’re in, different orderings of the same set count as the same outcome. Luckily, we already have a way to deal with this! If we have four people, there are 4! = 24 ways to order the set, so we could divide our total by this, to find the number of unique sets to choose. Now our formula is 10!/(4! ×(10-4)!) (here, the 4! goes on the bottom of the fraction), giving the answer 5040/24 = 210.

The general formula is then N!/(K!(N-K)!). While this does look like an attempt to write down the noise a goose makes, it’s a very useful and important formula in mathematics – the function is sometimes called the Choose function, and called ‘N Choose K’ – the number of ways to have N things, and choose K of them.

It’s sometimes also called the Binomial formula, as it crops up in the coefficients of binomial expansions like (a + b)^{n}, and by association, the numbers in Pascal’s triangle (mentioned, and briefly explained, in my December post).

With these simple tools, mathematicians (specifically, combinatorics…-acists?) have the power to count and enumerate increasingly complicated sets of things.

*How did you get on with the puzzles? And have you ever had combinatorial problems like these confront you in your day-to-day life? You might!*

Der Beitrag Combinatorics Puzzles erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag Karen Uhlenbeck: 2019 Abel Prize Laureate erschien zuerst auf Heidelberg Laureate Forum.

]]>Karen Keskulla Uhlenbeck was born in 1942 in Cleveland Ohio, and received her B.A. in 1964 from the University of Michigan. She went on to study for an MA and PhD at Brandeis University in Waltham, Massachusetts, and thereafter worked at a range of other institutions, including University of Texas at Austin where she is now an emeritus professor.

She has been awarded the Abel prize after a long and productive career in mathematics. The prize announcement from the Norwegian Academy of Science and Letters reads:

“Karen Uhlenbeck receives the Abel Prize 2019 for her fundamental work in geometric analysis and gauge theory, which has dramatically changed the mathematical landscape. Her theories have revolutionized our understanding of minimal surfaces, such as those formed by soap bubbles, and more general minimization problems in higher dimensions.”

Uhlenbeck’s work concerns the calculus of variations – studying functions, and how they behave, by finding their minima and maxima – changing the input value a small amount to see what happens to the output. It can be used to find the shortest path between two points when constrained to the surface of an object, or the shape a cable will make if left to hang, as well as problems concerning elasticity, solid and fluid mechanics, electromagnetism and gravitation. Calculus is particularly useful when considering and modelling physical systems, and Uhlenbeck worked in the crossover between calculus and geometry.

Geometric analysis connects differential geometry – studying 3D shapes and surfaces by considering the functions that define them – with differential equations. It was founded by Uhlenbeck and her collaborators in the early 1980s, when they published a paper on minimal surfaces, setting out tools and ideas that became the foundation of the topic. A minimal surface is one in which the surface area is minimal given the constraints.

A nice example of a minimal surface is given by soap bubbles. In the absence of any constraints, bubbles will naturally form a spherical shape – one with minimal surface area – but if the bubble is attached to a surface, or a wire, the minimal surface can look different. Depending on the situation, it can be quite hard to mathematically describe what such a surface looks like, and Uhlenbeck’s work included ways to deal with singularities (non-smooth points) on such surfaces.

Karen’s initial major was in physics, and her work often crosses over between maths and physics. After hearing a talk by fellow Abel prize winner and HLF regular, the late Sir Michael Atiyah, Uhlenbeck became interested in **gauge theory** – the study of forces such as electromagnetic charge, by making changes to related variables and observing (gauging) the result. Some of Uhlenbeck’s most noted work has been on this topic, and laid the groundwork for some of the most exciting ideas in modern physics.

In her life, Karen Uhlenbeck has made huge contributions to many areas of maths and physics, and has been recognised for her work with many awards – the latest of which is her Abel prize. As well as the 2007 Steele Prize for a Seminal Contribution to Research, she has been awarded a MacArthur Fellowship in 1983, and the National Medal of Science in 2000. She was elected to the U.S. National Academy of Sciences in 1986 and to the American Philosophical Society in 2012.

Uhlenbeck is keen to pass on her love of mathematics through outreach. As part of her work as a visiting senior research scholar at the Institute for Advanced Study in Princeton, she has co-founded the Park City Mathematics Institute, a three-week residential summer conference held in Utah for students, researchers and maths teachers, as well as being involved in the IAS’s Women and Mathematics Program.

