tag:blogger.com,1999:blog-58111244408382835022017-12-13T10:13:14.250+00:00Tony's Maths BlogA blog about maths things which interest me.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.comBlogger77125tag:blogger.com,1999:blog-5811124440838283502.post-28212044595232234512017-11-19T17:47:00.000+00:002017-11-19T17:47:53.557+00:00MathsJamHere is my overdue post on last weekend's MathsJam gathering. MathsJam is always a wonderfully exciting and enjoyable weekend, and this one was particularly good. I can honestly say that I enjoyed every talk, and was entertained, surprised and educated in roughly equal measures.<br /><br />I'm nervous about selecting highlights, because almost everything was a highlight, and my list inevitably leaves out many excellent talks. There is a list of all the talks on <a href="https://mathsjam.com/gathering/" target="_blank">the MathsJam website</a> but here are some I particularly remember (in the order in which they were presented):<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowFullScreen='true' webkitallowfullscreen='true' mozallowfullscreen='true' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwxgMIENZXvwzkl_8kWCltaiUXKUN7sSRv436H-L2IFMcBjvtpUd-zd_iarvTVIsKbl0pkwEXQfDbvE5eOOTg' class='b-hbp-video b-uploaded' FRAMEBORDER='0' /></div><br /><br /><ul><li>Simon's 3D-printed robot which solves Rubik's cube (time-lapse video shown here with Simon's permission);</li><li>Matt on logical deduction games, which brought back memories of playing Eleusis when I was a student and introduced others I need to find out about;</li><li>Noel-Ann on data and how to it can be represented (and misrepresented);</li><li>Zoe's poem about <i>e</i>, which (understandably) seems to be on everybody's highlight list;</li><li>Matthew's amazing recreation of a problem from <i>Captain Scarlet</i> about the bongs of Big Ben;</li><li>Andrew's paradoxical balloon monkey, which although made from a single balloon, has an underlying graph which is not semi-Eulerian;</li><li>Angela's poem;</li><li>Rachel on spinning yarn;</li><li>Alison on illogical units, and Dave on illogical scales;</li><li>Will on non-binary cellular automata;</li><li>Miles finding striking similarities between mountaineering and mathematics;</li><li>Glen showing how many holes a constructed object (equivalent to a T-shirt) possessed (most of MathsJam seem to have got it wrong, going for four rather than three!);</li><li>Sue on Ada Lovelace;</li><li>Paolo using a pack of cards to find two numbers from their sum and difference;</li></ul><br />And of course the "extra-curricular" puzzles, games and magic, Tiago showing me how to tie a knot with one hand, and the spectacular mathematical cakes.<br /><br />So once more a memorable MathsJam, with an excellent range of speakers and talks, friendly atmosphere and fascinating and surprising mathematics. The organisers once again did an amazing job!<br /><br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-55004153374050931832017-11-08T20:22:00.002+00:002017-11-08T20:26:57.212+00:00Looking forward to MathsJamWith only three days to go till the <a href="https://mathsjam.com/gathering/" target="_blank">MathsJam Gathering</a> - the best weekend of the year, I've been thinking of some of my favourite MathsJam discoveries. Sticking to pure mathematics, here are my memories of three gems. I could have chosen many others, but perhaps because these happen to relate to my current teaching, and I showed two of them to my graph theory students immediately upon my return from the gathering, they are the first that come to mind. Since I believe all MathsJam presentations are available online, further details should be readily available. <br /><br /><br /><ul><li>Colin Wright's amazing talk on graph colouring, which started by asking us to complete a partially-completed 3-colouring of a small graph, and turned into a more-or-less complete proof, within a 5-minute talk, that there is no polynomial-time algorithm for 3-colouring a graph.</li><li>Ross Atkins's talk about Braess's Paradox - a simple situation in which adding an extra road to a network, with no increase in traffic, results in longer average journey times. I should have known about this counter-intuitive result so I'm very glad to have found out about it, and especially with the wonderful demonstration with a network of springs that showed a mechanical realisation of the paradox.</li><li>David Bedford's "What's my polynomial?" I love this because it is arguably what the late Raymond Smullyan called a "monkey trick". David asked you to think of a polynomial <i>p</i>(<i>x</i>) with non-negative integer coefficients, and, for a single value of <i>x</i> of your choice, greater than any of the coefficients, tell him both <i>x</i> and <i>p</i>(<i>x</i>). He would then tell you your polynomial. Knowing that one needs <i>n</i> values to determine a polynomial of degree <i>n</i>, I was taken in by this!</li></ul><br /><br />I could have chosen many more examples: I'm certainly not ranking these presentations or any others. On another day I might have chosen a completely different set! But I'm certainly looking forward to coming across more wonderful mathematics this weekend!Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-70255610580448587382017-11-01T19:38:00.002+00:002017-11-01T19:38:50.931+00:00Black Mathematician MonthOne of the best things to have happened for mathematics in the UK recently is the arrival of <a href="http://chalkdustmagazine.com/" target="_blank">Chalkdust</a> magazine - an exciting, witty magazine with a unique style. (It's very different in feel from the equally admirable, and much missed, <i>iSquared</i>, which is happily <a href="https://www.stem.org.uk/resources/collection/4088/isquared-magazine" target="_blank">preserved online</a><i>.</i>)<br /><br />And the best thing that Chalkdust has done is the Black Mathematician Month which has just finished - a month of interviews, conversations and activities "promoting black mathematicians, and talking about building a more representative mathematical community". The stories that were told were sometimes shocking, sometimes horrifying, often inspiring, and very important. I was lucky enough to be one of the large audience for the final event, an excellent talk about the Black Heroes of Mathematics by <a href="http://nirachamberlain.com/" target="_blank">Nira Chamberlain</a> (and I was particularly pleased that several undergraduates from the University of Greenwich were also there). Nira told us about a number of great black mathematicians: despite his own negative experiences as a young black man wishing to become a mathematician, and the obstacles in his way, his presentation was overwhelmingly positive in tone and his passion communicated strongly with the audience.<br /><br />I myself was a very privileged mathematics student. I had an adequate grant and did not need to work while I was studying. I had a supportive family. Both my parents went to university (probably quite unusual for the time although I didn't realise that), as did my father's sister (I believe the first woman from her school to do so) and all my siblings. I was supported not only by their academic expectations but by their understanding of university education. I was well prepared by excellent schoolteachers. Careers advisers encouraged me to study maths, not to forget that ambition and aim to be a boxer (as Nira was advised) or a singer (as Nira's son, alarmingly recently, was told).<br /><br />I understood some of that privilege at the time. But of course, I was also white and male. It is only now, when I look at the achievement of people like Nira, and many of our students at Greenwich who have overcome enormous obstacles, that I am beginning to understand just how that contributed to my privilege. My mathematics cohort as an undergraduate was almost all white (possibly even entirely white: I don't remember any exceptions) and largely male. When I look at my classes (and colleagues) at Greenwich, I feel very glad to have the opportunity to work with such diverse people.<br /><br /><a href="http://chalkdustmagazine.com/black-mathematician-month/closing-first-black-mathematician-month/" target="_blank">Chalkdust's reflections on Black Mathematician Month</a> deserve wide circulation. This feels like an important initiative, which hopefully will help all potential mathematicians, whatever their race or gender, have the opportunity to follow their dreams, inspired by people like Nira and the other mathematicians featured.