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MENACE at Manchester Science Festival<div class='paragraph'>A few weeks ago, I took the copy of <a href='http://www.mscroggs.co.uk/blog/19' >MENACE</a> that I built to Manchester Science Festival, where it played around 300 games against the public while learning to play Noughts and Crosses. The group of us operating MENACE for the weekend included <a href="http://standupmaths.com" target="new">Matt Parker</a>, who made <a href="https://youtu.be/R9c-_neaxeU" target="new">two</a> <a href="https://youtu.be/KcmjOtkULi4" target="new">videos</a> about it. Special thanks go to Matt, plus<a href="https://twitter.com/stecks" target="new">Katie Steckles</a>,<a href="https://twitter.com/aaclarke4" target="new">Alison Clarke</a>,<a href="https://twitter.com/Andrew_Taylor" target="new">Andrew Taylor</a>,<a href="https://twitter.com/AshFrankland" target="new">Ashley Frankland</a>,David Williams,<a href="https://twitter.com/aPaulTaylor" target="new">Paul Taylor</a>,<a href="https://twitter.com/samheadleand" target="new">Sam Headleand</a>,Trent Burton, and<a href="https://twitter.com/zoelgriffiths" target="new">Zoe Griffiths</a> for helping to operate MENACE for the weekend.</div><div class='paragraph'>As <a href='http://www.mscroggs.co.uk/blog/19' >my original post about MENACE</a> explains in more detail, MENACE is a machine built from 304 matchboxes that learns to play Noughts and Crosses. Each box displays a possible position that the machine can face and contains coloured beads that correspond to the moves it could make. At the end of each game, beads are added or removed depending on the outcome to teach MENACE to play better.</div><h3>Saturday</h3><div class='paragraph'>On Saturday, MENACE was set up with 8 beads of each colour in the first move box; 3 of each colour in the second move boxes; 2 of each colour in third move boxes; and 1 of each colour in the fourth move boxes. I had only included one copy of moves that are the same due to symmetry.</div><div class='paragraph'>The plot below shows the number of beads in MENACE's first box as the day progressed.</div><a class='zoom' href='javascript:showlimage("sat-same-axes.png")'><img src='http://www.mscroggs.co.uk/img/320/sat-same-axes.jpg'></a><h3>Sunday</h3><div class='paragraph'>Originally, we were planning to let MENACE learn over the course of both days, but it learned more quickly than we had expected on Saturday, so we reset is on Sunday, but set it up slightly differently. On Sunday, MENACE was set up with 4 beads of each colour in the first move box; 3 of each colour in the second move boxes; 2 of each colour in third move boxes; and 1 of each colour in the fourth move boxes. This time, we left all the beads in the boxes and didn't remove any due to symmetry.</div><div class='paragraph'>The plot below shows the number of beads in MENACE's first box as the day progressed.</div><a class='zoom' href='javascript:showlimage("sun-same-axes.png")'><img src='http://www.mscroggs.co.uk/img/320/sun-same-axes.jpg'></a><h3>The Data</h3><div class='paragraph'>You can download the full set of data that we collected over the weekend <a href="https://www.dropbox.com/s/0f9ukxxvh0pgdui/MENACE-data.zip?dl=0" target="new">here</a>. This includes the first two moves and outcomes of all the games over the two days, plus the number of beads in each box at the end of each day. If you do something interesting (or non-interesting) with the data, <a href='http://www.mscroggs.co.uk/contact' >let me know</a>!</div>
http://www.mscroggs.co.uk/blog/45
http://www.mscroggs.co.uk/blog/4514 Nov 2017 11:36:22 GMTSunday Afternoon Maths LXI
<h2>XYZ</h2><div class='paragraph'>Which digits \(X\), \(Y\) and \(Z\) fill this sum?</div>$$\begin{array}{cccc}&X&Z&Y\\+&X&Y&Z\\\hline&Y&Z&X\end{array}$$
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http://www.mscroggs.co.uk/puzzles/LXI20 Aug 2017 12:00:00 GMTSunday Afternoon Maths LX
<h2>Where is Evariste?</h2><div class='paragraph'>Evariste is standing in a rectangular formation, in which everyone is lined up in rows and columns. There are 175 people in all the rows in front of Evariste and 400 in the rows behind him. There are 312 in the columns to his left and 264 in the columns to his right.</div><div class='paragraph'>In which row and column is Evariste standing?</div>
