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.

]]>Der Beitrag The legacy of Emmy Noether erschien zuerst auf Heidelberg Laureate Forum.

]]>Emmy Noether was a German mathematician who lived from 1882 to 1935, and made huge contributions to the worlds of mathematics and physics. She is a shining example of an excellent mathematical mind, but also a sad example of the struggles women faced in mathematics during her lifetime.

Emmy grew up in Erlangen, Bavaria, in a fairly academic household with two other mathematicians (her father Max, and brother Fritz). Her aptitude for the subject meant she wanted to study it at university – which nowadays would seem like a logical step, but in those days it wasn’t heard of. Instead of formally enrolling as a student at the University of Erlangen, Emmy was permitted to audit classes, meaning that with permission from the professors giving the lectures, she could sit in the room and watch – and she seized the opportunity to learn. Emmy was one of only two women in a university of 986 students.

Noether progressed with her studies despite these obstacles: she studied for a year at the University of Göttingen, then returned to Erlangen to teach – unpaid – and built up relationships with leading mathematicians including David Hilbert and Felix Klein. She was invited by Hilbert to teach at Göttingen in 1915, in spite of the university’s protests, but proved to be a worthwhile addition.

Throughout, Noether was consistently mathematically brilliant and her enthusiasm for mathematics was obvious. She was a helpful and patient teacher, worked well with others, and was not egotistical – despite all the disadvantages she faced, she was more concerned about injustices towards her students.

During her mathematical career, Noether worked in three main areas, as outlined by Hermann Weyl: on the calculus of invariants, leading to Noether’s Theorem – a hugely important idea in mathematical physics, linking symmetry to conservation laws; in abstract algebra, developing the theory of ideals in commutative rings; and in representation theory, bringing modules and ideals to the theory of group representations. She was also credited with assisting people and working in many other areas, without credit – including having a hand in the origins of the field of algebraic topology.

When Hitler came to power in Germany in 1933, Emmy was forced to leave her university job due to her Jewish background. She found a post at Bryn Mawr University in the US, lecturing there and at Princeton until she died a few years later, after suffering from an ovarian cyst.

Noether has been described by many as the most important woman in the history of mathematics – but it might be fairer to say that she’s one of the most important mathematicians of the last century, and the fact that she was a woman and had to deal with the prejudices she did just makes this more impressive. Unlike many of her male counterparts, Noether’s name isn’t as well-known as it should rightly be, and some have taken steps to try to rectify this.

Last November, I joined forces with Constanza Rojas-Molina, who’s been part of the HLF blog team and written blogs posts and done sketches of talks at the HLF for the last few years. Coni suggested we should do something in the month of November, and rename it Noethember. Inspired by the #inktober hashtag, used by illustrators and artists to motivate themselves to post an ink drawing every day in October, we decided #noethember would be a chance to celebrate Emmy Noether every day in November – with a drawing, a sketch, a story or anything else that remembers her life and work.

We picked a list of Noether facts, one for each day of the month and challenged the internet to respond – which they did. The hashtag was busy every day with sketches, photos, links, posts and all kinds of tributes, and we all learned a huge amount about Noether and her wonderful mathematical mind. We’ve written a round-up post collecting some content and other write-ups of the month, and you can still search for #noethember on Twitter to see what happened each day.

Some people drew an image every day, and others meant well to start with but tailed off (myself included!) while many just posted a few times during the month. The keenest posters got the most out of it, and some have even started an initiative to have a suggested theme for a creative mathematical drawing or craft every week in 2019 – look for the hashtag #mathyear to see how they’re getting on, or to join in visit Marlene Knoche’s blog where they’re sharing the topics each month.

Der Beitrag The legacy of Emmy Noether erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag Public Key Cryptography erschien zuerst auf Heidelberg Laureate Forum.

]]>Whitfield Diffie and Martin Hellman, both winners of the ACM Turing Award (among many other prestigious honours) are both past visitors to the Heidelberg Laureate Forum, and valued contributors to the programme. Their names are well-known in cryptography classrooms the world over, because of their eponymous encryption system. But what is Diffie-Hellman Key Exchange?

In any communication system, it’s safe to assume that a malicious third party is capable of tapping your communications and reading the messages you’re sending back and forth. This means if you were to send your message in a readable form, they’d be able to intercept and read it. This means it’s necessary to encrypt your message.

There are plenty of simple ways to encrypt data – some of which are so simple, a child could use them (and I often did). For example, the **Caesar (Shift) cipher** is a simple form of encryption where you substitute each letter for one three along in the alphabet – A goes to D, B goes to E and so on. Suhwwb vlpsoh, kxk? It’s easy enough to apply the code to your message, but this kind of code is easily broken by a mildly intelligent interceptor.

