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 n ranging from 1 to infinity.
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.
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]]>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) = 0
- 2 – 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:
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.
]]>Der Beitrag The Twelve Facts of Christmas: Pascal’s Triangle erschien zuerst auf Heidelberg Laureate Forum.
]]>At this time of year, you’ll see a lot of decorations around and a lot of shapes – stars, trees, snowmen, and so on. One shape you might not see many of though, is the triangle, and I think that’s a shame. So I’m going to share with you my 12 facts of Christmas, about one of the greatest triangles in mathematics: Pascal’s triangle.
To generate Pascal’s triangle, start with a one (the number of partridges in a pear tree) and imagine it’s sitting in an infinite row of zeroes going off to either side. Then you can generate the next row by writing the sum of each pair of numbers in the row in the gap underneath them.
This will give you an infinite row of zeroes with two ones in the middle, then repeating again will give you that with 1, 2, 1 in the middle, and so on. The rest of the triangle is obtained by simply repeating this process for the rest of time.
Although Pascal’s triangle is named after mathematician Blaise Pascal, many other mathematicians knew about the triangle hundreds of years earlier. Chinese mathematician Jia Xian (c. 1050) supposedly “[used] the triangle to extract square and cube roots of numbers,” and Persian mathematician Omar Khayyam (c. 1048–1113) seemed to also have knowledge of the structure.
If you have a bracket containing a sum, (a+b), and you want to raise that bracket to a power n, you will find that the different powers of a and b in the resulting expansions have different coefficients in front of them.
(a + b)^{2} = (1)a^{2} + 2ab + (1)b^{2}
(a + b)^{3} = (1)a^{3} + 3a^{2}b + 3ab^{2} + (1)b^{3}
(a + b)^{4} = (1)a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + (1)b^{4}
These coefficients are exactly the numbers in the n^{th} row of Pascal’s triangle! (The first row of the triangle is called row 0, so when I say the n^{th} row, I mean the one that starts ‘1, n’.)
This is connected to the fact that the triangle also encodes the ‘choose’ function in mathematics: If you have a pile of n things, and you want to choose k of them, the number of possible ways to do this will be the k^{th} entry in the n^{th} row of Pascal’s triangle (again starting from the 0^{th} in rows and columns).
For example, if I have 5 gold rings, and I want to give one of them to each of my two true loves, the number of different ways to choose two rings from five is 10 – which can be found in the fifth row, second column.
Apart from the outside edge, which is all made up of 1s, the other diagonals all have nice properties – the first stripe is just the whole numbers counting upwards; the next is the triangular numbers (sums like 1, 1+2, 1+2+3, 1+2+3+4 etc) – if you had a triangular number of Christmas puddings, you could arrange them into a triangle on the table.
The next stripe is the 3D triangle numbers, or tetrahedral numbers: the number of spherical Christmas puddings you can stack in a triangular-based pyramid, and each such number is a sum of the first triangular numbers. In fact, this pattern continues – the next set is 4D tetrahedral numbers, called pentatope numbers, and so on.
Taking a less steep diagonal slice through the triangle, and adding the resulting numbers, you’ll find they each sum to a Fibonacci number – each one is the sum of the previous two.
These numbers have many mysterious and deep connections in mathematics, and also occur in hypothetical rabbit population dynamics.
If you add the numbers in each horizontal row together, you’ll find there’s an interesting pattern in the numbers you get. Bonus points if you can work out why!
If you multiply together all the numbers in each row, that’ll give you a sequence of numbers. If you write S_{n} for the n^{th} number in that sequence, then the following is true:This says that if you take the product of a row, multiply it by the product of the row two below that, and then divide that by the square of the product of the row in between, this value gets closer and closer to the exponential constant e (as your value for n gets bigger and bigger). It gets everywhere, the number e…
If you have a square grid, and are allowed to move from one box to another vertically or horizontally adjacent box, you can challenge yourself to find a route from anywhere in the grid to the top left square. If you write the numbers of Pascal’s triangle diagonally across a square grid, you’ll find that the number in each square gives the number of different routes you can take from that square to the top left square.
Colouring in the even numbers of Pascal’s triangle one colour, and the odd numbers another, results in an exact copy of Sierpinski’s triangle, the triangular fractal. The more rows you colour down the triangle, the more detailed a fractal you get (not much can be seen in this example, but this video shows the process nicely).
