tag:blogger.com,1999:blog-58111244408382835022017-06-22T04:05:15.886+01:00Tony's Maths BlogA blog about maths things which interest me.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.comBlogger72125tag: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.com1tag:blogger.com,1999:blog-5811124440838283502.post-64215426509788899052015-01-18T18:35:00.003+00:002015-01-18T18:35:57.287+00:00Logical paradoxesI'm talking about logical paradoxes (<i><a href="http://www.gresham.ac.uk/lectures-and-events/this-lecture-will-surprise-you-when-logic-is-illogical" target="_blank">This lecture will surprise you: when logic is illogical</a></i>) at Gresham College on Monday 19th January, which is tomorrow as I write this on Sunday afternoon. I've been fascinated by these for years, thanks to writers like Martin Gardner, Raymond Smullyan, Douglas Hofstadter. <br /><br />It's nice to prove things: suppose I want to prove something which is slightly doubtful (like "Arsenal will beat Manchester City this afternoon" - a very unlikely proposition). Here's a proof from Martin Gardner. Consider these two statements:<br /><br />A: Both these statements are false<br />B: Arsenal will beat Manchester City this afternoon.<br /><br />Clearly A cannot be true, since if it were it would contradict itself. So A is false, and if B were also false, then A would be true, So B must be true.<br /><br />One proof isn't always enough, So here's another - this one is Curry's Paradox. Consider the statement:<br /><br />If this statement is true, then Arsenal will beat Manchester City this afternoon.<br /><br />Is this statement true? It's of the form "If A, then B", and we test that by seeing what happens when A is true. So assume that the first part of the statement above is true - which means that the whole statement is true, because that is what that clause asserts. And if that whole statement is true, and the first part is true, then the second part is true. So we have established the truth of the statement above, And if it is true, then Arsenal will win.<br /><br />So I've proved in two different ways that Arsenal will win, despite almost all the pundits and 76% of the BBC poll thinking the opposite. <br /><br />ADDED AT 6pm: Arsenal did win. Which proves the power of mathematical logic.<br /><br /><br /><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com4tag:blogger.com,1999:blog-5811124440838283502.post-52001972184891436532014-12-31T21:42:00.001+00:002014-12-31T21:42:06.973+00:00Why I can't have the perfect cup of coffeeApologies to any regular readers for the paucity of recent posts, due to pressure of work. A suitable New Year Resolution would be to post more often next year.<br /><br />This final post of 2014 comes from my presentation at MathsJam and was inspired by a puzzle in (I think) <i>Which Way Did the Bicycle Go</i> by Konhauser, Velleman and Wagon.<br /><br />Like most mathematicians coffee is important to me. I like my breakfast coffee to be strong and black, and (since one wants as much as possible of a good thing) I want my cup to be full.<br /><br />So how much coffee does my cup hold? Well, it is basically cylindrical, so you might assume the volume of coffee it can contain is pi(r^2)h. But just as the surface of the ocean is not flat but part of the surface of a sphere, so is the surface of my coffee. And the volume extra bit I get over the top of the cup depends on the curvature of te sphere - the more curved, the more coffee I get.<br /><br />What is the curvature? It depends how near to the centre of the earth we are. The closer to the centre of the earth, the sharper the curvature and the more coffee in the cup. (If my cup were infinitely far from the centre of the earth then the surface of my coffee would be flat.)<br /><br />So the higher my cup is, the less it can contain. Which means that if I fill my coffee cup to the brim, as soon as I lift it to drink from it, it will spill.<br /><br />Physicists may tell me that I am ignoring effects like surface tension or changes in density with altitude. But I'm a pure mathematician and such incidental matters don't interest me. What is annoying is that in an idealised mathematical universe I can't drink from a full cup of coffee without spilling some. Which is one more way in which the world doesn't work as it should.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-91904960152113413242014-08-25T15:27:00.002+01:002014-08-25T15:27:57.625+01:00Some nice puzzlesHere are some rather nice wooden puzzles found by Noel-Ann on recent trips. I'm not commenting on solutions here!<br /><h3 style="text-align: center;">Two Marbles</h3><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-55EZ557v4OM/U_tFdySwr3I/AAAAAAAAAMs/zRTFp_ES92Q/s1600/Marbles1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Two marbles - start position" border="0" src="http://1.bp.blogspot.com/-55EZ557v4OM/U_tFdySwr3I/AAAAAAAAAMs/zRTFp_ES92Q/s1600/Marbles1.jpg" height="176" width="320" /></a></div><br />Noel-Ann found this one in Texas. The object is to get the two marbles into the holes simultaneously, like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-dq8uk2g1W5k/U_tFec9LrhI/AAAAAAAAANA/ELMOJFGUbyU/s1600/Marbles2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Marbles puzzle solved" border="0" src="http://1.bp.blogspot.com/-dq8uk2g1W5k/U_tFec9LrhI/AAAAAAAAANA/ELMOJFGUbyU/s1600/Marbles2.jpg" /></a></div><br /><h3 style="text-align: center;">A Secret Box </h3>The other puzzles were brought back from a market in the Dordogne. This, amazingly, is a secret box which can be opened by performing a series of operations.