A more secure method would be to use a **key** to encrypt the message. A **Vigenère cipher** uses a keyword applied to a Caesar cipher – so instead of shifting the alphabet along by three for every letter, it’ll shift it by a different amount for each letter in the message, according to a pre-agreed keyword. For example, if I used the keyword HEIDELBERG, I would shift the first letter in the message along by H (8) spaces, then the second letter by E (5) spaces, then I (9) and so on. The keyword is repeated as many times as needed along the length of the message.

While such a code can still be cracked, using a method called a **Kasiski attack** (roughly, this involves looking for repeated strings of characters in the message, and using them to determine the length of the keyword), it’s more secure than a simple shift cipher.

It also creates an interesting new challenge, which still remains a challenge when you move to more secure cryptography systems: if you need to agree on a keyword with your co-conspirator, and your only method of communication is via a tapped line, how do you communicate the keyword?

Prior to the 1960s, any key information used in encoding secret messages had to be shared separately, in a private channel – or passed on a piece of paper. During the Second World War, the ‘unbreakable’ Enigma code was decoded partly due to intercepting enemy codebooks and key lists, containing the settings and codewords for each day, which gave the codebreakers huge insights into how to crack it.

Diffie and Hellman’s answer to this problem was **public key cryptography** – one of the first practical implementations of a cryptography system where the key could be shared on a public channel, without risk of interception by a third party. Based on mathematical ideas, it completely changed the way cryptography was conducted, and paved the way for further developments leading to forms of encryption still widely used today.

One simple way to understand the process is through modular arithmetic – calculating within a fixed range of numbers. For example, the numbers on a clock are 1-12, but if I asked you what’s four hours after 10 o’clock, the answer is 2 – you can wrap around. Modular arithmetic reduces each number to its remainder on division – for example, modulo 7, 12 is equivalent to 5, and 14 is equivalent to 0.

In Diffie-Hellman key exchange, two parties exchanging a message start by agreeing on a pair of numbers. One number is chosen as the **modulus**, which all the arithmetic will be conducted modulo, and the second is called the **base**. We’ll stick to standard cryptographical conventions, which is to say that our two messaging parties should be called Alice and Bob.

Alice and Bob publicly agree to use modulus 23, and base 5.

It’s important that the base is **coprime** to the modulus – that is to say, if you list all the multiples of the base (5, 10, 15, 20, 25…) and calculate them all modulo 23, they will run across all the possible values 0-23. This is equivalent to saying that 23 shouldn’t be divisible by 5, or have a common multiple less than 23×5 = 115.

Next, each of the two participants chooses a secret number, and raises the base to the power of their secret number: for example, Alice might choose a = 4, and Bob chooses b = 3, meaning Alice calculates 5^{4} mod 23 = 4, and Bob calculates 5^{3} mod 23 = 10.

Since all of this is modulo 23, there are any number of things Alice and Bob could have chosen to give the answers 4 and 10 – for example, if Alice had chosen a = 26, it would have given the same result – so it’s safe to communicate these two answers publicly, and it won’t give away what Alice and Bob’s secret numbers *a* and *b* are.

The next step relies on a clever bit of mathematics. If Alice receives Bob’s value for 5^{b} mod 23, and raises that to the power of her own secret number *a*, she’ll get 10^{4} mod 23 = 18. If Bob takes Alice’s value for 5^{a} mod 23 and raises it to the power of *b*, he’ll get 4^{3} mod 23 = 18. It’s not a coincidence that these values are the same!

In fact, it’s a property of this kind of modular arithmetic that in general,

(g^{a} mod p)^{b} mod p = (g^{b} mod p)^{a} mod p

This means that Alice and Bob both now know the number 18, and can use this as a secret key – for example, in our naive cipher system, this could mean using the 18th word from a text they both own (and they can publicly specify which text, since nobody intercepting will know they’re looking for the 18th word).

In reality, this kind of key exchange can be used for much more secure communications – the maths works at any scale, and it this is an example of a calculation that’s much easier for Alice and Bob than it is for a malicious attacker – raising a number to a power modulo a modulus is relatively easy to compute, but working out the numbers given the answers is in general a very hard problem (called the Discrete Logarithm Problem). Once your numbers are big enough (if your modulus is at least 600 digits long) even the faster computers can’t check all the possibilities in a short enough time to make it useful.

The shared values Alice and Bob calculated and sent (5^{4} mod 23 = 4 and 5^{3} mod 23 = 10) are called the **public keys**, and Alice and Bob’s secret numbers (a=4 and b=3) are called the **private keys**. Each public key set is only used once – since Alice and Bob’s calculation is computationally cheap, they can do it again easily by picking new private keys.