Even though we know many things about the numbers in Pascal’s triangle, there are still mysteries to be solved. An open conjecture, called Singmaster’s Conjecture, muses on the question of how many times, at most, any given number N > 1 can occur in the triangle. The most frequently occurring number known is 3003, which is in there 8 times, but there’s no exact value known for a maximum, if one even exists. I wrote about it for The Aperiodical a few years ago, if you’d like to read more.
The 3D triangle is a tetrahedron of numbers, and encodes trinomial coefficients – found when you raise a bracket of the form (a+b+c) to the n^{th} power. The numbers on each layer of the tetrahedron are the sum of the three adjacent numbers in the layer above it. Makes you wonder what a 4D tetrahedron looks like, if you weren’t already from Fact 4.
In case you were worried this post was veering away from being mildly Christmas-themed, don’t worry – if you draw the shape of a Christmas stocking and overlay it on Pascal’s triangle, there’s a nice pattern to be found there as well. Also called the hockey-stick identity, it states that the a sequence of numbers moving diagonally in towards the middle of the triangle will add up to the final number just off the diagonal at the bottom. Festive!
Der Beitrag The Twelve Facts of Christmas: Pascal’s Triangle erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Right on the money erschien zuerst auf Heidelberg Laureate Forum.
]]>The new £50 will be the latest to join the new series of plastic-rather-than-paper notes started by the new £5 note in 2016, and will be the last one to do so – a new polymer £20 note to be issued from 2020 will feature a portrait of the artist JMW Turner. The choice of Turner followed a lengthy selection process undertaken by the Banknote Character Advisory Committee, who consulted with the public to suggest options and chose Turner from over 30,000 nominations.
It’s been announced that the new £50 will definitely feature someone from the realm of science – which is exciting for fans of science, like me. But what I really want to know is, will it be a mathematician?
Up until now, mathematicians did occasionally feature on UK banknotes – as long as you count people who weren’t famous for specifically being mathematicians. For example, the current £20, which will be replaced in 2020, features economist Adam Smith.
Smith was also an author and philosopher, and his book The Wealth of Nations is considered to be the first modern book addressing the topic of economics. Smith discussed how self-interest motivates people in economic situations – ideas which presaged modern game theory – and that under competition, equilibria are established to optimise profitability. The note features a reference to his analysis of the division of labour in manufacturing. His ideas have formed the basis of much modern economics.
The current £50 note features two portraits – business partners Matthew Boulton and James Watt. Watt was an inventor, chemist and mechanical engineer, famous for his steam engines, and Boulton was an entrepreneur and manufacturer who helped Watt go into business selling them. Watt’s engines designs were ingenious and more efficient than contemporary models, and represented a step forward in the technology which powered the industrial revolution.
Watt wasn’t the only steam engine pioneer to have been featured on a UK banknote – a £5 note in circulation from 1993 to 2003 featured George Stephenson, inventor of the famous Stephenson’s Rocket (also featured on the note), and a talented engineer whose company manufactured engines and built railway lines all over the country, revolutionising rail transport.
Probably the closest we’ve historically had to a mathematician on a banknote was Florence Nightingale, who appeared on the £10 note from 1975 to 1994. While Nightingale is probably best known for her work as a nurse in the Crimean War, and for revolutionising the field of nursing, mathematicians think of her fondly as a great early example of the use of a pie chart in communicating statistics – her polar area diagram used a circular chart to demonstrate how much more effective her methods were at preventing disease, and effectively convinced politicians to adopt her approach.
The new banknote has to feature someone from science (along with a couple of other criteria – they have to be dead, and not a fictional character) who has contributed to the field, and ‘shaped thought, innovation, leadership or values in the UK’ as well as being an inspirational and non-divisive figure. So what are the chances it’ll be a mathematician?
The selection process opened on 2nd November, and has so far received over 170,000 names suggested by members of the public, and nominations are still open. They’ve released a list of over 800 names from the first week, having filtered out any that aren’t valid candidates for any of the reasons described above. While over at The Aperiodical (another blog I write for) Christian has done a full analysis of the list, I’ve picked out my favourites – although I’m sure there are many other worthy candidates I’ve missed:
So who would you choose to put on a banknote? And which other countries have famous figures on their notes? I wonder which country has the most mathematicians…?