She also gave the plenary lecture at the International Congress of Mathematicians in Kyoto in 1990, on Applications of Non-Linear Analysis in Topology. The only other ICM plenary given by a female mathematician was in 1932, when it was given by Emmy Noether (who I wrote about back in February).

It’s a sad state of affairs that in the ICM’s entire history, only two women have ever been their plenary speaker, but it’s not a surprise given how few women are included in the higher ranks of mathematics. While Uhlenbeck was considering where to study for her MA and PhD, she deliberately chose against prestigious schools such as Harvard, as she knew the competition there would be fierce, and this illustrates one of the many barriers women face in mathematics.

Karen Uhlenbeck has created a wonderful legacy of mathematics, as well as being a strong advocate and role model promoting greater gender diversity in mathematics and in science. Maybe given time we’ll see more female ICM speakers, and the world will have more fantastic mathematicians like Karen Uhlenbeck.

Der Beitrag Karen Uhlenbeck: 2019 Abel Prize Laureate erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag How to drink infinite beers without getting too drunk erschien zuerst auf Heidelberg Laureate Forum.

]]>Mathematically, this joke is very funny – but sadly, outside of mathematics, that doesn’t always correspond with jokes being actually funny. My suspicion would be that in order to get this joke, there’s a bit of mathematical information you’ll need first.

The series of drinks orders the mathematicians make might seem strange – certainly, in the UK, it’s normal to order a pint of beer, and a half-pint is also a pretty standard measure. Some places which serve particularly strong ales will also offer a ¼-pint option, but there aren’t many places you can get away with ordering an eighth of a pint, and assuming that the pattern was going to continue in the obvious way, none of the other drinks orders would be standard either.

The pattern goes:

1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 …

Each time, the number on the bottom of the fraction doubles, meaning the amount of beer being ordered halves. We can define the sequence as *1/2** ^{n}*, for values of

Let’s assume, for the sake of argument, that it’s possible to pour each of these measures, and that you can do so with a reasonable degree of accuracy – this might be difficult, since the tenth mathematician’s order is 1/512 of a pint, which is just under 1 millilitre. From experience, measuring quantities this small is quite difficult, especially with liquids like beer, where more than 1 millilitre of it will stick to the side of the glass while you’re pouring.

But this is maths, and we can conduct mathematical experiments (and jokes) in a perfect hypothetical world, where properties like surface tension and fluid adhesion don’t factor in. So, we have our measures of beer, which carry on forever in increasingly tiny glasses – each mathematician still orders a drink, even if it is a very small one for all but the first few – and the barkeeper seems to think they’ll be satisfied with just two pints of beer. Surely not?

The first mathematician’s drink accounts for one of the pints, so the second pint glass is where all the action takes place. Starting from an empty pint glass, we can pour in the half-pint, leaving exactly half a pint of space. Then the next drink is a quarter pint, filling half the remaining space and leaving a quarter pint gap – and so on.

As each drink is added to the glass, it takes up precisely half the space remaining in the glass – and this will always continue, no matter how many drinks we add. So the infinite series of drinks ordered, lined up along an infinite bar stretching along into the distance, can all fit into this one glass.

I wasn’t kidding when I said this was something I know from experience – I’ve performed this trick on stage, at comedy shows and on TV, in order to demonstrate this idea. Seeing it physically happen in front of you makes it much more comprehensible that you can have an infinite number of drinks which add up to a finite number. The trick is that the numbers are getting smaller, and doing so quickly enough that they get closer and closer to a finite number, without actually reaching it.

There are many examples of series like this, which are said to **converge** – that is, the sequence of sums of successive numbers of terms get closer and closer to a particular finite value. Convergent series include, for example, the series of sums of any sequence where the number on the bottom is an increasing power of the same fixed value. In general, the series of sums of numbers of the form *1/k*^{n}*,* where k is a whole number, converges to *k/(k-1)*.