<h2>Bending a Straw</h2><div class='paragraph'>Two points along a drinking straw are picked at random. The straw is then bent at these points. What is the probability that the two ends meet up to make a triangle?</div>
http://www.mscroggs.co.uk/puzzles/LX
http://www.mscroggs.co.uk/puzzles/LX02 Jul 2017 12:00:00 GMTBig Ben Strikes Again<div class='paragraph'>As a child, I was a huge fan of <em>Captain Scarlet and the Mysterons</em>, Gerry Anderson's puppet-starring sci-fi series.Set in 2068, the series follows Captain Scarlet and the other members of Spectrum as they attempt to protect Earth from the Mysterons.One of my favourite episodes of the series is the third: <em><a href="http://www.dailymotion.com/video/x3kegry" target="new">Big Ben Strikes Again</a></em>.</div><div class='paragraph'>In this episode, the Mysterons threaten to destroy London. They do this by hijacking a vehicle carrying a nuclear device, and driving it to a car park. In the car park, the driver of the vehicle wakes up and turns the radio on. Then something weird happens: Big Ben strikes thirteen!</div><a class='zoom' href='javascript:showlimage("radio.jpg")'><img src='http://www.mscroggs.co.uk/img/320/radio.jpg'></a><div class='caption'>The driver turning on the radio. Good to know that BBC Radio 4 will still broadcast at 92-95FM in 2068.</div><div class='paragraph'>Following this, the driver is knocked out again and wakes up in a side road somewhere. After hearing his story, Captain Blue works out thatthe car park must be 1500 yards away from Big Ben. Using this information, Captains Blue and Scarlet manage to trackdown the nuclear device and save the day.</div><a class='zoom' href='javascript:showlimage("scarlet_map.jpg")'><img src='http://www.mscroggs.co.uk/img/320/scarlet_map.jpg'></a><div class='caption'>A map of London with a circle of radius 1500 yards drawn on it.</div><div class='paragraph'>After rewatching the episode recently, I realised that it would be possible to recreate this scene and hear Big Ben striking thirteen.</div><h3>Where Does Big Ben Strike Thirteen?</h3><div class='paragraph'>At the end of the episode, Captain Blue explains to Captain Scarlet that the effect was due to light travelling faster than sound: as thedriver had the radio on, he could hear Ben's bongs both from the tower and through the radio. As radio waves travel faster thansound, the bongs over the radio can be heard earlier than the sound waves travelling through the air.Further from the tower, the gap between when the two bongs are heardis longer; and at just the right distance, the second bong on the radio will be heard at the same time as the first bong from the tower.This leads to the appearance of thirteen bongs: the first bong is just from the radio, the next eleven are both radio and from the tower, andthe final bong is only from the tower.</div><div class='paragraph'>Big Ben's bongs are approximately 4.2s apart, sound travels at 343m/s, and light travels at 3×10<sup>8</sup>m/s (this is so fast that itcould be assumed that the radio waves arrive instantly without changing the answer). Using these, we perform the following calculation:</div>$$\text{time difference} = \text{time for sound to arrive}-\text{time for light to arrive}$$$$=\frac{\text{distance}}{\text{speed of sound}}-\frac{\text{distance}}{\text{speed of light}}$$$$=\text{distance}\times\left(\frac1{\text{speed of sound}}-\frac1{\text{speed of light}}\right)$$$$\text{distance}=\text{time difference}\div\left(\frac1{\text{speed of sound}}-\frac1{\text{speed of light}}\right)$$$$=4.2\div\left(\frac1{343}-\frac1{3\times10^8}\right)$$$$=1440\text{m}\text{ or }1574\text{ yards}$$<div class='paragraph'>This is close to Captain Blue's calculation of 1500 yards (and to be fair to the Captain, he had to calculate it in his head in a few seconds).Plotting a circle of this radius centred at Big Ben gives the points where it may be possible to hear 13 bongs.</div><div class='attributedimage'><a class='zoom' href='javascript:showlimage("scarlet_my_map.jpg")'><img src='http://www.mscroggs.co.uk/img/320/scarlet_my_map.jpg'></a><a href="https://www.openstreetmap.org/" target="new">OpenStreetMap</a></div><div class='paragraph'>Again, the makers of <em>Captain Scarlet</em> got this right: their circle shown earlier is a very similar size to this one.To demonstrate that this does work (and with a little help from <a href="https://twitter.com/televisionduck" target="new">TD</a> and her camera),I made the following video yesterday near Vauxhall station. I recommend using earphones to watch it as the later bongs are quite faint.</div><video width='512' height='288' controls poster='/video/13.jpg'> <source src='http://www.mscroggs.co.uk/video/13.webm' type='video/webm'> <source src='http://www.mscroggs.co.uk/video/13.ogg' type='video/ogg'> <source src='http://www.mscroggs.co.uk/video/13.mp4' type='video/mp4'>Your browser does not support the video tag.</video>
