## Plus Blog

 December 1, 2013 Here's a strange fact: if you look up some numbers, for example the numbers in your tax return, population sizes of Chinese provinces, or the length of the world's rivers, then most likely around 30% of these numbers start with the digit 1, around 18% start with the digit 2, 12.5% start with a 3, and so on, all the way to 9 (which only heads up around 5% of the numbers) - the larger the digit, the fewer numbers in your list start with it. This fact, known as Benford's law, applies to so many different kinds of data sets that it's often used to detect fraud. But why does it work? Well, if the processes that give rise to your list of numbers do produce a universal distribution of first digits, then this distribution should apply no matter what units you use. It should work no matter if you do your tax return in pounds or in euros, or measure your rivers in metres or miles - it's universal after all. This means that the distribution of first digits remains the same when you multiply your numbers by whatever constant you need to change between units. And it turns out that the only distribution with this property of scale invariance is precisely the Benford distribution. As an example, imagine that your first digits are distributed equally (roughly the same proportion of numbers begin with the digit 1, 2, 3...) – so NOT according to the Benford distribution. Is this distribution scale invariant? Let's see what happens when we multiply by 2. All numbers starting with 5, 6, 7, 8, and 9, when multiplied by 2, give a number starting with 1. By contrast, the only way to end up with a number beginning with, say, 3, is to start out with a number starting with 1. In other words, the resulting distribution of first digits, after multiplying by 2, is skewed towards 1. It's not uniform, so your original distribution is not scale invariant. It's not too hard to show that in order to be scale invariant, the first digits have to be distributed in the way stipulated by the Benford distribution. It's worth noting though that Benford's law only applies to data sets that are neither too random, nor too constrained: alas, it doesn't work for lottery numbers. To find out more about Benford's law read our article Looking out for number one. This news story explores an application of Benford's law to uncover potentially fraudulent elections. Return to the Plus Advent Calendar 1 comments November 29, 2013 Oooo.... We're getting excited! It's nearly time for our Plus Advent Calendar! The first door won't open until Sunday 1 December, but here's a sneek peek! November 22, 2013 As intelligent life forms in many universes join together to mark the 50th anniversary of Doctor Who, we thought we'd add our own little anniversary acknowledgement by revisiting the Plus posters' moment of fame! We were delighted when the BBC approached us back in 2005 to ask if some of the Plus posters we had created for schools could be used as set dressing in an episode of Doctor Who with a storyline revolving around maths classrooms. And accordingly, in April 2006, the Plus posters made their TV debut in the Doctor Who episode School Reunion, alongside tenth Doctor David Tennant. The Doctor has infiltrated a school under cover as a science teacher, and discovers an alien plot to use the enhanced intelligence of the school students to solve the Skasis Paradigm, the key to the fundamental secrets of the universe. Solving the Paradigm requires imagination as well as intelligence - and when you're looking for maths students who are both imaginative and intelligent, of course they'll be reading Plus... See those colourful posters in the background? That's them! Image © BBC The posters which featured in the episode were launched in 2004, funded by EPSRC, in order to showcase the wide range of careers open to people with a background in maths. As Plus readers know know, the huge range of subjects featured in our articles and career interviews is proof of the amazing range of possibilities open to mathematicians : from avalanche researcher to audio software engineer, almost every field you can think of is represented on the Careers with Maths posters. The posters were intended to spread this message and to be a resource for teachers and career advisors dealing with mathematically-minded students. All the Plus posters We are extremely proud of the appearance of the Plus posters on Doctor Who, and very grateful to Charles Trevelyan, Plus's graphic designer. The posters were hugely successful: at one point we were receiving over 1000 requests for them a day! Although the posters are long out of print and no longer being distributed you can still download and print out a copy for yourself. With The Doctor's help, maths fame has now spread throughout the TV-viewing world and beyond, reaching intelligent civilisations of all ages and dimensions. But then, if they're intelligent they probably already got the message. Enjoy the 50th anniversary episode tomorrow night, whichever time and relative dimension in space you find yourself! November 20, 2013 In the eighteenth century the city we now know as Kaliningrad was called Königsberg and it was part of Prussia. Like many other great cities Königsberg was divided by a river, called the Pregel. It contained two islands and there were seven bridges linking the various land masses. A famous puzzle at the time was to find a walk through the city that crossed every bridge exactly once. Many people claimed they had found such a walk but when asked to reproduce it no one was able to. In 1735 the mathematician Leonhard Euler explained why: he showed that such a walk didn't exist. Euler's solution is surprisingly simple — once you look at the problem in the right way. The trick is to get rid of all unnecessary information. It doesn't matter what path the walk takes on the various land masses. It doesn't matter what shape the land masses are, or what shape the river is, or what shape the bridges are. So you might as well represent each land mass by a dot and a bridge by a line. You don't have to be geographically accurate at all: as long as you don't