Diffie and Hellman were the first to publish this idea, and did so in the early 1970s, although mathematicians at GCHQ had managed to prove this kind of system was possible in 1969 (but their work was classified and couldn’t be published until later). Diffie and Hellman are clear that their idea was based on a concept developed by Ralph Merkle, and they insist it should really be called Diffie-Hellman-Merkle key exchange.

It was rightly hailed as a huge breakthrough at the time, and some were even concerned that such a secure new system might be too much power to allow just anyone to use – in the late 1970s Hellman found himself in a fight with the NSA over whether he should be publishing his results in international journals, as written about by Ben Orlin following Hellman’s 2017 HLF talk.

On its own, Diffie-Hellman-Merkle isn’t perfect – without authentication (being able to prove that the person you’re receiving the message from is indeed Bob, and not an interloper pretending to be Bob) it’s possible for someone to send you their own public key in place of Alice’s, which would allow them to break your code. This is known as a man-in-the-middle attack, and using D-H-M as a basis, authenticated public key systems have been developed which prevent it.

The principles behind Diffie-Hellman-Merkle key exchange became the basis of RSA encryption – developed by Rivest, Shamir and Adleman, and using a slightly different type of calculation that’s even more difficult to undo (see: the Factoring Problem) – making it practically so difficult to crack that it might as well be impossible, given current computer technology. RSA is used widely for transmitting data securely, and this underlying mathematics has made it possible to build all the complex structures and systems we now rely on for finance, the internet and telecommunications all over the world. Thanks, Diffie and Hellman (and Merkle)!

Der Beitrag Public Key Cryptography erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag A Puzzle ‘Four’ the New Year erschien zuerst auf Heidelberg Laureate Forum.

]]>It’s a New Year, and with it comes a new four-digit number. When faced with a number like 2019, it’s the mathematician’s natural instinct to do maths with it. Having quickly checked whether the year is prime (it’s sadly divisible by 3) the next obvious step is to discover interesting facts about the number, and to create and share number puzzles which use it.

Alex Bellos posted a set of number puzzles on New Year’s Eve, including Ed Southall’s lovely fact that 2019 is the smallest number that can be written in 6 ways as the sum of the squares of 3 primes; and Matt Parker’s posted a YouTube video listing some interesting number facts about 2019, in 2 minutes and 19 seconds.

One of Alex Bellos’ puzzles is a real classic of the genre – I see the same puzzle popping up every year, each time using the digits of the year in question:

Using only the digits 2, 0, 1 and 9 [exactly once each], create expressions that equal all of the numbers from 0 to 12. The expressions can include any of the arithmetical symbols +, –, x, ÷ and √, and brackets.

I’ll start you off:

0 x (2 + 1 + 9) = 02 – 1 + (0 x 9) = 1

If you’d like to spend some time working on this problem, go ahead – you can check your answers against Alex’s solution post. This type of problem – “Using only the given digits and certain operations, which numbers can be made?” – will be familiar to viewers of Channel 4’s Countdown in the UK as the regular **Numbers Game**.

In the show, a mathematician places a selection of randomly chosen numbers across the top of the board, and then presses a button to generate a random number as a target. Contestants have 30 seconds to work out how to combine them, using addition, subtraction, multiplication and division only, to reach the target.

It’s a lovely challenge, and still hugely popular after many years of broadcast. For a nice example of an out-of-left-field solution, watch this excellent example from 1997, and at the other end of the spectrum, watch this clip from 2009 in which the random number generator lands on a particularly easy one (featuring a beautiful 30 seconds of British people sitting around in awkward silence, as we do so well).

The rules of the Countdown Numbers Game have been well formulated – it only allows the use of the four basic mathematical operations, and the numbers are chosen from a predetermined set (the ‘top row’ referred to in the clip always includes 25, 50, 75 and 100 and the rest of the board – ‘small ones’ – consists of the numbers 1-10, twice each). Contestants can choose any configuration of six numbers they like, picked randomly from the large and small (e.g. ‘two large and four small’).

Such strict rules work well in a competition setting, but in our 2019 puzzle context, it wouldn’t be as interesting. Alex’s puzzle specifies that your expressions for the numbers 1-12 “**can include any of the arithmetical symbols +, –, ×, ÷ and √, and brackets**” – he’s included the square root symbol here, since it doesn’t involve writing any numbers, so it’s not as much like cheating as it would be if you included the ‘squared’ symbol as well.

In order to reach the numbers 13-20, Alex allows a little more leeway (presumably because some of the numbers in this range aren’t possible with the initial set of operations) – you’re now allowed to **concatenate numbers** together (for example, you can put the digits 1 and 9 together to make 19) and put numbers as **exponents** – so you would be allowed to square something, since 2 is one of your digits (but you couldn’t raise something to a power you don’t have a digit for).