http://www.mscroggs.co.uk/blog/44
http://www.mscroggs.co.uk/blog/4403 Jun 2017 09:59:03 GMTSunday Afternoon Maths LIX
<h2>Turning Squares</h2><div class='paragraph'>Each square on a chessboard contains an arrow point up, down, left or right. You start in the bottom left square. Every second you move one square in the direction shown by the arrow in your square. Just after you move, the arrow on the square you moved from rotates 90° clockwise. If an arrow would take you off the edge of the board, you stay in that square (the arrow will still rotate).</div>
<div class='paragraph'>You win the game if you reach the top right square of the chessboard. Can I design a starting arrangement of arrows that will prevent you from winning?</div>
<h2>Elastic Numbers</h2><div class='paragraph'><i>Throughout this puzzle, expressions like \(AB\) will represent the digits of a number, not \(A\) multiplied by \(B\).</i></div><div class='paragraph'>A two-digit number \(AB\) is called <i>elastic</i> if:</div><ol><li>\(A\) and \(B\) are both non-zero.</li><li>The numbers \(A0B\), \(A00B\), \(A000B\), ... are all divisible by \(AB\).</li></ol><div class='paragraph'>There are three elastic numbers. Can you find them?</div>
<h2>Square Pairs</h2><div class='paragraph'>Can you order the integers 1 to 16 so that every pair of adjacent numbers adds to a square number?</div><div class='paragraph'>For which other numbers \(n\) is it possible to order the integers 1 to \(n\) in such a way?</div>
http://www.mscroggs.co.uk/puzzles/LIX
http://www.mscroggs.co.uk/puzzles/LIX23 Apr 2017 12:00:00 GMTThe End of Coins of Constant Width<div class='paragraph'>Tomorrow, the new 12-sided one pound coin is released.</div><div class='attributedimage'><a class='zoom' href='javascript:showlimage("one-pound.png")'><img src='http://www.mscroggs.co.uk/img/320/one-pound.jpg'></a><a href="http://www.royalmint.com/aboutus/policies-and-guidelines/advertising-guidelines" target="new">Royal Mint</a></div><div class='paragraph'>Although I'm excited about meeting this new coin, I am also a little sad,as its release ends the era in which all British coins are shapes of constantwidth.</div><h3>Shapes of Constant Width</h3><div class='paragraph'>A shape of constant width is a shape that is the same width in every direction,so these shapes can roll without changing height. The most obvious such shapeis a circle. But there are others, including the shape of the seven-sided 50pcoin.</div><div class='attributedimage'><a class='zoom' href='javascript:showlimage("50p2008.jpg")'><img src='http://www.mscroggs.co.uk/img/320/50p2008.jpg'></a><a href="http://www.royalmint.com/aboutus/policies-and-guidelines/advertising-guidelines" target="new">Royal Mint</a></div><div class='paragraph'>As shown below, each side of a 50p is part of a circle centred around the opposite corner.As a 50p rolls, its height is always the distance between one of the corners andthe side opposite, or in other words the radius of this circle. As these circlesare all the same size, the 50p is a shape of constant width.</div><div class='attributedimage'><a class='zoom' href='javascript:showlimage("50p2008circle.jpg")'><img src='http://www.mscroggs.co.uk/img/320/50p2008circle.jpg'></a><a href="http://www.royalmint.com/aboutus/policies-and-guidelines/advertising-guidelines" target="new">Royal Mint</a></div><div class='paragraph'>Shapes of constant width can be created from any regular polygon with anodd number of sides, by replacing the sides by parts of circles centred at theopposite corner. The first few are shown below.