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-52063999282897686872017-10-14T17:18:00.000+01:002017-10-14T17:18:16.834+01:00Monty HallTwo weeks ago, on Saturday 30 September, two big names in mathematics died. Vladimir Voevodsky, who was only 51, made huge contributions to mathematics. I became aware of his importance to contemporary mathematics when reading Michael Harris's wonderfully stimulating book <i>Mathematics without Apologies </i>and regret that I do not know much about him and his work.<br /><br />The other, Monty Hall, was not a mathematician but a game show host, who has given his name to one of the most famous recreational mathematical puzzles. A lot has been written about the Monty Hall Problem: I recommend Jason Rosenhouse's book (called, surprisingly enough, <i>The Monty Hall Problem</i>, which gives an excellent account of the embarrassing (for male mathematicians) <i>l'affaire Parade</i> which brought the puzzle to public notice - see <a href="http://marilynvossavant.com/game-show-problem/" target="_blank">http://marilynvossavant.com/game-show-problem/</a> for the correspondence.<br /><br />In Monty Hall's game show, a contestant had to choose one of three boxes. One contained a car: the other two each contained a goat. After the contestant had made their choice, Monty (who knew which box contained the car) would sometimes open the door of an unchosen box to reveal a goat, and then offer the contestant the chance to change their choice. Should the contestant switch?<br /><br />I remember, as a schoolboy, discussing with my friends a problem in one of Martin Gardner's books. Three prisoners, A, B and C, are told that on the next day two of the three will be executed: which two has already been decided randomly. (As I get older I increasingly find the rather bloodthirsty settings of puzzles like this in very poor taste: why do so many mathematical puzzles involve the abuse and execution of prisoners?) A knows that his chance of survival is 1/3. The guard won't answer any question which would give him information about whether or not he has been chosen for execution. But A points out to the guard that at least one of the other two is going to die, so if the guard identifies to A one of the others who will die, then that cannot give any information about A's fate: whichever two have been selected, the guard can answer this question without revealing whether A has also been chosen.<br /><br />So the guard tells A that C is going to die. A is now happy: his survival chance was 1/3 but has now gone up to 1/2 since it is either him or B who will survive.<br /><br />Of course (on certain assumptions) A is wrong: it is B whose survival chance has gone up to 2/3. A's chance is unchanged at 1/3. If A and B were selected to die, the guard would tell A that B was ill-fated. If it was A and C to die, then the guard would answer "B". Bit if B and C are both going to die, then the guard could answer either "B" or "C", and if one assumes the guard chooses randomly which to name, then enumerating the cases shows that when the guard answers "C", two times out of three it was A rather than B who is also selected for death.<br /><br />This led me to get the Monty Hall Problem wrong when I first read about it in a newspaper article (the <i>Independent</i>, perhaps around Christmas 1990?) Knowing that in the Gardner problem A's chances haven't changed, I assumed that the quiz show contestant can't improve their chance of winning by switching. This is plain wrong, but I wonder if memories of Gardner's puzzle led astray many of the mathematicians who on first seeing it got the Monty Hall Problem wrong? Although my initial answer was wrong, on careful reading of the analysis I did quickly come to agree that the contestant should switch, and verified this by computer simulation.<br /><br />In fact, the problems are very closely related. What Monty Hall is doing is essentially saying to A, "C is going to die - would you like to change places with B?" And since B has a 2/3 chance of survival, A should certainly accept that offer. (On certain assumptions.)<br /><br />But the assumptions are critical (and most recent presentations of the Monty Hall Problem do make this clear.) The 2/3 probability of winning if the contestant switches assumes that the host will always carry out the procedure, and that, when the contestant has initially chosen the box with the car, that the host will choose one of the other boxes to open with equal probability. (If the host simply always opens the nearer box with the goat, then on the occasions when the host opens the further away box, the contestant will know that a switch guarantees success.) And the host might not go through this procedure every time. If the host wants to save his employers money, then he might only offer the switch option on those occasions when the contestant has initially chosen the winning box. If the host likes the contestant, he might only offer the switch option when the initial choice is losing.<br /><br />In fact (according to, for example, <a href="https://en.wikipedia.org/wiki/Monty_Hall" target="_blank">the Wikipedia entry for Month Hall</a>, in the real game show Monty did not always offer the choice. He was playing a psychological game with the viewer, and, when "The Monty Hall Problem" became famous, he was well aware that the conditions necessary for the mathematical puzzle did not in fact apply to his game show. I find it very pleasing that the game show host had a better understanding of the mathematics problem than many of the mathematicians whose instinctive answer, like mine, was wrong. <br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-11220750679086645462017-09-23T16:00:00.001+01:002017-09-23T16:00:19.685+01:00How I won a jar of sweetsI was the proud winner of the <a href="http://picbear.com/greenwichmathsoc" target="_blank">Greenwich University MathSoc</a> "guess how many sweets in the jar" competition at the Welcome Fair this week.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-p0sldGug7Ig/WcZ1PxovkrI/AAAAAAAAARw/wjgBRwsYTWEs7F6fi00xmB1LOhKEMabogCLcBGAs/s1600/TM%2Bsweets.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="664" data-original-width="768" height="276" src="https://3.bp.blogspot.com/-p0sldGug7Ig/WcZ1PxovkrI/AAAAAAAAARw/wjgBRwsYTWEs7F6fi00xmB1LOhKEMabogCLcBGAs/s320/TM%2Bsweets.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">I was delighted to win because of the method I used, based on the "Wisdom of Crowds", which provided far superior to my colleagues' misguided attempts to estimate the volume of the jar and the average volume of the sweets. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">With a problem like "how many sweets in the jar", no individual is likely to be able to make a particularly accurate estimate. Some will guess in the right ballpark, while some will wildly over- or under- estimate. The theory is, however, that if one averages many independent guesses, the result is likely to be close to the solution.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I tried this for my competition entry. I could see seventeen previous entries, so I added them up (rounding to simplify the addition of seventeen three-digit numbers while talking to the MathSoc team at their stand) and divided by 17. This gave me 337, which was my entry. The exact solution was 336, which is a striking vindication of the Wisdom of Crowds.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Of course, this method isn't guaranteed to work. For example, if one had asked people to guess how many points Leicester City would win in the English Premier League in 2015/16, and taken the average, one would not have been close!</div><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-69448029535501530202017-06-12T19:41:00.002+01:002017-06-12T19:42:09.498+01:00Greenwich Maths Time - a Festival of MathematicsSince it's only two weeks until the <a href="http://www.gre.ac.uk/mathsfestival" target="_blank">IMA Festival of Mathematics and its Applications</a> at the University of Greenwich, which I am helping to organise, it's time I wrote something about it here. This national Festival presents, over two days, about 50 talks, workshops and activities, showcasing the diversity of mathematics and mathematicians. Visitors can learn about applications of mathematics in statistics, cryptography, numbers, fashion, medicine, fire safety engineering, and many more: they can learn how to make boomerangs and they can walk on custard. There is something for everybody!<br /><br />The Festival has been generously sponsored by the IMA, the University of Greenwich, FMSP, GCHQ, the OR Society, the London South East and Kent & Medway Maths Hubs, and FDM, For information about the sponsors, click on the logos on <a href="http://www.gre.ac.uk/mathsfestival" target="_blank">the Festival website</a>.