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-lsdZRfDO8_g/U_tFdGUTUTI/AAAAAAAAAMk/U0LTYqVzOKg/s1600/Box.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Secret Box" border="0" src="http://3.bp.blogspot.com/-lsdZRfDO8_g/U_tFdGUTUTI/AAAAAAAAAMk/U0LTYqVzOKg/s1600/Box.jpg" /></a></div><br /><h3 style="text-align: center;">A packing problem </h3>This one starts off as a rather nice box with a red slab on top.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-jD9XFhf8wqM/U_tFdLDPwDI/AAAAAAAAANM/BNF4b6OvWSA/s1600/Box2closed.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="A packing puzzle box" border="0" src="http://4.bp.blogspot.com/-jD9XFhf8wqM/U_tFdLDPwDI/AAAAAAAAANM/BNF4b6OvWSA/s1600/Box2closed.jpg" height="299" width="320" /></a></div><br />When one slides the lid open one finds that it is packed full with more wooden blocks. How can one add the red block and close the lid?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-GYo0bE89zeM/U_tFdEOzuDI/AAAAAAAAAMo/7e69AiC6YTk/s1600/Box2open.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Inside the box" border="0" src="http://4.bp.blogspot.com/-GYo0bE89zeM/U_tFdEOzuDI/AAAAAAAAAMo/7e69AiC6YTk/s1600/Box2open.jpg" height="320" width="209" /></a></div><h3 style="text-align: center;">Another Packing Problem</h3>This one has four pieces each composed of two overlapping slabs. Can one fit them into the tray?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-kts52j0CTBQ/U_tFelh8r6I/AAAAAAAAAM4/jsMOFTCfOjU/s1600/Packing.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Another packing problem" border="0" src="http://4.bp.blogspot.com/-kts52j0CTBQ/U_tFelh8r6I/AAAAAAAAAM4/jsMOFTCfOjU/s1600/Packing.jpg" height="191" width="320" /></a></div><h3 style="text-align: center;"></h3>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-6528636637101338632014-06-07T21:47:00.000+01:002014-06-07T21:47:12.153+01:00LabanotationI was fascinated that one of the Secret Mathematicians discussed in <a href="http://www.gresham.ac.uk/lectures-and-events/the-secret-mathematicians" target="_blank">Marcus du Sautoy's wonderful Gresham College / London Mathematical Society lecture</a> a couple of weeks ago was Rudolf Laban. Some years ago a friend of mine was studying dance theatre at the Laban Centre (now part of Trinity Laban Conservatoire). She was studying hard for her exam on Labanotation, which is Laban's method of notating dance. I looked over her shoulder and was struck by the beautiful mathematics behind the notation. (Apparently this wasn't the right thing to say to somebody who needed to pass an exam in a subject with which she had no natural affinity - which I'm happy to say she did.)<br /><br />Mathematicians tend to like good systems of notation, and notating something as fluid and instantaneous as dance is a huge challenge. I'm not qualified to say how effective Labanotation is for recording dance - my friend and her fellow dance students certainly didn't find it intuitive. But it has been used for one invaluable project. <br /><br />Alec Finlay has notated Archie Gemmill's goal for Scotland against Holland in the 1978 World Cup. (You can <a href="http://www.amazon.co.uk/Labanotation-Archie-Gemmill-Goal-Pocketbooks/dp/0748663258" target="_blank">buy the book from Amazon</a> - I can't find a publishers' website.) That the notation which drive my dancer friend to distraction has been used to record the greatest goal ever scored in a football match is a demonstration of the importance of notation (which I regard as a branch of mathematics).Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-89404213878741170782014-04-08T09:44:00.002+01:002014-04-09T08:18:52.272+01:00109th Carnival of Mathematics<h2><span style="font-size: large;">Welcome to the 109th <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank">Carnival of Mathematics</a>!</span></h2><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-aWo9xT58IKU/U0T0JV0xkBI/AAAAAAAAAMQ/drdNvXHPHB4/s1600/shibuya_2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Shibuya 109 department store, Tokyo, Japan" border="0" src="http://3.bp.blogspot.com/-aWo9xT58IKU/U0T0JV0xkBI/AAAAAAAAAMQ/drdNvXHPHB4/s1600/shibuya_2.jpg" height="320" width="237" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">109 is a prime. I'm pleased to say that it is a happy, polite and amenable number (for definitions see, for example, <a href="http://www.numbersaplenty.com/109" target="_blank">www.numbersaplenty.com/109</a>). It is a Chen prime - that is a prime number <i>p</i> such that <i>p+2</i> is either prime or the product of two primes: in this case 111 = 3 times 37. You may remember that from <a href="http://www.whitegroupmaths.com/2014/02/107th-carnival-of-mathematics.html" target="_blank">Carnival 107</a>, since 107 was also a Chen prime.