</div><div class='attributedimage'><a class='zoom' href='javascript:showlimage("Reuleaux_polygons.png")'><img src='http://www.mscroggs.co.uk/img/320/Reuleaux_polygons.jpg'></a><a href="https://en.wikipedia.org/wiki/File:Reuleaux_polygons.svg" target="new">en wiki user LEMeZza</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en" target="new">CC BY-SA 3.0</a></div><div class='paragraph'>It's also possible to <a href="https://commons.wikimedia.org/wiki/File:How_to_make_mathematical_roller_curve_based_on_triangle.svg" target="new">create shapes of constant width from irregular polygons with an odd number</a>,but it's not possible to create them from polygons with an even number of sides.Therefore, the new 12-sided pound coin will be the first non-constant width British coin sincethe (also 12-sided) threepenny bit was phased out in 1971.</div><div class='paragraph'>Back in 2014, I <a href="http://www.mscroggs.co.uk/blog/7" target="new">wrote to my MP</a> in an attempt to findout why the new coin was not of a constant width. He forwarded my letter tothe Treasury, but I never heard back from them.</div><h3>Pizza Cutting</h3><div class='paragraph'>When cutting a pizza into equal shaped pieces, the usual approach is tocut along a few diameters to make triangles. There are other ways to fairlyshare pizza, including the following (that has appeared here before as an answer to <a href="http://www.mscroggs.co.uk/puzzles/86" target="new">this puzzle</a>):</div><a class='zoom' href='javascript:showlimage("pizza-answer.png")'><img src='http://www.mscroggs.co.uk/img/320/pizza-answer.jpg'></a><div class='paragraph'>The slices in this solution are closely related to a triangle of constantwidth. Solutions can be made using other shapes of constant width,including the following, made using a constant width pentagon and heptagon (50p):</div><a class='zoom' href='javascript:showlimage("pizza-answer-pent.png")'><img src='http://www.mscroggs.co.uk/img/320/pizza-answer-pent.jpg'></a><a class='zoom' href='javascript:showlimage("pizza-answer-hept.png")'><img src='http://www.mscroggs.co.uk/img/320/pizza-answer-hept.jpg'></a><div class='paragraph'>There are many more ways to cut a pizza into equal pieces. You can find them in <ref1>.</div><div class='paragraph'>You can't use the shape of a new pound coin to cut a pizza though.</div><div class='edit'>Edit: Speaking of new £1 coins, I made this stupid video with <a href="https://twitter.com/pecnut" target="new">Adam "Frownsend" Townsend</a> about them earlier today:</div><references><1><t>Infinite families of monohedral disk tilings</t><a>Joel Haddley and Stephen Worsley</a><d>December 2015</d><l>https://arxiv.org/pdf/1512.03794.pdf</l></1></references>
http://www.mscroggs.co.uk/blog/43
http://www.mscroggs.co.uk/blog/4327 Mar 2017 18:28:11 GMTDragon Curves II<div class='note'>This post appeared in <a href="http://chalkdustmagazine.com/read/issue-05" target="new">issue 05 of <i>Chalkdust</i></a>. I stronglyrecommend reading the rest of <i>Chalkdust</i>.</div><div class='paragraph'>Take a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper \(n\) times, then the edge will make an order \(n\) dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of <a href="http://chalkdustmagazine.com/read/issue-05" target="new">issue 05 of <i>Chalkdust</i></a> is an order 10 dragon curve.</div><a class='zoom' href='javascript:showlimage("folding-chalkdust.jpg")'><img src='http://www.mscroggs.co.uk/img/320/folding-chalkdust.jpg'></a><a class='zoom' href='javascript:showlimage("labelled-chalkdust.jpg")'><img src='http://www.mscroggs.co.uk/img/320/labelled-chalkdust.jpg'></a><div class='caption'>Top: Folding a strip of paper in half four times leads to an order four dragon curve (after rounding the corners). Bottom: A level 10 dragon curve resembling a dragon.</div><div class='paragraph'>The dragon curves on the cover show that it is possible to tile the entire plane with copies of dragon curves of the same order. If any readers are looking for an excellent way to tile a bathroom, I recommend getting some dragon curve-shaped tiles made.</div><div class='paragraph'>An order \(n\) dragon curve can be made by joining two order \(n-1\) dragon curves with a 90° angle between their tails. Therefore, by taking the cover's tiling of the plane with order 10 dragon curves, we may join them into pairs to get a tiling with order 11 dragon curves. We could repeat this to get tilings with order 12, 13, and so on... If we were to repeat this <i>ad infinitum</i> we would arrive at the conclusion that an order \(\infty\) dragon curve will cover the entire plane without crossing itself. In other words, an order \(\infty\) dragon curve is a space-filling curve.