<br /><br />I've been delighted that so many top presenters have agreed to take part in the Festival. If you can get to Greenwich on either of these days, you'll have a wonderful time exploring lots of exciting mathematics!Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-11010941177876119392017-05-09T21:28:00.001+01:002017-05-09T21:28:33.864+01:00A Champions League Mathematical curiosity(Apologies for any reader who doesn't share my interest in football. You don't have to read on.)<br /><br />When I'm fed up marking, like so many other middle-aged men, I turn to the computer game Football Manager. And a disaster for my team last night made me aware of a curiosity regarding the tie-break rules in the Champions League.<br /><br />This is the situation. It's the last minute of the last match of the group stage and my team, Arsenal, are losing 2-1 at home to Olympiakos (we haven;t had much luck and the red card early in the first half for our midfielder Victor Wamyama has cost us). But it's OK - Bayern Munich (who are losing in Lyon) will win the group and we will come second, qualifying for the knock-out stages, while Olympiakos will be out.<br /><br />But what's this? Bayern have scored a last-minute equaliser against Lyon! They were going to win the group anyway, so presumably we still finish in that all-important second place.<br /><br />But no - Olympiakos are now above us: we are down to third and we are out! A goal in the other game, which doesn't change the position of the teams in that game, has caused Arsenal and Olympiakos to swap positions, with disastrous results.<br /><br />It's all to do with <a href="https://en.wikipedia.org/wiki/2015%E2%80%9316_UEFA_Champions_League_group_stage#Tiebreakers" target="_blank">tie-break rules</a>. Without Bayern's late goal, all four teams would have had eight points, The tie-break rule looks at the scores in matches involving all the teams which have tied on points. In this case Bayern (who scored six at home to Olympiakos) had the best goal difference, with Arsenal (who had played well until the last match, despite early red cards ion three matches) second best, and so they finish first and second. But Bayern's equaliser means they have nine points, and Arsenal and Olympiakos are tied for second place: and since the two teams drew in Greece and Olympiakos won in London, the Greeks are above Arsenal. That last-minute goal in the other match really has moved Olympiakos above Arsenal. (If you want to check for yourself, I give all the scores below.)<br /><br />This consequence of the mathematics of the tie-break rules is something I was vaguely aware of, though I didn't realise the peril my team were in until about ten game-minutes before Bayern's goal. But it is slightly counter-intuitive that a goal in match B can cause the teams in match A to change positions. (In fact in my game, before the goal Lyon had been in third place above Olympiakos, so conceding the goal caused them to drop to fourth, but a slightly different set of results could have meant that a game in match B could leave the two teams in that match in the same position while inverting the order of the other two teams.)<br /><br />Here, for anyone who cares, are the complete results:<br />Game 1 - Arsenal (2) 2 Bayern (0) 0; Lyon (0) 1 Olympiakos (0) 0<br />Game 2 - Bayern (0) 0 Lyon (0) 0; Olympiakos (0) 1 Arsenal (0) 1<br />Game 3 - Lyon (0) 1 Arsenal (1) 3; Olympiakos (0) 1 Bayern (0) 1<br />Game 4 - Arsenal (0) 0 Lyon (0) 0; Bayern (3) 6 Olympiakos (0) 0<br />Game 5 - Bayern (0) 1 Arsenal (0) 0; Olympiakos (0) 1 Lyon (0) 0<br />Game 6 - Arsenal (0) 1 Olympiakos (0) 2; Lyon (3) 3 Bayern (0) 3<br /><br /><br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-69216811801940986052017-05-01T17:06:00.001+01:002017-05-01T17:06:02.297+01:00Something I'd forgottenOne of my parents daily rituals was to change the date on a device on top of the bureau which displayed the day, date and month. My equivalent is to change the date on my Rubik's cube-style calendar , which also gives the day, ate and month: the month in three-letter form ("Jan", "Feb", "Mar" etc). It's nice that the names of the months in English make this possible (though www.puzl.co.uk, where I got my calendar cube, have also sold French and German versions).<br /><br />But I was astonished to find in the cellar yesterday that I had made something similar myself, 35 years ago. I have absolutely no recollection of this at all (but the handwriting is mine), but I am quite impressed by my ingenuity.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-1dIHH36lQ8U/WQdbs5YUSEI/AAAAAAAAARY/sePeYOVbSfAOrJR36LQahvc7tOjBnLnVwCEw/s1600/Rubik%2BCube%2BBackgammon%2Bscorer.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="296" src="https://4.bp.blogspot.com/-1dIHH36lQ8U/WQdbs5YUSEI/AAAAAAAAARY/sePeYOVbSfAOrJR36LQahvc7tOjBnLnVwCEw/s320/Rubik%2BCube%2BBackgammon%2Bscorer.JPG" width="320" /></a></div><br />In my final year as a student, I played backgammon regularly with my friend Karen. It appears that I constructed this device to keep track of the cumulative score. It's rather the worse for wear, and has lost one of the corner cubies. But I assume that it still records the final state of play when we had played our last game - which shows that, if you want to win at a gambling game, it is best not to play against a future professor of psychology.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-2963603032898006522017-04-01T09:49:00.002+01:002017-04-01T09:49:24.009+01:00How to use number theory to help bus travellersLiving in London, I use buses a lot. Each day I catch a 180 or 199, ignoring the 188, 286 and others whcih don't take me home. I have a good memory for all the different buses in the parts of central London that I frequent, but wouldn't it be easier if the bus number told me where it was going? If, rather than 180 arbitrarily designating a route between Lewisham and Belvedere, the number itself indicated where the bus goes?<br /><br />So today I am going to unveil my scheme for taking advantage of the properties of numbers to do exactly that. I will start off with my first idea, and then show enhancements.<br /><br />The Fundamental Theorem of Arithmetic tells us that any number can be expresssed as the product of prime numbers in an essentially unique way. So my first scheme allocates to every destination a different prime number. For example, we might assign 2 to Waterloo, 3 to Euston and 5 to London Bridge. Then the bus route serving these three places would be number 30. All you need to do, as a passenger, is know the number assigned to your destination. If you want to go to Waterloo, you know that any even-numbered bus will take you there: if you are heading for London Bridge then any bus whose number ends with 0 or 5 will do,<br /><br />You might object that it is easy to tell when a number is divisible by 2 or 5, but less easy if your destination's number is, say, 17. But <a href="https://en.wikipedia.org/wiki/Divisibility_rule" target="_blank">there are tricks</a>: to test whether a number is divisible by 17, one simply tests the number obtained by subtracting five times the last digit from the rest (so for 374, we subtract 20 from 37, getting 17: that tells us 17 divides 374).<br /><br />But this method has a glaring weakness. It doesn't tell us the order of the stops. I want to be able to distinguish the route Euston - Waterloo - London Bridge from Euston - London Bridge - Waterloo since I want to travel direct from Euston to Waterloo, and going via London Bridge would take much longer. So my improved proposal uses Godel numbering. If a bus visits destinations a, b, c, ... in that order, its number is 2^a times 3^b times 5^c times ... And this can be dynamically adjusted during the journey: when the bus has left Euston, the number changes to reflect that the next stop is Londo Bridge. Now, I can find out not only whether the bus takes me to Waterloo, but how many stops there are first. In my example, if I am at Euston, bus number 288 (2^5x3^2) will take me to London Bridge and then Waterloo, while 1944 (2^3*3^5) will go to London Bridge via Waterloo.<br /><br />But my final scheme is even better. In this one, The bus number is the product of the primes representing the places it visits, each raised to the power of the number of minutes it is expected to take to get there. If a bus doesn't call at destination <i>p</i>, then the number is not a product of <i>p</i>. So if I want to get to Waterloo, and bus number <i>n </i>arrives, I check to see what is the largest power of 2 which divides <i>n</i>: that is how many minutes it will take to get there. Of course, this is adjusted dynamically, in the same way as London's bus stops now tell us how long it will be before the next bus arrives.