Der Beitrag Right on the money erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Measuring the Change erschien zuerst auf Heidelberg Laureate Forum.
]]>You might think there are certain universal constants that never change – the length of a second, the weight of a kilogram, the length of a metre – and to some extent that’s true. But measuring things accurately, and being confident that your measurements are accurate, has been a difficult task for a very long time.
One unit of measurement the ancient Romans used was carob seeds (the origin of the modern units carat and karat, used by jewelers to measure the weight of gemstones and the purity of precious metals) – small, even-sized seeds that had a vaguely consistent weight. But while this was accurate enough for their purposes, there was obviously slight variation in the size of the seeds, meaning their measurements could differ wildly.
I’m reminded of a recent challenge in an episode of the UK comedy challenge show Taskmaster, in which contestants had to measure the circumference of a caravan in baked beans – given only three minutes, some tried to frantically lay beans out around the caravan, but one contestant cleverly measured a single bean, then measured the caravan and calculated how many beans would fit around. As Lucy Rycroft-Smith wrote up on her blog, this actually gave a wildly inaccurate answer, as the beans varied in size – and the one chosen must have been a bit of an outlier.
Since Roman times, methods have become gradually more sophisticated. Standards were established – in England, alongside legal and judicial rules, the Magna Carta of 1215 laid out standards for measurement, and in 1266 the weight of 1lb was defined as the weight of 7680 grains of wheat – less reliant on a single variable item’s weight, but still not ideal. It took until 1791 for the International System of Units to be established – originating at the French Academy of Sciences, the Système International (SI) defined the length of 1 metre as one ten millionth of the distance from the pole to the equator, and one kilo as the weight of one cubic decimetre of water (at 4 degrees centigrade).
The SI has been adopted in almost every country in the world as the standard system for measuring units: in 1875 the Metre Convention was signed by 17 countries, establishing committees and organisations to oversee standards, and the International Bureau of Weights and Measures officially established the initial six base SI units at their 11th conference in 1960. SI has since grown to encompass many different types of measurement, now based on seven base units – the ampere, kelvin, second, metre, kilogram, candela and mole. It also includes 22 derived units for other physical quantities like the watt and the lumen.
However, these units still needed to be defined in a more rigorous way. The distance from the equator to the pole is not easy to measure – and such large measurements are prone to inaccuracy. Not to mention that the exact location of the earth’s magnetic poles is slowly moving! If your unit of mass is defined in terms of this – one cubic decimetre is a cube 10 cm on each side, which would change if the length of a metre varied – that will also be affected. The challenge became to find standards that continue to keep the same value independent of any changes to the things they’re based on.
Some of the SI units can be defined relative to fixed values – for example, the speed of light in vacuum is a universal constant which can be measured and doesn’t change. This allows us to define the metre as the distance travelled by light in a fixed amount of time (as of 1983, 1/299792458 of a second). Similarly, the triple point of water – the temperature at which it can be a solid, a liquid and a gas depending on the ambient conditions – is used as a definition of 0 degrees centigrade, and since the kelvin was adopted as the official SI unit of temperature, 273.16 K (as of 1954). Since 1960, only one SI unit has been fixed relative to a variable physical quantity, and that’s the kilo.
Another problem SI units have is the difficulty of actually measuring them. Even if you have a fixed definition of a unit – once these universal definitions for the metre and kelvin are established, you can weigh a decimetre of water at the right temperature – it’s still difficult to propagate this measurement. In 1889, the International Prototype of the Kilogram (IPK) was created – a 39 mm high and 39 mm wide cylinder of a platinum-iridium alloy, designed to be as stable as possible, and to be stored in a secure vault in France.
There are official copies of the kilogram, called témoins (witnesses) that are stored in other locations around the world, and used as local references – but very occasionally they’re returned to Paris to check they’re still the same weight – three times in the IPK’s history.
There are many problems with the official definition of a unit being tied to a physical real-world object. Primarily, if the mass of the IPK changes, that would change the definition of the mass of a kilogram to match. That means if someone were to accidentally damage it and remove a chunk of the metal, the weight of a kilogram would change accordingly – literally, around the world, people paying a fixed price per kilo for goods would suddenly find their costs increasing. Obviously, in reality it might not actually happen so quickly – but this is the official definition. Since many of the other SI units are defined based on the kilogram, the mole, the ampere, and the candela would all be affected by any change to Le Grand K (an affectionate nickname).