</div><div class='paragraph'>Like so many other interesting bits of recreational maths, dragon curves were popularised by <a href="http://chalkdustmagazine.com/biographies/mathematical-games-martin-gardner/" target="new">Martin Gardner</a> in one of his <i>Mathematical Games</i> columns in <i>Scientific American</i>. In this column, it was noted that the endpoints of dragon curves of different orders (all starting at the same point) lie on a logarithmic spiral. This can be seen in the diagram below.</div><a class='zoom' href='javascript:showlimage("dragons_with_spiral.jpg")'><img src='http://www.mscroggs.co.uk/img/320/dragons_with_spiral.jpg'></a><div class='caption'>The endpoints of dragon curves of order 1 to 10 with a logarithmic spiral passing through them.</div><div class='paragraph'> Although many of their properties have been known for a long time and are well studied, dragon curves continue to appear in new and interesting places. At last year's <a href="http://www.mathsjam.com/conference/" target="new">Maths Jam conference</a>, <a href="http://aperiodical.com/author/paul/" target="new">Paul Taylor</a> gave a talk about my favourite surprise occurrence ofa dragon.</div><div class='paragraph'>Normally when we write numbers, we write them in base ten, with the digits in the number representing (from right to left) ones, tens, hundreds, thousands, etc. Many readers will be familiar with binary numbers (base two), where the powers of two are used in the place of powers of ten, so the digits represent ones, twos, fours, eights, etc.</div><div class='paragraph'>In his talk, Paul suggested looking at numbers in base -1+i (where i is the square root of -1; you can find more adventures of i <a href='http://chalkdustmagazine.com/features/variations-fermat-agony-four-fits/'>here</a>) using the digits 0 and 1. From right to left, the columns of numbers in this base have values 1, -1+i, -2i, 2+2i, -4, etc. The first 11 numbers in this base are shown below.</div><table><tr><td>Number in base -1+i</td><td>Complex number</td></tr><tr><td align='right'>0</td><td>0</td><tr><td align='right'>1</td><td>1</td><tr><td align='right'>10</td><td>-1+i</td><tr><td align='right'>11</td><td>(-1+i)+(1)=i</td><tr><td align='right'>100</td><td>-2i</td><tr><td align='right'>101</td><td>(-2i)+(1)=1-2i</td><tr><td align='right'>110</td><td>(-2i)+(-1+i)=-1-i</td><tr><td align='right'>111</td><td>(-2i)+(-1+i)+(1)=-i</td><tr><td align='right'>1000</td><td>2+2i</td><tr><td align='right'>1001</td><td>(2+2i)+(1)=3+2i</td><tr><td align='right'>1010</td><td>(2+2i)+(-1+i)=1+3i</td></table><div class='paragraph'>Complex numbers are often drawn on an Argand diagram: the real part of the number is plotted on the horizontal axis and the imaginary part on the vertical axis. The diagram to the left shows the numbers of ten digits or less in base -1+i on an Argand diagram. The points form an order 10 dragon curve! In fact, plotting numbers of \(n\) digits or less will draw an order \(n\) dragon curve.</div><a class='zoom' href='javascript:showlimage("complex_dragon_10.jpg")'><img src='http://www.mscroggs.co.uk/img/320/complex_dragon_10.jpg'></a><div class='caption'>Numbers in base -1+i of ten digits or less plotted on an Argand diagram.</div><div class='paragraph'>Brilliantly, we may now use known properties of dragon curves to discover properties of base -1+i. A level \(\infty\) dragon curve covers the entire plane without intersecting itself: therefore every Gaussian integer (a number of the form \(a+\text{i} b\) where \(a\) and \(b\) are integers) has a unique representation in base -1+i. The endpoints of dragon curves lie on a logarithmic spiral: therefore numbers of the form \((-1+\text{i})^n\), where \(n\) is an integer, lie on a logarithmic spiral in the complex plane.</div><div class='note'>If you'd like to play with some dragon curves, you can download the <a href="https://github.com/mscroggs/dragon_curves" target="new">Python code used to make the pictures here.</a></div>
http://www.mscroggs.co.uk/blog/42