<br /><br />You might object that this system assumes bus passengers can carry out mental arithmetic. But of course, there will be apps to do this for those who are not confident. I will just point my smartphone at the front of a bus, and the app will read the number (say 7200) and tell me that it will be at Waterloo (destination 2) in 5 minutes (since the highest power of 2 dividing 7200 is 2^5 = 32).<br /><br />Yet another way in which mathematics can make our lives easier!<br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-68447531747021354242017-03-12T17:56:00.002+00:002017-03-12T17:56:43.993+00:00Puzzles from my grandmotherI have been reading the autobiography of one of my heroes - the great popular mathematics writer, Martin Gardner, who along with my school maths teachers Jimmy Cowan and Ivan Wells, inspired me with the excitement of mathematics. I have come late to Gardner's autobiography, <i>Undiluted Hocus-Pocus</i>, which was published a few years ago, partly because of luke-warm reviews, and while I enjoyed many of the anecdotes, I wouldn't regard it as essential reading even for those who, like me, admire Gardner enormously. But I'm glad to have read it.<br /><br />One of Gardner's stories reminded me of my own childhood. Gardner recounts his uncle telling him a riddle. "There was one duck with two ducks behind it; one duck with two ducks in front of it; and one duck between two ducks. How many ducks were there?"<br /><br />SPOILER ALERT: Gardner notes that his uncle began by saying, "There were three ducks", which gave away the answer.<br /><br />This reminded me of my paternal grandmother giving me two puzzles when I must have been perhaps in the upper levels of primary school. I had to make sense of the following:<br /><br />First riddle:<br /><br />11 was a race-horse.<br />22 was 12.<br />1111 race.<br />22112.<br /><br />Second riddle:<br /><br />If the B MT put : .<br />If the B . putting : .<br /><br />(I have just googled the first of these, and curiously it is described on one website as a tongue-twister, which it isn't!)<br /><br />SOLUTIONS<br /><br />The first riddle reads as "One-one was a race-horse. Two-two was one too. One-one won one race. Two-two won one too."<br /><br />The second is "If the grate be empty, put coal on. If the grate be full, stop putting coal on."Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-26562708310282770822017-01-02T17:37:00.002+00:002017-01-02T17:37:10.769+00:00Why (some) mistakes are interesting<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-b9oQe0xp-eQ/WGqOSbhwLAI/AAAAAAAAAQs/iHZOd_HJs0w-BFLBLStsbJjlSeoB1TG7wCLcB/s1600/event92.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Image of Hull Paragon crash from www.railwaysarchive.co.uk" border="0" src="https://1.bp.blogspot.com/-b9oQe0xp-eQ/WGqOSbhwLAI/AAAAAAAAAQs/iHZOd_HJs0w-BFLBLStsbJjlSeoB1TG7wCLcB/s1600/event92.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Having recently put up here a couple of posts about mistakes, I feel I should perhaps say why I find (some) mistakes fascinating. <br /><br />As you would expect, there are a number of reasons.<br /><br />Perhaps a mistake casts interesting light on someone's thought processes, revealing the way a mathematician was approaching a problem or what was in their had as they tackled it.<br /><br />Perhaps it is simply a cautionary example: seeing how someone else has erred helps one avoid making the same mistake.<br /><br />Yet another reason is not exactly schadenfreude (though that may be part of it) but that seeing better people than me make errors is encouraging: I make lots of mistakes and it's helpful to realise that most other people do too!<br /><br />Now, I worked for many years as a software engineer, working on safety-critical systems, and I have taught software engineering.<br /><br />Errors occur too often in software, and the better we understand how we make errors, the more likely we are to be able to reduce their frequency.<br /><br />When I was writing software, I felt that there were lessons to be learnt from railway accidents, particularly those where a remarkable combination of circumstances defeated what had seemed to be an infallible system.<br /><br />You might have had considerable faith in the Tyer electric tablet system, which for many years after its introduction prevented the dreadful collisions on single-track lines which had occurred when two trains travelling in opposite directions entered the same section: but <a href="https://en.wikipedia.org/wiki/Abermule_train_collision" target="_blank">at Abermule in 1921 </a>the combination of many tiny lapses by several individuals subverted what had appeared to be an infallible system.<br /><br />Equally unlucky was <a href="https://en.wikipedia.org/wiki/Hull_Paragon_rail_accident" target="_blank">the Hull Paragon accident in 1927</a>, when two apparently independent slips by signalmen interacted in an extremely unlikely way to subvert the signalling system which protected the trains (the photo above comes from www.railwaysarchive.co.uk).<br /><br />As a software engineer,I felt there was a lot to be learnt from thinking about such system failures: could I be confident that my own system could not fail in some unlikely combination of circumstances, when the accidents at Abermule and Hull Paragon show how even apparently the most secure systems can fail?<br /><br />Rhese are some reasons I think (some) mistakes are worth our study.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com2tag:blogger.com,1999:blog-5811124440838283502.post-30733200824075900092016-12-29T18:36:00.002+00:002016-12-29T18:38:43.394+00:00A curious cat and another curious errorThe first mathematics book written in English was the snappily-titled, anonymous <i>An introduction for to lerne to recken with the pen, or with the counters accordyng to the trewe cast of Algorisme, in hole numbers or in broken, newly corrected...</i>, first published in the 1530s. A few years ago the British Library paid £95,000 for a copy of the first edition. Happily, a facsimile (of the second edition) is now cheaply available in the wonderful series produced by <a href="http://www.renascentbooks.co.uk/" target="_blank">TGR Renascent Books</a>.<br /><br />The book concludes by presenting lots of interesting problems and their solutions. One is a version of the Josephus problem, in which fifteen out of thirty merchants are to be cast overboard to save an overloaded galley in a storm: the reader is given a mnemonic for arranging the merchants so that the right fifteen (the Christians rather than the Saracens, as one would expect for the time) are saved. The mnemonic is a Latin verse, which seems a little odd for a book whose selling point was that it was in English! Presumably the author was padding his book out with whatever came to hand, not very carefully, as we shall see.<br /><br />Another example asks about travellers going in opposite directions between London and Paris and when they will meet, or at least I think that is the intention: but since in the book one traveller is going between Paris and London, and the other between Paris and Lyon, they are unlikely to meet unless one of them gets badly lost. It seems that our author was trying to make the problem more relevant to an English readership but didn't carry his intention through.<br /><br />The most curious of the problems, to my mind, is "The rule and questyon of a Catte". The problem presented (transcribed from a 1546 edition) is,<br /><span style="font-family: inherit;"><br /></span><br /><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><span style="font-family: inherit;">"There is a catte at the fote of a tre the lēght of 300 fote / this catte goeth upwarde eche day 17 fote, and descendeth the nyghte 12 fote. I demaunde in howe ōge tyme that she be at ŷ toppe."</span></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><span style="font-family: inherit;"><br /></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><span style="font-family: inherit;">Or in today's spelling, "</span>There is a cat at the foot of a tree of height 300 feet. This cat goes upward each day 17 feet, and descends each night 12 feet. I ask, how long a time will she take to reach the top?" It's good to see that even as far back as the sixteenth century, authors of maths textbooks were presenting realistic real-life problems to their students.</div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;">Luckily our author provides the solution:</div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><span style="font-family: inherit;">"Answere. Take by and abate the nyghte of the day, that is 12 of 17 and there remayneth 5, there fore the catte mounteth eche daye 5 fote / deuyde now 300 by 5 and thereof cometh 60 dayes then she shall be at the toppe. And thus ye maye do of all other semblable."