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Sunday 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 GMTIs MEDUSA the New BODMAS?<div class='note'>I wrote this post with, and after much discussion with <a href="http://adamtownsend.com" target="new">Adam Townsend</a>. It also appeared on the <a href="http://chalkdustmagazine.com/blog/medusa-new-bodmas/" target="new">Chalkdust Magazine blog</a>.</div><div class='paragraph'>Recently, <a href="https://twitter.com/icecolbeveridge" target="new">Colin "IceCol" Beveridge</a> blogged about something that's been irking him for a while: those annoying social media posts that tell you to work out a sum, such as \(3-3\times6+2\), and state that only $n$% of people will get it right (where \(n\) is quite small). Or as he calls it <a href="http://www.flyingcoloursmaths.co.uk/new-years-resolution-genius-sic/" target="new">"fake maths"</a>.</div><a class='zoom' href='javascript:showlimage("fake-maths.jpg")'><img src='http://www.mscroggs.co.uk/img/320/fake-maths.jpg'></a><div class='caption'>A classic example of <a href="http://www.flyingcoloursmaths.co.uk/new-years-resolution-genius-sic/" target="new">"fake maths"</a>.</div><div class='paragraph'>This got me thinking about everyone's least favourite primary school acronym: BODMAS (sometimes known as BIDMAS, or PEMDAS if you're American). As I'm sure you've been trying to forget, BODMAS stands for "<b>B</b>rackets, (to the power) <b>O</b>f, <b>D</b>ivision, <b>M</b>ultiplication, <b>A</b>ddition, <b>S</b>ubtraction" and tells you in which order the operations should be performed.</div><div class='paragraph'>Now, I agree that we all need to do operations in the same order (just imagine trying to explain your working out to someone who uses <i>BADSOM</i>!) but BODMAS isn't the order mathematicians use. It's simply wrong. Take the sum \(4-3+1\) as an example. Anyone can tell you that the answer is 2. But BODMAS begs to differ: addition comes first, giving 0!</div><div class='paragraph'>The problem here is that in reality, we treat addition and subtraction as equally important, so sums involving just these two operations are calculated from left-to-right. This caveat is quite a lot more to remember on top of BODMAS, but there's actually no need: Doing all the subtractions before additions will always give you the same answer as going from left-to-right. The same applies to division and multiplication, but luckily these two are in the correct order already in BODMAS (but no luck if you're using PEMDAS).</div><div class='paragraph'>So instead of BODMAS, we should be using <i>BODMSA</i>. But that's unpronounceable, so instead we suggest that from now on you use <b>MEDUSA</b>. That's right, <b>MEDUSA</b>:</div><ul><li><b>M</b>abano (<i>brackets</i> in Swahili)</li><li><b>E</b>xponentiation</li><li><b>D</b>ivision</li><li><b>U</b>kubuyabuyelela (<i>multiplication</i> in Zulu)</li><li><b>S</b>ubtraction</li><li><b>A</b>ddition</li></ul><div class='paragraph'>This is big news. MEDUSA vs BODMAS could be this year's <a href="https://www.youtube.com/watch?v=ZPv1UV0rD8U" target="new">pi vs tau</a>... Although it's not actually the biggest issue when considering sums like \(3-3\times6+2\).</div><div class='paragraph'>The real problem with \(3-3\times6+2\) is that it is written in a purposefully confusing and ambiguous order. Compare the following sums:</div>$$3-3\times6+2$$ $$3+2-3\times6$$ $$3+2-(3\times6)$$<div class='paragraph'>In the latter two, it is much harder to make a mistake in the order of operations, because the correct order is much closer to normal left-to-right reading order, helping the reader to avoid common mistakes. Good mathematics is about good communication, not tricking people. This is why questions like this are "fake maths": real mathematicians would never ask them. If we take the time to write clearly, then I bet more than \(n\)% of people will be able get the correct answer.</div>
http://www.mscroggs.co.uk/blog/40
http://www.mscroggs.co.uk/blog/4013 Jan 2017 04:32:41 GMTSunday Afternoon Maths LVIII
<h2>Factorial Pattern</h2>$$1\times1!=2!-1$$$$1\times1!+2\times2!=3!-1$$$$1\times1!+2\times2!+3\times3!=4!-1$$<div class='paragraph'>Does this pattern continue?</div>
<h2>Placing Plates</h2><div class='paragraph'>Two players take turns placing identical plates on a square table. The player who is first to be unable to place a plate loses. Which player wins?</div>
http://www.mscroggs.co.uk/puzzles/LVIII