If you take a sequence like the **triangular numbers** (numbers of dots that can be arranged in a triangle, like pins in a bowling alley, including 1, 3, 6, 10, 15, and so on) and consider the series of fractions 1, 1/3, 1/6, 1/10, 1/15 …, these sums also converge to 2. And the series of sums of fractions of **1 divided by a square number**, 1/n^{2}: 1, 1/4, 1/9, 1/16, 1/25 and so on, converges to π^{2}/6 (of course). But even though these series keep getting closer and closer to this value, they’ll never actually reach it – there’s always a little gap left in the top of the pint glass.

In fact, the only way the second pint glass will actually be 100% full is if we do continue pouring drinks in to infinity – mathematically, we say that the **limit** of the series, adding the numbers together to infinity, is 2. This explains the additional punchline you can give to the original joke, in which that the barkeeper adds ‘The problem with you mathematicians is, you need to know your limits!’

Der Beitrag How to drink infinite beers without getting too drunk erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag What impedes science communication? Results of an Extensive Survey with Young Researchers erschien zuerst auf Heidelberg Laureate Forum.

]]>The demand for science communication to the public (the so-called external science communication) is currently becoming louder and louder against the background of the debates on “fake science” and populism. But by no means are all scientists responding to the call for science communication. So what impedes this sort of engagement? What do top young researchers – also considered the “next generation of professors” – see as obstacles to science communication? And how strongly do they agree with common prejudices such as that science communication is only something for the showmen among scientists or that it makes science itself more shallow?

From 2014 to 2018, we surveyed the participants of the Heidelberg Laureate Forum and the Lindau Nobel Laureate Meetings. After quality control, we were left with 988 complete datasets of young scientists. These scientists carried out their research in 89 countries and none were older than 35 years of age. 41.5% of the interviewees were women, while 57.5% were men. Fig. 1 shows the distribution by discipline.

In the two years preceding the survey, most had conducted their research in Asia, Europe, and the USA (Fig. 2). In the analyses, stratified by continent, we have essentially limited ourselves to these three well-represented groups.

In another blog post we analyzed the commitment to and the attitudes towards science communication to the public. As far as their personal commitment to science communication is concerned, we found that differences between continents and scientific cultures are greater than those between disciplines – always provided that the selection process of young researchers for the major events we investigated was based on the same criteria in all countries.

Apparently, whether a scientist works in Asia or in Europe makes a difference in how they speak about science with the public and what they think about science communication. Whether they are e.g. biologists or physicists did not matter as much. However, we also asked the participants if they agreed or disagreed with statements such as that “lack of time on the part of scientists is a great obstacle in communicating science.” Of three possible obstacles, the lack of time factor was clearly considered the most important. 65.2% of the young researchers surveyed totally or partially agreed with the corresponding statement, whereas only 17.2% partially disagreed or totally disagreed (Fig. 3). While the amount of those who agreed to this statement did not significantly differ between young researchers working on different continents, differences between researchers in different disciplines could be observed. Compared to researchers in other disciplines, mathematicians and economists see time pressure as less of an obstacle for science communication (however, more than half of them still agree that time trouble is an obstacle). In particular, the life scientists surveyed see the lack of time as an obstacle for science communication.

There was no clear consensus among the young researchers whether or not “the insecurity scientists have when dealing with laypersons is a great obstacle in communicating science.” 33.0% agreed, while 36.0% disagreed (Fig. 4). Researchers from Asia particularly see this insecurity as a problem. However, in this question, the differences between disciplines are also somewhat more prominent. Chemists, life scientists and engineers saw more of a problem here than physicists and computer scientists.

As another possible obstacle for science communication, we asked about a possible lack of public interest. Across the entire board, this statement was more strongly disagreed upon than agreed upon (42.1% to 32.7%; see Fig. 5). However, the answers differ considerably with subgroups. Young researchers from Asia and the USA saw greater disinterest in science among the public than their colleagues from Europe. And mathematicians, economists and chemists agreed much more with this statement than physicists and life scientists.

Some believe that a major obstacle in effectively conveying science is that scientists believe that “communicating science is mostly something for showmen.” We wanted to know whether the young researchers in our survey agree with that prejudice. The answer is: mostly not. Only 17.0% of the young researchers agreed, whereas a large majority, 59.2%, disagreed with it (Fig. 6). With 77.3%, researchers in the USA disagreed the most with this statement. When comparing the response behavior by disciplines, it is noticeable that the least resistance to the statement comes from scientists in the field of engineering.