</span></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><span style="font-family: inherit;"><br /></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><span style="font-family: inherit;">That is, "</span>Subtract the night from the day, that is 12 from 17: this gives 5, therefore the cat mounts each day 5 feet. Divide now 300 by 5 and you get 60 days: then she shall be at the top. And thus you may do all other similar problems."</div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;">Which is very neat and useful. But unfortunately the answer is wrong. After sliding down 12 feet on the 57th night, the cat will be at a height of 285 feet and will reach the top of the tree after 58 days, not 60.</div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;">What is curious about this is that the author seems to have missed the point of the puzzle. It is interesting, surely, only because it is a trick question, but the author has fallen for the trick. What has happened?</div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; unicode-bidi: embed; vertical-align: baseline;">A historian friend with whom I discussed this had a plausible idea. The author was probably taking his problems from a continental book. This book may have presented the problem of the cat, and first derived the incorrect solution as above, but then went on to say something like, "But in fact this is not the correct answer, because ..." and explained the trick. However the unwary writer of <i>An introduction ...</i> didn't read any further, and reproduced the problem with the incorrect solution. Who knows how many generations of English students were bemused by their cats gaining the top of the tree two days earlier than they had calculated as a result of this carelessness?</div>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-10245400085252789972016-12-02T19:49:00.002+00:002016-12-02T19:49:40.782+00:00Learning from the audience at my Prisoners' Dilemma talkToday I had the privilege of taking part in an excellent "<a href="http://www.thetrainingpartnership.org.uk/study-days/subjects/maths/" target="_blank">Mathematics in Action</a>" day at which around 700 school students heard a series of talks about maths. I was talking about one of my favourite subjects, the Prisoners' Dilemma (PD) (I gave <a href="http://www.gresham.ac.uk/lectures-and-events/when-maths-doesnt-work-what-we-learn-from-the-prisoners-dilemma" target="_blank">a rather different talk about the same material at Gresham College</a> a few years ago.) I was in amazing company: the other speakers were David Acheson, James Grime and Hannah Fry, and we were wonderfully compered by Sarah Wiseman. It was a delight to talk to such an enthusiastic audience.<br /><br />This was my second such event, and this time I had enough confidence to ask the audience (by show of hands) how they would play the games I was discussing. I had hoped to win more than my fair share of the "Rock, Paper and Scissors" games, but the audience out-thought me (it probably didn't help my cause that, thanks to a clicker malfunction, my choice was revealed to the audience earlier than I had intended).<br /><br />But the really interesting thing was when I asked whether the audience would co-operate or defect in the Prisoners' Dilemma. To my surprise, the vast majority of the audience chose to co-operate: to my even greater surprise, those who defected were loudly hissed by many in the audience! This made a point that I was coming to (very briefly) at the end of my talk: games like the PD give us insights not just into games but into issues like trust and reputation. If defecting in a PD results in this kind of opprobrium, then the benefit of a shorter prison sentence may be negated by the damage to one's reputation, and this kind of peer pressure makes co-operation a more profitable choice than defection.<br /><br />But then when I asked the audience about the "Cold War arms race" PD - should a superpower invest its resources in more and bigger nuclear weapons rather than in health research and education? - the response was different. People who would co-operate in the standard PD, rather than betray their friend, chose to build up their nuclear arsenals. Furthermore, there was no hissing. (To be fair, the way I asked the questions may have had a lot to do with the answers, so I am not claiming that the audience behaviour proves anything at all, only that it is suggestive.) <br /><br />So basically it appears that we frown upon people who are selfish in their dealings with individuals, but when it is selfishness at national level, our response is quite different. If this reading is correct, then there ia a real challenge to achieving co-operation between nations because we perceive that kind of co-operation as fundamentally different from people interacting as individuals, and we don't feel the same social behaviour to be nice to other nations as we do when we interact with people.<br /><br />As Martin Nowak says in his fascinating book <i>Super Co-operators</i>, "<span style="font-family: inherit;">… Our analysis of how to solve the [Prisoners'] Dilemma will never be completed. This Dilemma has no end."</span>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-89441056894745713022016-11-29T20:30:00.000+00:002016-11-29T20:30:00.102+00:00A curious errorOne of the happy outcomes from the recent MathsJam conference (See my previous post) was that I was able to get my copies of two recent maths books signed by their authors. (It would have been three had my copy of <i>Matemagia</i> not already been signed.) One was <i>Problems for Metagrobologists</i> by the mathematical puzzles expert David Singmaster. This is a collection of mathematical puzzles, with a lot of fascinating insights and historical comments. (I'm not convinced David's title will maximise sales of this excellent book: using a word which many of your target audience may not understand is probably not the most effective marketing technique.)<br /><br />I was interested in Singmaster's discussion of his puzzle 196 ("Not so likely"). He quotes a 1977 magic book which indicates a good bet - you ask your dupe to cut a pack of cards into three and bet even money that there is an Ace, a Two or a Jack at the bottom of one of the piles. Apparently the original author indicates that there is about a two-thirds probability that you will win - Singmaster not only asks you whether this is right (it isn't!) but also invites you to speculate on how the author went wrong.<br /><br />I was interested because I had recently acquired a 1964 book on mathematical magic, and had opened it at exactly the same problem (well, in this case the winning cards are an Ace, a Four and a Jack). But this author (whose blushes I will spare by not naming him) gives a different answer, albeit also wrong. This book indicates that the chance of winning is 36 in 52, There are twelve ways the bottom card of the first pile can win, twelve winning cards for the bottom card of the second pile, and twelve for the third pile: a total of 36. I will leave the reader to identify the fallacy in this argument.<br /><br />The author goes on to say that, of one cut the deck into four piles, one would have an overwhelming 48 in 52 chance of winning. (Did he try it? It wouldn't take too many attempts to cast doubt on that claim.)<br /><br />I find it astonishing that the author didn't go one step further, and comment that if one divided the deck into five piles, one would have 60 chances of winning out of 52. At that point it becomes clear that something is very wrong with the argument. <br /><br />So why didn't the author take that step, and see that he had made an error? Did he do so, realise there was a problem, and give up, hoping his readers wouldn't notice? Did the author really believe his answers? Is the whole thing some sort of joke? Probability is a difficult subject and it is easy to go wrong. But this author is not only well-read but also well-connected (he thanks Martin Gardner for advice in his introduction). Could he really be unaware that he had got this example so badly wrong? Whereas Singmaster in his book gives a plausible guess as to how his author arrived at his erroneous two-thirds figure, I find it puzzling that my author didn't carry his argument that one step further and see that he had gone wrong. <br /><br />I'm not quite sure what moral to draw from this!Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-89920235134620971682016-11-15T20:14:00.001+00:002016-11-15T20:18:18.336+00:00The impossibility of blogging about MathsJamI thought I would do a blog post on the <a href="http://www.mathsjam.com/conference/" target="_blank">MathsJam weekend conference</a> which I have just attended. This was two days of short (five-minute maximum) talks about interesting maths, delivered by a wide variety of speakers. Although by definition anything presented at MathsJam is recreational, topics varied from very pure mathematics to very applied, with statistics, operational research, computing and communication all included (and also art and poetry). I think I say this every year, but 2016 was the best MathsJam yet.<br /><br />My plan for this blog post was to describe three or four highlights and ideas I had taken away. But after a quick look at my notes I find that that would be impossible, There were too many highlights to mention: no small selection could be fair. So all I'm going to do is refer readers to the website which, we are promised, will in due course make the presentations available, and thank the organisers and participants for providing such an amazing, inspiring, friendly weekend.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-10093726543860204152016-10-15T20:12:00.005+01:002016-10-15T20:12:45.287+01:00Puzzles - to be solved, or to be admired?The composer Howard Skempton once said to me that there are three kinds of piano music: music to be read, music to be played, and music to be listened to. I think something similar is true of mathematical puzzles.<br /><br />Certainly one ought to attempt most mathematical puzzles for oneself, before looking at the solution. But I think there are some examples where one can admire the solution without attempting the puzzle oneself. One example might be the <a href="https://en.wikipedia.org/wiki/100_prisoners_problem" target="_blank">100 prisoners problem</a> - the solution is beautiful and I don't think that I would have gained anything by spending a long time thinking about the problem before looking it up. I don't feel too bad about looking up how to solve the 5x5x5 Rubik cube - I did work out how to solve the 3x3x3 one by myself (albeit almost 40 years ago: I might not be able to do that now) so I didn't feel that I had to prove anything to myself, and I felt that I had better things to do with my time. (On the other hand, the fact that I am writing this self-justifying post may suggest that I do feel some guilt about this!)<br /><br />Anyway, here is one problem that is certainly in the "to be solved for oneself" category. It was knew to me: I came across it, surprisingly, in a literary novel - Ethan Canin's <i>A Doubter's Almanac</i>, one of the small category of novels in which the principal character is a Fields Medallist. (The only other one I can immediately think of is Peter Buwalda's <i>Bonita Avenue</i> - if you know of any others, please tell me!)<br /><blockquote class="tr_bq">A mathematician buys a lottery ticket, choosing six different integers between 1 and 46. She (it's "he" in the book) chooses her numbers so that the sum of their base-ten logarithms is an integer. How many possible choices are there?</blockquote>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-91099687820396643622016-07-24T20:16:00.003+01:002016-07-24T20:16:31.913+01:00The Murdered Mathematician<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Sn4ZRt_wfNE/V5UTh45tspI/AAAAAAAAAQA/ARzCzRuTkngAK_xtoBva-qwKdp0aZhXWgCLcB/s1600/Mathematician500.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://4.bp.blogspot.com/-Sn4ZRt_wfNE/V5UTh45tspI/AAAAAAAAAQA/ARzCzRuTkngAK_xtoBva-qwKdp0aZhXWgCLcB/s320/Mathematician500.jpg" width="320" /></a></div><div><br /></div><div><br /></div>One of my interests is mathematicians in fiction - fictitious mathematicians tell us something about how the extra-mathematical world views us (of course there have been great novels by mathematicians too). There is crime fiction in which mathematicians are murderers or detectives. I particularly like Hector Hawton's <i>Murder by Mathematics</i> (HT John Sharp who told me about it), in which the Head of the Mathematics Department in a London university gets murdered, and it turns out (implausibly, I hope) that everybody in his professional and personal life wanted him dead.<div><br /></div><div>Perhaps the strangest such novel is Harry Stephen Keeler's <i>The Murdered Mathematician</i>. Keeler (1890 - 1967) was an eccentric novelist many of whose (decidedly unusual) works are, wonderfully, available from <a href="http://www.ramblehouse.com/" target="_blank">Ramble House Press</a> (http://www.ramblehouse.com/) whose service is excellent. There is <a href="https://en.wikipedia.org/wiki/Harry_Stephen_Keeler" target="_blank">an entertaining Wikipedia article</a>.</div><div><br /></div><div>In <i>The Murdered Mathematician </i>the victim is an eccentric professor, "Radical Luke" (whose radicalness is exemplified by his refusal to use Greek letters in doing mathematics). The murder is solved by Quiribus Brown, who is 7 foot 6 tall, and has been taught higher mathematics by his father. The book contains an exam paper of Radical Luke's, and Brown uses some fairly sophisticated mathematics to identify the murderer. It is certainly unlike any other book I have ever read! I'm now reading the further adventures of Quiribus Brown in <i>The Case of the Flying Hands.</i></div>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-40496494489659320092016-07-09T18:03:00.002+01:002016-07-09T18:03:50.319+01:00Maths and Tennis (Again)<a href="http://tonysmaths.blogspot.co.uk/2012/07/mathematics-and-tennis.html" target="_blank">I wrote about mathematics and tennis at this time of year</a> four years ago: specifically on the decisions about reviews and about the curious fact that, against a stronger opponent, you are more likely to break serve from 30-15 down than from love all. I had completely forgotten about that post, but was stimulated to return to the topic by a comment on the BBC web coverage of yesterday's semifinal between Raonic and Federer. Raonic was 40-love up on his service, and according to the report, rather than follow the standard tactic of playing a safe second service that is almost certain to avoid a double fault, Raonic chose to attempt to serve aces on first and second serves, backing himself to be successful in one of his six opportunities.<br /><br />So is this a good strategy when you are 40-love up on your service? How much better than your second serve does your first serve have to be to make this tactic optimal?Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-53809180370158362612016-06-19T19:56:00.004+01:002016-06-19T19:56:49.374+01:00Maths my father taught meToday being Father's Day, I thought I would write about a piece of mathematics my father showed me many years ago.<br /><br />I have been reading Erica Walker's inspiring book <i>Beyond Banneker: Black Mathematicians and the Paths to Excellence</i>, a study of three generations of Black mathematicians in the USA, the obstacles they faced, and the networks and structures which supported them. It was interesting that, although few of them had any professional mathematicians in their families, many of the mathematicians Walker writes about were stimulated in their early childhood by a family member with an interest in puzzles, or engineering, or some kind of applied, non-academic mathematics.<br /><br />I've had a very privileged life. I had access to excellent schools and the best universities and was taught by outstanding teachers. But why was I interested in mathematics as a child? I had no close relatives who had studied mathematics (or indeed science) beyond school level to influence me. (My father and grandfathers studied classics and law, my mother social work, my aunt history, and (unlike me) everyone was very musical.)<br /><br />As a child I was obsessed by football. I was no good as a player but I loved games about football. I played Subbuteo Table Football with my friends (who were all better than I was, but I didn't mind losing: in retrospect my interest was in the modelling. I used to simulate tournaments with random numbers, trying to get realistic results. (It really upset me that a football game we had called "Wembley", in which match scores were decided by dice, gave lower division teams playing away in the cup a dice with a 5 on it, while no other team could score more than 4. A one in six chance of Rochdale scoring five at Old Trafford? I couldn't take that game seriously.)