http://www.mscroggs.co.uk/puzzles/LVIII01 Jan 2017 12:00:00 GMTChristmas (2016) is Over<div class='paragraph'>More than ten correct solutions to this year's <a href='http://www.mscroggs.co.uk/blog/36' >Advent calendar puzzle competition</a> were submitted on Christmas Day, so the competition is now over.(Although you can still submit your answers to get me to check them.) Thank-you to everyone who took part in the puzzle, I've had a lot offun watching your progress and talking to you on Twitter, Reddit, etc. You can find all the puzzles and answers (from 1 January) <a href='http://www.mscroggs.co.uk/puzzles/advent2016' >here</a>.</div><div class='paragraph'>The (very) approximate locations of all the entries I have received so far are shown on this map:</div><a class='zoom' href='javascript:showlimage("advent2016map.jpg")'><img src='http://www.mscroggs.co.uk/img/320/advent2016map.jpg'></a><div class='paragraph'>This year's winners have been randomly selected from the 29 correct entries on Christmas Day. They are:</div><table><tr><td>1</td><td>Jack Jiang</td></tr><tr><td>2</td><td>Steve Paget</td></tr><tr><td>3</td><td>Joe Gage</td></tr><tr><td>4</td><td>Tony Mann</td></tr><tr><td>5</td><td>Stephen Cappella</td></tr><tr><td>6</td><td>Cheng Wai Koo</td></tr><tr><td>7</td><td>Demi Xin</td></tr><tr><td>8</td><td>Lyra</td></tr><tr><td>9</td><td>David Fox</td></tr><tr><td>10</td><td>Bob Dinnage</td></tr></table><div class='paragraph'>Your prizes will be on their way in early January.</div><div class='paragraph'>Now that the competition has ended, I can give away a secret. Last year, <a href="https://twitter.com/zbvif/status/674235088901681152" target="new">Neal</a>suggested that it would be fun if a binary picture was hidden in the answers. So this year I hid one. If you write all the answers in binary, witheach answer below the previous and colour in the 1s black, you will see this:</div><a class='zoom' href='javascript:showlimage("advent2016secret.jpg")'><img src='http://www.mscroggs.co.uk/img/320/advent2016secret.jpg'></a><div class='paragraph'>I also had a lot of fun this year making up the names, locations, weapons and motives for the final murder mystery puzzle. In case you missed them these were:</div><table><thead><td>#</td><td>Murder suspect</td><td>Motive</td></thead><tr><td>1</td><td>Dr. Uno (uno = Spanish 1)</td><td><b>O</b>beying <b>n</b>ameless <b>e</b>ntity</td></tr><tr><td>2</td><td>Mr. Zwei (zwei = German 2)</td><td><b>T</b>o <b>w</b>orry <b>o</b>thers</td></tr><tr><td>3</td><td>Ms. Trois (trois = French 3)</td><td><b>T</b>o <b>h</b>elp <b>r</b>eally <b>e</b>vil <b>e</b>lephant</td></tr><tr><td>4</td><td>Mrs. Quattro (quattro = Italian 4)</td><td><b>F</b>or <b>o</b>ld <b>u</b>nknown <b>r</b>easons</td></tr><tr><td>5</td><td>Prof. Pum (pum = Welsh 5)</td><td><b>F</b>or <b>i</b>ndividual <b>v</b>iolent <b>e</b>nd</td></tr><tr><td>6</td><td>Miss. Zes (zes = Dutch 6)</td><td><b>S</b>topping <b>i</b>diotic <b>x</b>enophobia</td></tr><tr><td>7</td><td>Lord Seacht (seacht = Irish 7)</td><td><b>S</b>uspect <b>e</b>spied <b>v</b>ictim <b>e</b>ating <b>n</b>ewlyweds</td></tr><tr><td>8</td><td>Lady Oito (oito = Portuguese 8)</td><td><b>E</b>pic <b>i</b>nsanity <b>g</b>ot <b>h</b>im <b>t</b>oday</td></tr><tr><td>9</td><td>Rev. Novem (novem = Latin 9)</td><td><b>N</b>obody <b>i</b>n <b>n</b>ewsroom <b>e</b>xpected</td></tr></table><br /><table><thead><td>#</td><td>Location</td><td>Weapon</td></thead><tr><td>1</td><td>Thr<b>one</b> room</td><td>Wrench (1 vowel)</td></tr><tr><td>2</td><td>Ne<b>two</b>rk room</td><td>Rope (2 vowels)</td></tr><tr><td>3</td><td>Benea<b>th ree</b>ds</td><td>Revolver (3 vowels)</td></tr><tr><td>4</td><td>Edge o<b>f our</b> garden</td><td>Lead pipe (4 vowels)</td></tr><tr><td>5</td><td><b>Five</b>s court</td><td>Neighbour's sword (5 vowels)</td></tr><tr><td>6</td><td>On the <b>six</b>th floor</td><td>Super banana bomb (6 vowels)</td></tr><tr><td>7</td><td>Spar<b>se ven</b>ue</td><td>Antique candlestick (7 vowels)</td></tr><tr><td>8</td><td>W<b>eight</b>lifting room</td><td>A foul tasting poison (8 vowels)</td></tr><tr><td>9</td><td>Mathematics mezza<b>nine</b></td><td>Run over with an old Ford Focus (9 vowels)</td></tr></table><div class='paragraph'>Finally, well done to Scott,Matthew Schulz,Michael Gustin,Daniel Branscombe,Kei Nishimura-Gasparian,Henry Hung,Mark Fisher,Jon Palin,Thomas Tu,Félix Breton,Matt Hutton,Miguel,Fred Verheul,Martine Vijn Nome,Brennan Dolson,Louis de Mendonca,Roni,Dylan Hendrickson,Martin Harris,Virgile Andreani,Valentin Valciu,and Adia Batic for submitting the correct answer but being too unlucky to win prizes this year. Thank you all for taking part and I'll see younext December for the next competition.</div>
http://www.mscroggs.co.uk/blog/39
http://www.mscroggs.co.uk/blog/3928 Dec 2016 06:49:35 GMT