Do public outreach and societal dialogue have a negative effect on science itself? Finally, we confronted the young researchers with the statement “communicating science leads to making science itself shallower.” A total of 67.0% completely or partially disagreed with this statement. They were opposed by only 11.6% who totally or partially shared this view (Fig. 7). 83.6% of young researchers who conducted their research in the USA disagreed with this statement, which is particularly high. Subdivided by discipline, economists agreed the most with 26.0%, who clearly stand out from their colleagues in all other disciplines. While one can assume that the scientific culture in the USA simply encompasses communicating science and thus induces the particular response behavior of researchers in the USA, reasons for the differing response behavior of the economists are difficult to discern.

In brief: When asked about obstacles to and typical prejudices against science communication, a lack of time can be identified as the strongest factor. Uncertainty in dealing with laypersons and a lack of interest on the other hand are far behind, although there are interesting disciplinary differences when looking at it in detail. In addition, our data suggest that prejudices against science communicators and lay audiences are rather less present in young researchers in the USA compared to their colleagues in Europe and Asia.

**Carsten Könneker** is editor-in-chief of “Spektrum der Wissenschaft”, the German edition of “Scientific American”. From 2012 to 2018, he headed the Chair of Science Communication and Science Studies at the Karlsruhe Institute of Technology. He also is the founding director (2012-2015) of the National Institute for Science Communication (NaWik) in Karlsruhe.

**Philipp Niemann** is the
scientific head of the National
Institute for Science Communication (NaWik).

**Christoph Böhmert** recently
completed his PhD at the Faculty of Humanities and Social Sciences at the
Karlsruhe Institute of Technology.

Der Beitrag What impedes science communication? Results of an Extensive Survey with Young Researchers erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag This Maths Puzzle is Baffling Facebook erschien zuerst auf Heidelberg Laureate Forum.

]]>One such puzzle is given below – using symbols to represent some unknown numbers, can you work out the answer?

In this case the symbols are burgers, fries and drinks; the same problem has also been presented in terms of burgers, bottles of beer and glasses of beer, but any three emoji, pictures or symbols could be substituted in for them. In fact, it’s something mathematicians do all the time, except we tend to use more boring prosaic symbols like ‘x’, and ‘π’. (I guess using emoji would open up our options further – and there is already an emoji LaTeX package, for those who want to typeset it properly.)

The reason why people find these problems so ‘baffling’ is that the setters have deliberately made them so – using some tricks to nudge people towards incorrect answers, then allowing a stream of commenters to berate them when they make a mistake. One such technique is using different numbers of items in different lines. Above, the in the third line are in pairs, whereas the fourth line has a single packet of . While it’s easy to see this when you look carefully, you might not notice it on a quick scan of the image.

Another trick is combining operations without specifying the order – for example, the fourth line above reads “ + × = ?”. But does this mean you add the and , then multiply by , or do you multiply by then add ? In processing a sequence of operations like this, the natural thing might be to work from left to right, processing each command one at a time: start with a , add , then multiply by . But in school we learn that operations have a standard order, and if you’ve heard the word BODMAS, or BEDMAS, you’ll know that multiplication (M) comes before addition (A), and should be resolved first.

The thing is, the way the problem is written seems to deliberately be trying to trip you up. The setter could have written “ × + = ?”, which wouldn’t have been quite so difficult to parse. Or, they could have done the proper mathematical thing, and used brackets: “ + ( × ) = ?” Most people working properly in mathematics will tell you that BODMAS is irrelevant, because you can always use brackets to disambiguate.

The worst culprits are those who do even tricksier things to confuse you. The example below, shared as a problem given to Chinese school children which subsequently ‘stumped the internet’, includes a picture of a cat – and if you look closely you’ll see the cat sometimes is wearing a whistle, and other times isn’t. The whistle is one of the unknowns in the puzzle, so ‘cat with whistle’ has a different value to ‘cat without whistle’. Most people won’t spot this on first glance, meaning they’ll get the wrong answer even if their maths is flawless.

