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.
]]>Der Beitrag Women in Maths Exhibition – a graphic summary erschien zuerst auf Heidelberg Laureate Forum.
]]>For more details about this exhibition, check the HLF’s news article.
Der Beitrag Women in Maths Exhibition – a graphic summary erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Impressions from Friday’s Session at SAP-I erschien zuerst auf Heidelberg Laureate Forum.
]]>Here are some impressions from the talks by Wendelin Werner, Konstantinos Daskalakis and Martin Hellman. Don’t forget to watch the lectures from Werner, Daskalakis and Hellman in full on the HLF YouTube channel.
Der Beitrag Impressions from Friday’s Session at SAP-I erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Poster Session at HLF erschien zuerst auf Heidelberg Laureate Forum.
]]>On Thursday, a Poster Flash session, where the young participants gave 2-3 minute presentations of their research, was followed by a Poster Session. We talked briefly to some of the young scientists, and got the feeling of what’s to come in the future in terms of research. For example: machine learning, robotics, mathematical models with high impact applications (like fingerprint identification), and graph theory applied to cryptography.
Der Beitrag Poster Session at HLF erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag HLF 2018 Comes to an End erschien zuerst auf Heidelberg Laureate Forum.
]]>As a journalist, it was fascinating for me to watch the students grow more confident as the week went on. By Friday, they were able to go up to the laureates and strike up a conversation about their work, Heidelberg, or even the dessert that we were eating. The laureates themselves really felt like they knew the students, sometimes referring to them by their first name in the hallway. This is what the HLF does so well. It provides so many wonderful opportunities for students and laureates to meet and mingle that, by the end of the week, conversations flow easily, collaborations naturally develop, and friendships are made.
I spent last week blogging about the scientific talks and my conversations with the students but there was also an element of personal advice and mentorship throughout the week. For my last 2018 HLF blog, I wanted to highlight some examples, particularly talks given by two computer scientists. These talks really drove home the message that these laureates did not set out to be the best in their field, but with a little self-confidence and determination, they achieved just that.
Here are some examples of the personal advice given to the students:
Dave Patterson’s “What Worked Well For Me”
Martin Hellman’s “Ingredients for Successful Research”
If you are interested in applying for the 7^{th} annual HLF (September 22-27^{th}, 2019), please keep checking this website. The application will be posted at the end of November. So “embrace your fears” and “swing for the fences” and do not miss this opportunity next year!
Der Beitrag HLF 2018 Comes to an End erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Quotable Women in Mathematics erschien zuerst auf Heidelberg Laureate Forum.
]]>This reflects an existing gender imbalance in mathematics, which will hopefully improve as more women choose it for a career. But mathematics is done by all kinds of people all over the world, and many of them have occasion to say interesting and/or profound things about their subject – even the ones who aren’t men. Since Wikiquote is part of the Wikimedia network and can be edited and contributed to, we held a small editing day back in May where we added a number of quotes from female mathematicians to the page, to try to balance out the score.
Walking around the Women in Mathematics exhibit here at the HLF this week, I was struck by the amazing quotes taken from the interviews with each of the mathematicians featured. I’ve picked out some of my favourites – and yes, I’ve added them to the Wikiquote page. Enjoy!
“In doing mathematics, I express something personal. It is a source of joy to know that, despite this personal aspect, the fruit of my work can be of interest to other mathematicians.” – Nalini Anantharaman
“Mathematics offers a common language across borders. It is a real joy.” – Alice Fialowski
“My earliest mathematical memory is my father explaining to me the theorem that three angles in a triangle add up to 180 degrees. The idea that something could be proved to be always true was very appealing to me.” – Frances Kirwan
“I used to fear I was not made for mathematics and would look for people to tell me I was on the right track. You need to develop a personal conviction that you are a mathematician, and that what you are doing makes sense.” – Katarzyna Rejzner
“I enjoy being surprised by mathematics and its intrinsic difficulty. The moment I enjoy best is when the pieces of the puzzle fall into one coherent whole.” – Katrin Wendland
“You should not choose to do mathematics if you want to make money; your salary as a mathematician will never correspond to the amount of time and energy invested in your work.” – Margarida Mendes Lopes
“I like to find out as much about a mathematical object as possible, just as you might want to understand a person as well as possible.” – Oksana Yakimova
Der Beitrag Quotable Women in Mathematics erschien zuerst auf Heidelberg Laureate Forum.
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