<br /><br />And so I needed to create fixture lists for football leagues - <i>n</i> teams each having to play every other team home and away. How could I draw up the weekly fixtures? Trial and error wasn't going to work for the 18 teams in the then Scottish First Division,<br /><br />Clearly if <i>n </i>is even, if teams play every Saturday (only), it requires at least (<i>n</i>-1) Saturdays for every team to play every other once. Perhaps the first sophisticated mathematical question I asked was "Can it always be done in <i>n</i>-1<i> </i>Saturdays? I remember thinking it wasn't clear to me that we could always draw up a fixture list that worked, with every team playing every week,<br /><br />Anyway, I asked my father, and he told me how to do it. Number the teams 1 to <i>n.</i> In week 1, team 1 plays team 2, team 3 plays team <i>n</i>-1, team 4 plays team <i>n</i>-2, team 5 plays team <i>n</i>-3 and so on. This pairs all the teams except team <i>n</i>/2 and team <i>n </i>who play each other. In week 2, team 1 plays team 3, team 4 plays team <i>n</i>-1, team 5 plays team <i>n</i>-2, and so on: team 2 is left to play team <i>n. In week </i>k, team 1 plays team <i>k</i>, team 2 plays team <i>k</i>-1, team 3 plays team <i>k-</i>2, and so on: then team <i>k</i>+1 plays team <i>n</i>-1, <i>k</i>+2 plays <i>n</i>-2, <i>k+</i>3 plays <i>n</i>-3, and so on. The unmatched team in the middle of one of these sets of pairings plays team <i>k. </i><br /><i><br /></i>This algorithm works, and shows that it is always possible to play all the matches in <i>n</i>-1<i> </i>rounds. If <i>n</i> is odd, then one team is necessarily idle each week: the algorithm can be modified by adding an extra team called "bye", and we see that a league with 2<i>n</i>-1 teams can play all their fixtures in 2<i>n-</i>1 weeks. (A league with an odd number of teams is unusual but in the Scottish League of my childhood, Division Two contained 19 teams.)<br /><br />It's only recently that it has occurred to me that this solution to a childhood problem is a serious combinatorial algorithm. So where did my father get it from? Presumably not the mathematical literature! I asked him recently and he said that he worked it out for himself. <br /><br />So although I may not have had professional mathematicians in my family, But my father was capable of working out for himself a nice algorithm to solve a tricky combinatorial problem (even if he had no idea he was doing mathematics). So my own interest in mathematics didn't come from nothing: my father could think mathematically and solve mathematical problems, even although at the time neither he nor I knew that that was what he was doing.<br /><br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-82952859893551589612016-05-30T17:53:00.006+01:002016-05-30T17:53:54.071+01:00Game Theory and "Beau Geste"<span style="font-family: Arial, Helvetica, sans-serif;">I wrote <a href="http://tonysmaths.blogspot.co.uk/2013/07/the-game-theory-of-jane-austen.html" target="_blank">a post some time ago</a> about <span style="background-color: white; line-height: 18px;">Michael Suk-Young Chwe's book </span><i style="line-height: 18px;">Jane Austen, Game Theorist</i><span style="background-color: white; line-height: 18px;">, which argues that Austen's works are a systematic exploration of game theory ideas. I have to say that I was not entirely convinced. For me, game theory is about players thinking about their choices and their opponents' choices, and thinking about what opponent is thinking is important. The examples in Austen of strategic thinking didn't, for me, capture that aspect of game theory.</span></span><br /><span style="font-family: Arial, Helvetica, sans-serif;"><span style="background-color: white; line-height: 18px;"><br /></span></span><span style="font-family: Arial, Helvetica, sans-serif;"><span style="background-color: white; line-height: 18px;">But I recently reread <i>Beau Geste</i>, P.C. Wren's 1920s adventure story of the French Foreign Legion with a tragic denouement at Fort Zinderneuf. Let me say straight away that the book shows all the unpleasant racism of its time. It also presents an old-fashioned view of the code of duty which I fear, as a child, I took more seriously than it deserves. (Wren's sequel, <i>Beau Sabreur</i>, is much more ambivalent in this regard, or perhaps I just missed the irony in <i>Beau Geste</i>.)</span></span><br /><span style="background-color: white; font-family: Arial, Helvetica, sans-serif; line-height: 18px;"><br /></span><span style="background-color: white; font-family: Arial, Helvetica, sans-serif; line-height: 18px;">In these matters </span><i style="font-family: Arial, Helvetica, sans-serif; line-height: 18px;">Beau Geste </i><span style="font-family: Arial, Helvetica, sans-serif; line-height: 18px;">is very much of its time. In its own terms, it is a rattling good adventure story. And, unlike Austen, it presents real game theory dilemmas (I am trying not to give significant spoilers) . The problems the heroes face require them to think through the consequences of their actions and how others will react. Both in the matter of the theft of the jewel which sets up the adventure, and in taking sides in the potential mutiny at Fort Zinderneuf, they are thinking not only about their own actions but about the other parties'. The mutiny is interesting because, thanks to the characters' interpretation of the demands of their duty and loyalties to their comrades, everyone on both sides has full information, so it really is a nice example of strategic game theory thinking.</span>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-11987288236393411962015-12-13T17:45:00.002+00:002015-12-13T17:51:35.279+00:00The geometrical artwork of José de Almade Negreiros<div class="separator" style="clear: both; text-align: center;"><a -="" almada="" alt="" by="" href="http://2.bp.blogspot.com/-7V8MELdU45s/Vm2rMTGn3WI/AAAAAAAAAPw/BWNDcLYZ0RY/s1600/almada%2Bnegreiros.PNG" imageanchor="1" negreiros="" rawing="" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-7V8MELdU45s/Vm2rMTGn3WI/AAAAAAAAAPw/BWNDcLYZ0RY/s320/almada%2Bnegreiros.PNG" width="311" /></a></div><br />As always, the annual MathsJam conference was full of wonderful things, and if I ever find time to post on this shamefully neglected blog I may return to some of these topics. But one of the special highlights of the 2015 MathsJam was Pedro Freitas's talk about the geometrical art of the Portuguese painter José de Almada Negreiros (1893-1970). Almada Negreiros (who also wrote novels and poems) made a collection of drawings "Language of the Square" which mathematicians will find fascinating. Happily, Pedro and Simao Palmeirim Costa have written a book <i>Livro de Problemas de Almada Negeriros</i> (Sociedade Portuguesa de Matemática, November 2015) which contains excellent colour reproductions of twenty-nine drawings. The book is available from https://www.spm.pt/store/list/novidades.<br /><br />Sadly for me, however, the text is in Portuguese, but an essay by the same authors, in English, can be found at <span style="font-family: "calibri" , sans-serif; font-size: 14.6667px; line-height: 15.6933px;"><a href="http://webpages.fc.ul.pt/~pjfreitas/pdfs/AlmadaGeomCanon.pdf" target="_blank">webpages.fc.ul.pt/~pjfreitas/pdfs/AlmadaGeomCanon.pdf</a></span><br /><br />I'm delighted to have discovered these mathematical artworks - yet another MathsJam discovery!<br /><br /><span class="st" style="background-color: white; color: #545454; font-family: "arial" , sans-serif; font-size: x-small; line-height: 1.4; word-wrap: break-word;"></span>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-69791672945160311572015-11-01T18:08:00.001+00:002015-11-01T18:08:09.517+00:00Remembering Lisa JardineI was very sorry to hear of the death last week of the historian Lisa Jardine. Although it wasn't her main focus, she made a big contribution to our understanding of early modern mathematics and especially of key figures like Robert Hooke and Christopher Wren. Her books are wonderful - readable, full of insights, and giving a vivid picture of intellectual life in the seventeenth century.<br /><br />I was lucky enough to hear her talk, less than a year ago, at the BSHM Christmas meeting last December when she gave an inspiring talk about women in twentieth century mathematics -in particular <span style="font-family: 'Times New Roman', serif; font-size: 12pt; line-height: 115%;">Hertha Ayrton, Mary Cartwright and Emmy Noether.</span><br /><span style="font-family: 'Times New Roman', serif; font-size: 12pt; line-height: 115%;"><br /></span><span style="font-family: Times New Roman, serif;"><span style="font-size: 12pt; line-height: 115%;">Jardine;s scholarship was important, but so was her encouragement of others. I believe she was an exceptional research supervisor, and her writing certainly inspired many, myself included. I experienced her kindness several times, and enjoyed a few conversations with her in coffee breaks at conference. Twice I consulted her by email, and although she can have had no idea who I was, she replied quickly, enthusiastically and helpfully. (On the first occasion I was seeking </span><span style="line-height: 18.4px;">clarification</span><span style="font-size: 12pt; line-height: 115%;"> of a view attributed to her in someone else's book, and on the second I was hoping to persuade her to talk about the novelist Robert Musil at a conference I was organising - she agreed in principle but sadly the dates didn't work out.)