This kind of manipulation is wasting people’s time at best, and actively unhelpful at worst – given people are already predisposed to fear maths, giving them what looks like a simple problem, then exposing them to the cruelty of social media comments sections when they get the wrong answer is surely only going to make them feel even worse about it. This blog post from 2017, by a maths teacher and education professor, shares some more detailed thoughts on the phenomenon.

That said, it’s a good sign that people are prepared to have a go at a maths thing they see on Facebook – they might enjoy the tiny buzz of solving the puzzle, and appreciate a chance to flex their maths muscles. It’s also been noted by educators that using pictorial symbols or emoji instead of algebraic variables increases many students’ ability to understand and solve this kind of problem – the website solvemoji.com has hundreds of such problems, free for educators to use.

Also, students can apparently tackle much more difficult problems in this form than they would be prepared to in a traditional setting. The examples pictured above are both systems in three variables – much more complex than many school children would usually be expected to solve.

When mathematicians are faced with systems of equations like this – called **simultaneous equations** – they can be categorised and understood pretty well. If all the operations are addition or subtraction, or multiplying a variable by a number, the equations are called **linear**.

For example, “+ = ”, also known as “+ 2 = 5 ” is a linear equation, as the only multiplying that happens is between an emoji and a constant number (not a variable that we need to find the value of).

Linear equations behave in a nice predictable way – if all the equations in a system are linear, and you have 3 variables to find, you will need at least 3 equations that relate the variables to each other in order to find a single solution. For example, using only the second equation in the fast food example at the start, we couldn’t say for sure what either of the values are, but if we combine it with the other equations we can fix the values. In general, the number of equations relating the variables needs to be greater than or equal to the number of variables.

However, linear equations aren’t the whole story, and in the fast food and whistle cat examples above, there’s an extra complication introduced in the last line. When you multiply a cat by a whistle, or a by a , the equation becomes nonlinear – these are **polynomial equations**. In this case, the same number of equations as variables might not be enough to fix a single answer – for example, × = 4 has both = 2 and = -2 as a solution. If you have a mixture of linear and nonlinear equations, you can try solving the linear ones first and then substitute in the number values to make the nonlinear ones a bit easier.

One interesting point to consider here is that in these problems imply a secret additional set of restrictions on the answers, without actually saying it. In the previous paragraph, I casually mentioned the idea of = -2, but many people approaching this problem wouldn’t consider the possibility of the symbols representing anything other than a positive whole number. While = ½ is strictly possible, what if we ruled it out?

In the case where you’re only interested in whole number solutions, equations and systems of simultaneous equations like these take on a new name. Called **Diophantine equations**, and named after the 3rd century Greek mathematician Diophantus of Alexandria, they require the solutions to be integers (whole numbers). Diophantus was one of the first mathematicians to introduce the idea of symbolism into algebra, and only a few short millennia have taken us from there to+ + = 20.

Diophantine equations are slightly more restricted in their possible solutions than general equations – for example, if I told you that I had two different numbers that added to 3, you’d be able to find infinitely many sets of solutions – say, n, which can be any value (except 1.5, as the numbers must be different), and 3-n. But if I tell you they’re both whole numbers (and I require them to be positive), you can immediately tell me the answers are 1 and 2.

This type of equation also allows for interesting methods of attack. With a single equation in two variables, you can use the Euclidean algorithm, first finding the factors in common between the coefficients given and then working backwards to determine what combinations will give you a valid answer in whole numbers.

Other types of maths problem sometimes also turn out to be based on Diophantine equations. If you’ve ever encountered the classic water-jug-pouring type problems, where you have set sizes of containers and need to end up with an exact amount of water in one of them, that isn’t a whole multiple of one of the container sizes, this is actually a Diophantine equation.

For example, if you have one container that holds 3 litres and one that holds 5 litres, it’s possible to measure out 4 litres – and you’ll do this effectively by adding or subtracting full jugs of 3 and 5 litres. So the problem can be formulated as 3x + 5y = 4, where x and y will be positive (or negative) whole numbers representing the aggregate number of times each jug is filled or emptied. For the most elegant (if slightly sweary) solution to this particular problem, I direct you to the work of my fellow mathematicians Bruce Willis and Samuel L Jackson in one scene from Die Hard 3.

Der Beitrag This Maths Puzzle is Baffling Facebook erschien zuerst auf Heidelberg Laureate Forum.

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