Alex then invites you to take this even further, and add in other mathematical expressions to get all the numbers up to 100. There’s a project for a rainy day!

The genre of puzzles this 2, 0, 1, 9 digit puzzle falls in has its own classic version – originally printed in “Mathematical Recreations and Essays” by W. W. Rouse Ball back in 1914, the **Four Fours** puzzle challenges people to make each whole number using four of the number 4. Without giving too much away, the first few examples can be calculated as follows:

0 = 4 ÷ 4 × 4 − 4

1 = 4 ÷ 4 + 4 − 4

2 = 4 − (4 + 4) ÷ 4

3 = (4 × 4 − 4) ÷ 4

Of course, each of these can be expressed in several other ways, and all the numbers up to 72 are definitely possible, given the right set of operations – and many others are possible above that.

Operations you might permit include:

- The standard ‘Countdown’ mathematical operations of +, – , × and ÷
- Concatenation of digits to make two, three and four digit numbers
- Exponentiation – putting one number to the power of another
- Taking the square root, or the nth root, if n is a digit you’re allowed
- Factorial symbol, e.g. 4! = 4 × 3 × 2 × 1
- Decimal points, to produce e.g. .4 (point four, with an implied nought) = 4/10
- A dot or bar over a number to indicate an infinite recurring decimal, or a … at the end, so you can make .4… = 4/9

Some people also include some slightly questionable options – for example, the **subfactorial**, !n, also called the **derangement number**, is defined as the number of ways to rearrange a set of n objects so that none of them end up in their original places – for example, 1234 going to 2341. !4 = 9, and while this is a well-defined quantity, it’s rarely a function that’s seen or used outside of serious pure mathematics, and very few people have heard of it.

Similarly, the **gamma function**, Γ(x), is considered an extension of the factorial to non-integer values – for example, you could calculate Γ(0.4) = 2.21816… but it wouldn’t be much use in trying to make 5. However, the gamma function is still defined on whole numbers – but because of the way it’s defined, Γ(n) = (n-1)!. This means Γ(4) = 6, which might get you out of a tricky spot if you need a six and haven’t got enough fours left to make it. But it’s surely cheating!

Once you’ve decided which of these to allow and how constrained to make the challenge, you can attempt Four Fours (or its more challenging alternatives, **Five Fives** and **Six Sixes**, which I hope I don’t have to explain); or you could pick any four digits – your birth year might make for a nice personalised puzzle.

Some coders have taken it upon themselves to find ways to crunch the problem through software – given your permitted operations and starting digits, you can apply all the allowed functions to all the numbers (or pairs of numbers) you start with, then repeat this using your results – having done this enough times you’ll find all the possible numbers you can get using the given starting criteria. But that’s DEFINITELY cheating.

Given the idea of picking any four digits to play with, it’s also possible to turn this into a competitive game, to challenge your friends and see who’s the best at this. You wouldn’t be the first to do this – **Krypto**, a game designed by Daniel Yovichin 1963, involves dealing a set of cards in front of the players to pick your numbers, and players compete to find expressions for given sets of cards before their friends do.

If you’re not interested in buying a special set of cards to play, it’s also possible to play using a normal deck of playing cards. Number fighting fans in Shanghai developed **The 24 Game** (not to be confused with 24: The Game, which is the official board game of the Kiefer Sutherland TV series, and is very different) – using a standard deck of cards with pictures removed, four random cards are dealt onto the table in front of everyone, taking aces to be 1, and whoever can make 24 first is the winner.

24 has been chosen here as a number with plenty of factors, that can be reached in a variety of ways by multiplication and addition, and with a bit of quick thinking it’s possible to work out a way to get 24 from almost all of the 1820 four-card combinations (but not all: sadly, 1, 1, 1, 1 isn’t possible with the standard arithmetical operations – unless you allow factorials).

My friends and I have played a hand-based version, where each of four players is dealt 10 cards, and on each turn everyone places a single card into the middle – whoever shouts a way to make 24 first wins those four cards and scores them in a pile, and once all 10 cards from your hands have been played, the winner is whoever has the most scored cards. (By agreement, it’s occasionally been necessary to give up on a particular set of four numbers, when nobody has been able to get an answer – in which case you can score one each).

You might find the idea of this kind of number torture bemusing – mathematics is about much more than just mental arithmetic and crunching numbers, and since you can choose the rules you give yourself, the challenge is slightly arbitrary. Even so, blackboards in maths departments all over the world will find that if someone started writing “4 4 4 4 = 1, 4 4 4 4 = 2″ down one side of the board, leaving gaps to fill in the operators, it would quickly become a project with which many would join in.

Der Beitrag A Puzzle ‘Four’ the New Year erschien zuerst auf Heidelberg Laureate Forum.

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