It’s also impossible to keep the mass of a physical object constant – over time, the surface of the metal reacts with substances in the air, and particles of contamination adhere to the surface – adding up to around 1 microgram per year. The kilograms are cleaned, using a very careful specified procedure, to remove this surface contamination – and while it’s only a tiny difference, comparable to the weight of a single eyelash, this can mean the mass of the artefact fluctuates by around 50 parts in a billion.
Since the ‘witnesses’, and not the actual kilogram, are used to maintain other more local standard weights, and these are each used to create reference weights for use in labs, business and industry, these tiny errors can propagate, causing noticeable differences down the chain. Alternatives to the IPK have been suggested, including a spherical mass prototype made of silicon, but these still have the inherent risks and inadequacies of any physical reference mass.
So, how do we create a reference for the mass of the kilogram that doesn’t reply on a physical object? Enter the Kibble balance. Invented by Dr Bryan Kibble in 1975 (he called it a Watt balance, but it was renamed after his death), it’s used to measure the value of one of the other fundamental constants – h, the Planck constant.
The Planck constant, named after physicist Max Planck, relates the energy carried by a photon to its frequency – but since energy and mass are equivalent (thanks, Einstein) it also relates frequency to mass. The Kibble balance allows the Planck constant to be measured with incredible accuracy, meaning that we can now fix its value and use that to define the mass of a kilogram. The value is now officially going to be h = 6.62607015 × 10^{-34} joule-seconds, as of next May, when the SI system will officially change to redefine the kilogram.
The value of h has been chosen based on the current value of the kilogram, since you need to measure against a mass. But from May 2019 onwards, if anyone has their own Kibble balance (currently not a cheap option, but new designs might mean that smaller commercial versions will become widespread) they can use the new fixed value of h to get an accurate, universal kilogram – and gram and microgram, meaning the error previously introduced by using a fixed kilogram as a reference is also reduced.
This change will mean very little practically to anyone who doesn’t work in a physics lab – the value of a kilogram isn’t going to change (in fact, it’ll strictly change even less than it usually does) – but this new definition will become the international standard, and masses will be measured and calculated relative to it. The has potential implications in many fields, from quantum mechanics to mechanical engineering. The definition of the kelvin is also likely to change – it can also be defined relative to the Boltzmann constant, measured in Joules per kilogram, and now we have a fixed kilogram that can be tied to.
A vote at the general conference on weights and measures – involving representatives from 57 nations – took place on Friday 16th November in Versailles, and was unanimous in favour of the change: so it’s now official. As of next May, the golf-ball sized chunk of metal in a glass jar in France will become merely a nostalgic object of historical significance: part of the history of physics, and a symbol of humanity’s quest to measure the universe.
Der Beitrag Measuring the Change erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Permutations and Tribulations erschien zuerst auf Heidelberg Laureate Forum.
]]>While I definitely spent time at school rearranging objects (I just like it when things are in the right order, ok?), and often idly wondered about how many different ways there might be to do this, I never realised, until I learnt about it in a first-year maths lecture, that there is a whole mathematical structure around such rearrangements.
You can find the number of ways there are to rearrange N objects by multiplying together all the numbers from 1 up to N: this is called ‘N factorial’, written as ‘N!’. So for 3 objects there are 3! = 3 × 2 × 1 = 6, and with 3 objects there are six possible rearrangements: 123, 132, 213, 231, 312 and 321. For four things there are 4! = 24 arrangements, and so on. Mathematicians study ways to combine permutations, thinking of them as a function that takes a list of objects and rearranges them, but you can also just think of them as all the different ways to order a list of a given size.
As well as being a nifty presentation of an idea, and having lots of lovely properties, permutations also crop up in many other areas of maths – they are mainly studied in combinatorics, but anyone working in group theory or algebra will come across them all the time. I also found them popping up in my PhD research in topology, and they are incredibly useful in many contexts.
Imagine my excitement when I learned that as well as my beloved permutations, there’s also the concept in mathematics of superpermutations. These are sequences of numbers which contain every permutation of the numbers 1 to N, at least once somewhere within the sequence.
For example, if I wanted a superpermutation on the numbers 1 and 2, I could write 121. This contains 12, and 21. To go up to three, and have a sequence that includes all six of the three-item permutations listed above, I’d need 123121321.