</span></span><br /><span style="font-family: Times New Roman, serif;"><span style="font-size: 12pt; line-height: 115%;"><br /></span></span><span style="font-family: Times New Roman, serif;"><span style="font-size: 12pt; line-height: 115%;">Her contribution to the history of science, direct and indirect, is immense. She is a great loss to the history of mathematics.</span></span>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-33388762270946927102015-10-25T15:33:00.001+00:002015-10-25T15:33:25.617+00:00Outreach: Simon Singh's commentsThe <i>Times Higher</i> has reported a conference talk by the writer Simon Singh about public engagement in science under the heading <a href="https://www.timeshighereducation.com/news/simon-singh-criticises-wasteful-spending-in-science-outreach" target="_blank">"Simon Singh criticises wasteful science outreach"</a>. This has even led to the ultimate distinction of <a href="https://www.timeshighereducation.com/opinion/laurie-taylor-22-october-2015" target="_blank">a mention in Laurie Taylor's Poppletonian</a>, the comic column which is consistently the highlight of my week.<br /><br />Now, not only did I not hear Singh's talk, so that I am relying on the THE report, but Singh is one of my heroes for his contributions to mathematics and its public reputation. For one thing, his documentary film on Andrew Wiles's proof of Fermat's last theorem conveyed to the general public the emotion and the joy of doing mathematics, and what the profession is about, in a way which was almost unprecedented. His subsequent writings - the book on Fermat's Last Theorem, the book on codes (and the wonderful CD-rom he made with Nicholas Mee), his recent book on the maths of <i>The Simpsons</i> and others - have continued to inform the public and inspire young mathematicians. And his creation of the undergraduate ambassador scheme, which puts maths undergraduates into schools and colleges, is a hugely important contribution to maths education in the UK, directly benefiting school and university students and motivating many outstanding graduates to become maths teachers. So his views on public engagement certainly deserve to be taken seriously.<br /><br />But on this I disagree with Singh's comments (at least as they are reported). He says that in his view the best science outreach is "largely dirt cheap". Well, there is certainly a lot of excellent public engagement work on mathematics that is done on the cheap (such as the Royal Institution Masterclasses and the British Science Festival, the power of both of which I have seen at first hand), and a large number of people doing it more or less in their spare time for no reward other than the joy of communicating mathematics. But it shouldn't be like that, and it isn't only like that,<br /><br />Singh is critical of the funding of a ballet about relativity. "People hate physics, they hate ballet, all you've done is allowed people to hate things more efficiently." Well, I believe that science and mathematics are so important that they should feature prominently in art. There should be ballets, novels, operas about mathematics. Happily, thanks to people like Marcus du Sautoy, Scarlett Thomas, Dorothy Ker and a great many others, there are. Not everyone hates ballet: a dance piece about science is potentially reaching a valuable audience. When Singh says of the "Faces of Mathematics" project of portraits of mathematicians that "I don't quite understand how this is really going to have an impact", he is surely not using his imagination. We need to show the world the diverse nature of mathematicians to encourage the aspirations of the potential mathematicians of the future,<br /><br />Of course Singh is right to suggest that the value for money of any outreach project should be compared with the cost of a teacher. And we certainly need more teachers, and to pay them better. But imaginative (and expensive) public engagement projects are also important. They won't all succeed, Singh's TV programme about Wiles was of inestimable value in showing what mathematics is about. Dance and photography projects promoting public engagement with mathematics and science have similar potential. Even when they fail, they are not wasteful.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-76575554818339988652015-10-08T19:38:00.002+01:002015-10-08T19:38:35.497+01:00The Mpemba Paradox<span style="font-family: inherit;">As a mathematician I love mathematical paradoxes because they are disturbing and thought-provoking. For example, Parrondo's Paradox tells us something counter-intuitive about probabilistic games; Simpson's paradox reminds us that we have to think carefully about statistics; and Curry's paradox is just mind-bending.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Paradoxes in the sciences are important because they make us think about our theories and where they don't quite match reality, driving new scientific ideas. My favourites include Olbers' Paradox (why is the sky dark at night?) and the EPR Paradox which shows us just how surprising the world is.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">So I was delighted to come across, in an article by Oliver Southwick in <a href="http://chalkdustmagazine.com/" target="_blank">the excellent magazine <i>Chalkdust</i></a>, a paradox that was new to me, the Mpemba Paradox. <span style="background-color: white; color: #252525; line-height: 22.4px;"> </span><span style="background-color: white; color: #252525; line-height: 22.4px;">"If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?" (The background story is wonderful - read the article!) Not only is there no agreement on the answer, but it gives insights into the mathematical equations involved, and mathematical modelling may help us understand the effect.</span></span><br /><span style="background-color: white; color: #252525; line-height: 22.4px;"><span style="font-family: inherit;"><br /></span></span><span style="background-color: white; color: #252525; line-height: 22.4px;"><span style="font-family: inherit;">Like all the best paradoxes, this is amusing but tells us something surprising about our world.</span></span><br /><br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-10063666457162706772015-05-24T21:32:00.004+01:002015-05-24T21:32:49.677+01:00Greenwich's mathematician in residence<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-69uNlmC11mI/VWIzWTUsKFI/AAAAAAAAAO4/Ln1KDy-E7fg/s1600/IMG_1109.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Mathematician in Residence show" border="0" height="213" src="http://4.bp.blogspot.com/-69uNlmC11mI/VWIzWTUsKFI/AAAAAAAAAO4/Ln1KDy-E7fg/s320/IMG_1109.JPG" width="320" /></a></div><br />As I write, the mathematician <a href="http://www.katiesteckles.co.uk/" target="_blank">Katie Steckles</a> is in residence at the <a href="http://www.greenwichunigalleries.co.uk/mathematician-in-residence-katie-steckles/" target="_blank">Stephen Lawrence Gallery of the University of Greenwich</a>, where she is demonstrating some fun maths to the public and getting people to take part in various mathematical activities. It's a fascinating show, covering many different kinds of mathematics, from tessellation to random walks, fractals to Benford's Law, and some graph theory.<br /><br />For me, what has emerged is the visual connections. My photo above shows to the left images of blackboards relating to some of the maths research at Greenwich: even without a full explanation of the context the blackboards viewers are intrigued by the look of the mathematics. The photo below shows a wall of "doodles" by visitors, showing that all such doodles (formed by closed curves) can be coloured in only two colours so that regions with a common boundary are different colours.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-rn_4F5k3hwU/VWIzhf90W9I/AAAAAAAAAPA/Zz8VPSLP5T4/s1600/IMG_1108.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="graph theory doodles" border="0" height="320" src="http://3.bp.blogspot.com/-rn_4F5k3hwU/VWIzhf90W9I/AAAAAAAAAPA/Zz8VPSLP5T4/s320/IMG_1108.JPG" width="213" /></a></div><br />Katie has curated a marvellous collection full of visual and mathematical interest. Hopefully the show will persuade people that there is much more to maths than sums! Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1