disturb the connectivity of the dots, which is connected to which, you can distort your picture in any way you like without changing the problem. Transforming the problem. Image: Bogdan Giuşcă. Once you have represented the problem in this way, its features are much easier to see. After playing around with it for a while you might notice the following: when you arrive at a dot via a line (enter a land mass via the bridge), then unless it is the final dot at which your walk ends, you need to leave it again, by a different line as those are the rules of the game. That is, any dot that is not the starting and end-point of your walk needs to have an even number of lines coming out of it: for every line along which you enter there has to be one to leave. For a walk that crosses every line exactly once to be possible, at most two dots can have an odd number of lines coming out of them. In fact there have to be either two odd dots or none at all. In the former case the two correspond to the starting and end points of the walk and in the latter, the starting and end points are the same. In the Königsberg problem, however, all dots have an odd number of lines coming out of them, so a walk that crosses every bridge is impossible. Euler's result marked the beginning of graph theory, the study of networks made of dots connected by lines. He was also able to show that if a graph satisfies the condition above, that the number of dots with an odd number of lines is either zero or two, then there will always be a path through it that crosses every line exactly once. The result also marked the beginning of topology, which studies shapes only in terms of their connectivity, without taking note of distances and angles. The London tube map is a great example of the topological triumph. By distorting distances and angles it turns what would otherwise be an unintelligible mess into a map that every tourist can read effortlessly. You can find out more here. November 19, 2013 Michael Foale with a model of Mir. Image © Mathematical Sciences Research Institute. "That's when I thought, 'this could be the moment we all die'. The de-pressurisation alarm went off and this meant that there really was a hole in the station. I felt the pressure fall in my ears. And then it got cold. Really, really cold." No, it's not a quote from Gravity, but the words of astronaut Michael Foale recalling an incident in 1997 that nearly cost him his life. Foale's story has all the ingredients for a movie: a damaged space station, no ground communications, no power, and only one chance to get it right. The crew used the soyuz's thrusters — their only means of getting back home — to control the station's chaotic spin. Their main tools were their brains and the basic maths of motion. As Foale put it, "You cannot be an effective mission designer, astronaut, flight controller or engineer if you don't know maths." Read our interview with Foale for some real-life adventure, space and maths, or watch the Gravity trailer below. November 6, 2013 How do you balance a cardboard cut-out of a triangle on a pencil? Trial and error is one way, but maths can save you lots of bending down and picking it up. Take the pencil and a ruler and connect the mid-point of each side to the opposite corner. You'll find that the three lines intersect in a single point, which lies exactly a third of the way from the midpoint of each side to the opposite vertex. That point, called the centroid, is the centre of mass of the triangle. If the triangle is made from uniform material, so it's not more lumpy in some places than in others, then the centroid is the unique point on which you can balance it without it tipping over. Amazingly, the centroid would also be the centre of mass of the triangle if its mass was concentrated only at its corners, and evenly divided between them. The centroid of a triangle. Image created using an interactivity from Math Open Reference. Instead of drawing a line from the mid-point of a given side to the opposite corner, you could also draw the line which passes through the mid-point but forms a right angle with the side the mid-point is on. If you do this for each side you again get three lines, and again these meet at a single point, called the circumcentre of the triangle. If you now draw a circle with the circumcentre as its center passing through one of the triangle's corners, you will find that the other two corners of the triangle lie on the circle too! The circumcentre of the triangle is also the centre of the unique circle that contains the three corners of the triangle. But it doesn't need to lie inside the triangle — in fact, it only does if all the triangle's angles are less than 90 degrees (so the triangle is acute). If one angle is greater than 90 degrees (the triangle is obtuse) then the circumcenter lies outside the triangle, and if one angle is exactly equal to 90 degrees then it is the mid-point of the hypothenuse. The circumcentre of a triangle. Image created using an interactivity from Math Open Reference. But there's another point that qualifies as a centre of a triangle. You find it by drawing aline from each corner that is perpendicular to the opposite side. Amazingly, the three lines again meet in a single point, called the orthocentre. As for the circumcentre, the orthocentre lies inside the triangle if the triangle is acute and outside it if it is obtuse. If one of the angles is exactly equal to 90 degrees then the orthocenter will be one of the corners. The orthocentre of a triangle. Image created using an interactivity from Math Open Reference. And what links all these points together? A straight line! The beautiful fact that the centroid, circumcentre and orthocentre of a triangle all lie on a straight line was first noticed in the 18th century by Leonhard Euler, one of the most prolific mathematicians of all time. That line now carries his name: it's called the Euler line of a triangle. You can play around with the three different centres and the Euler line on Math Open Reference which has beautiful interactive demonstrations, from which we made the images illustrating these articles.