The thing mathematicians have been interested in is: what’s the shortest this sequence of digits could be, for a given number N? It’s obvious that if you were to just write the six permutations on three objects out in order: 123132213231312321, you’d have a superpermutation – but it’s not the shortest one. Starting from this simple upper bound of N! × N (the number of possible permutations multiplied by the length of each one), mathematicians have been working to try to get a better bound on the length of the minimal superpermutation. For values of N from 1 to 5, we know the answer – 1 (just 1), 3 (our example of 121), 9 (123121321, as seen above), for four things it’s 33 symbols long, and for five things the shortest string is 153 symbols long (eight different such strings exist, proved to be all of them in 2014). There’s a nice pattern in these values:
1 = 1!
3 = 1! + 2!
9 = 1! + 2! + 3!
33 = 1! + 2! + 3! + 4!
153 = 1! + 2! + 3! + 4! + 5!
But in mathematics, sadly not every pattern carries on forever. Beyond N = 5, the pattern breaks down, and for large values of N the number of possibilities becomes so huge, we don’t have much of an idea of what’s going on at all. For example, the minimal length for six objects, which following the pattern you might expect to be 1! + 2! + 3! + 4! + 5! + 6! = 873, turns out to be something else. In 2014, mathematician Robin Houston discovered a superpermutation for N = 6 of length 872. Others have since found hundreds more of this same length.
Many assumptions that had been made about this problem were suddenly thrown on their head. It’s possible to construct minimal superpermutations for N=1 to 5 using a known algorithm, building it at each stage using a superpermutation for one fewer objects, combined with the new symbol – but this method doesn’t give a minimal superpermutation beyond N = 5. But is 872 the shortest, or could a shorter one be out there?
All of this is covered in James Grime’s wonderful Numberphile video, which he made back in January to explain the problem. But there’s been a recent development in the story which has again taken quite a few people by surprise – and in the video, when James says ‘we don’t know if this is the shortest’ – he may now wish to revise that statement.
Back in 2011, an interesting post appeared on an internet message board called 4Chan, in a discussion about the anime TV series ‘The Melancholy of Haruhi Suzumiya’ – a show which originally aired its episodes in non-chronological order. They posed a problem: if you want to watch all the episodes of a show in all possible orders, what’s the fewest episodes you need to watch?
This is exactly the minimal superpermutation question again, but phrased in terms of TV show episodes. If your show had only 3 episodes, and you watched them in the order 123121312, you’d have seen all 3! = 6 possible orderings during your marathon binge-watch.
Apparently one anonymous user took the problem seriously, and earlier this month it was discovered that they’d in fact found a way to prove a lower bound on such numbers – a minimum length. Since internet forums aren’t the usual location for serious mathematical discourse, nobody thought to take it seriously – until now. Their proof does appear to hold up, and shows that the minimum length of a superpermutation on N objects has to be at least N! + (N−1)! +(N-2)! + N − 3.
This means for 6 objects, it is 6! + 5! + 4! + 6 – 3 = 867. In the wake of this discovery, algorithms have been constructed that can generate superpermutations with this length – and just in case this wasn’t a strange enough story already, the person who found the algorithms is Australian sci-fi author Greg Egan.
Robin Houston, along with some colleagues, has now written up the 4Chan proof as a formal paper, and they are working to integrate it with other proofs about the upper bound for the length of a minimal superpermutation to try to get a complete picture – having both an upper and lower bound brings us much closer to finding the exact value.
The proof method involves visualising all the permutations as part of a network, or graph, and considering how easy it is to move from one permutation to another by adding symbols to the end – for example, 123 can be followed by 231 just by adding one symbol to get 1231, but for 123 to be followed by 312 we’d need to add two extra symbols. Then, the ‘easiest’ path through the graph is determined, using known algorithms from graph theory to solve what’s called the ‘Travelling Salesman Problem’ – how to visit every point with the least possible effort.
This simple problem has now found at least a partial solution – and one which could have been known years ago, if only people had thought to look in the dark corners of the internet for it. At least you know that if you wanted to watch all 14 episodes of the Haruhi TV series in all the possible orders, you’d only need to watch at least 14! + 13! + 12! + 14 – 4 = 93,884,313,611 episodes, which at half an hour per episode would only take you 4.4 million years. Better get started, then!
Der Beitrag Permutations and Tribulations erschien zuerst auf Heidelberg Laureate Forum.
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