http://www.mscroggs.co.uk/blog/4208 Mar 2017 03:22:57 GMTThe Importance of Estimation Error<div class='paragraph'>Recently, I've noticed a few great examples of misleading uses of numbers in news articles.</div><div class='paragraph'>On 15 Feb, BBC News published a breaking news article with the headline<a href="https://web.archive.org/web/20170215094259/http://www.bbc.co.uk/news/business-38979582" target="new">"UK unemployment falls by 7,000 to 1.6m"</a>.This fall of 7,000 sounds big; but when compared to the total of 1.6m, itis insignificant. The change could more accurately be described as a fall from 1.6m to 1.6m.</div><div class='paragraph'>But there is a greater problem with this figure. In the<a href="https://www.ons.gov.uk/employmentandlabourmarket/peopleinwork/employmentandemployeetypes/bulletins/uklabourmarket/feb2017" target="new">original Office of National Statistics (ONS) report</a>,the fall of 7,000 was accompanied by a 95% confidence interval of ±80,000.When calculating figures about large populations (such as unemployment levels), it is impossible to ask every person in the UK whether theyare employed or not. Instead, data is gathered from a sample and this is used to estimate the total number. The 95% confidence intervalgives an idea of the accuracy of this estimation: 95% of the time, the true number will lie of the confidence interval. Therefore, we canthink of the 95% confidence interval as being a range in which the figure lies (although this is not true, it is a helpful way to thinkabout it).</div><div class='paragraph'>Compared to the size of its confidence interval (±80,000), the fall of 7,000 is almost indistinguishable from zero. This means that it cannot be said with any confidence whether the unemployment level rose or fell. This is demonstrated in the following diagram.</div><a class='zoom' href='javascript:showlimage("errorbars1.jpg")'><img src='http://www.mscroggs.co.uk/img/320/errorbars1.jpg'></a><div class='caption'>A fall of 7,000 ± 80,000. The orange line shows no change.</div><div class='paragraph'>To be fair to the BBC, the headline of the article changed to <a href="http://www.bbc.co.uk/news/business-38979582" target="new">"UK wage growth outpaces inflation"</a>once the article was upgraded from breaking news to a complete article, and a mention of the lack of confidence in the change was added.</div><div class='paragraph'>On 23 Feb, I noticed another BBC News with misleading figures: <a href="http://www.bbc.co.uk/news/uk-39062436" target="new">Net migration to UK falls by 49,000</a>.This 49,000 is the difference between322,000 (net migration for the year ending 2015) and273,000 (net migration for the year ending 2016).However both these figures are estimates: in the <a href="https://www.ons.gov.uk/peoplepopulationandcommunity/populationandmigration/internationalmigration/bulletins/migrationstatisticsquarterlyreport/feb2017" target="new">original ONS report</a>,they were placed in 95% confidence intervals of ±37,000 and ±41,000 respectively. As can be seen in the diagram below,there is a significant portion where these intervals overlap, so it cannot be said with any confidence whether or not net immigration actually fell.</div><a class='zoom' href='javascript:showlimage("errorbars2.jpg")'><img src='http://www.mscroggs.co.uk/img/320/errorbars2.jpg'></a><div class='caption'>Net migration in 2014-15 and 2015-16.</div><div class='paragraph'>Perhaps the blame for this questionable figure lies with the ONS, as it appeared prominently in their report while the discussion of itsaccuracy was fairly well hidden. Although I can't shift all blame from the journalists: they should really be investigating the quality of thesefigures, however well advertised their accuracy is.</div><div class='paragraph'>Both articles criticised here appeared on BBC News. This is not due to the BBC being especially bad with figures, but simply due to thefact that I spend more time reading news on the BBC than in other places, so noticed these figures there. I quick Google search reveals that the unemployment figure wasalso reported, with little to no discussion of accuracy, by <a href="https://www.theguardian.com/uk-news/2017/feb/23/net-migration-to-uk-falls-by-49000-after-brexit-vote" target="new">The Guardian</a>,<a href="https://www.ft.com/content/40a56e7d-24a3-3ec3-8195-92a338710efc" target="new">the Financial Times</a>, and<a href="http://news.sky.com/story/brexodus-net-migration-falls-below-300000-lowest-for-three-years-10778369" target="new">Sky News</a>.</div>
http://www.mscroggs.co.uk/blog/41
http://www.mscroggs.co.uk/blog/4125 Feb 2017 14:50:20 GMT