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">109 is the 24th term of <a href="http://en.wikipedia.org/wiki/Euclid%E2%80%93Mullin_sequence" target="_blank">the Euclid-Mullin sequence</a>, a curious sequence of which each term is the smallest prime dividing the product of all the previous terms plus 1. It's motivated by Euclid's proof that there are infinitely many prime numbers. It begins 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, ... and only the first 51 terms of the sequence are known, since the 52nd is the smallest prime factor of a 335-digit number of which the divisors haven't yet been found.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">If we express 1/109 as a decimal, the last six digits of the recurring sequence are 853211, which gives us the first six Fibonacci numbers in reverse order.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;">Now to the Carnival.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;">It's always good to start with humour. In the last few days, xkcd has been in excellent form: here is <a href="http://xkcd.com/1347/" target="_blank">one on teachers' statistics</a>. And having spent much of my life writing mathematical modelling software, I particularly enjoyed <a href="http://phdcomics.com/comics/archive.php?comicid=1693" target="_blank">PHD Comics on software testing</a> which, as so often, hits the mark.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">How about the amazing Domputer? Watch the <a href="http://aperiodical.com/2014/04/domputer-the-movie/#more-11870" target="_blank">video of the domino computer at the Aperiodical</a>.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">What does "real-world" mean with respect to maths in the classroom? <a href="http://blog.mrmeyer.com/2014/dear-mathalicious-fake-world-which-of-these-questions-is-real-world/" target="_blank">dy/dan</a>, <a href="http://blog.mathalicious.com/2014/03/19/the-real-worlds/" target="_blank">mathalicious</a> and <a href="http://www.mathgoespop.com/2014/03/keeping-it-real-an-addendum.html" target="_blank">Math goes Pop!</a> have been debating this difficult question.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;">Maria Droujkova has claimed in <a href="http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/" target="_blank">an interview in Atlantic Monthly</a> that "5-year-olds can do calculus": there is discussion at <a href="http://education.lms.ac.uk/2014/03/5-year-olds-can-do-calculus-by-maria-droujkova-collecting-comments/" target="_blank">the De Morgan Forum</a>.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;">GCHQ, the British Government Communications Headquarters which employs many mathematicians, has been in the news recently. Tom Leinster wrote an opinion piece for the April Newsletter of the London Mathematical Society entitled "Should Mathematicians Co-operate with GCHQ?", which can also be found (with discussion) at <a href="http://golem.ph.utexas.edu/category/2014/03/should_mathematicians_cooperat.html#more" target="_blank">the n-Category Café</a>.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><a href="http://www.theguardian.com/books/2014/apr/04/why-all-love-numbers-mathematics" target="_blank">Alex Bellos tells us why we all like numbers</a> in the Guardian.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The previous host of this Carnival, <a href="http://mathhombre.blogspot.co.uk/2014/03/archimedes-twin-circles.html" target="_blank">MathHombre, has been using Geogebra to investigate Archimedes's twin circles</a>.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://aperiodical.com/2014/03/how-to-solve-a-rubiks-cube-in-one-easy-step/" target="_blank">Paul Taylor examines Rubik's Cube at the Aperiodical</a>, examining the periods of small sequences of moves.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://jfhcullen.com/2014/03/22/Model-Checking-Beyond-Computer-Systems/" target="_blank">Jimi Cullen has been exploring model checking</a> - a tool from computer science with applications beyond that discipline.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://math-frolic.blogspot.co.uk/2014/03/magic-or-math.html" target="_blank">MathFrolic links humour with mathematics and magic</a> in reflecting on a mind-reading trick.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Another puzzle - can you see a pattern in apparently random data - leads <a href="http://welltemperedspreadsheet.wordpress.com/2014/03/26/puzzle-9-melting-fractals/" target="_blank">The Well-Tempered Spreadsheet to Melting Fractals</a>.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both;"><a href="http://thonyc.wordpress.com/2014/03/31/did-edmond-tells-robert-to-sling-his-hooke/" target="_blank">The Renaissance Mathematicus</a> tells us about a dubious account of the origins of Newton's <i>Principia</i> in the American TV series <i>Cosmos.</i></div><div class="separator" style="clear: both;"><i><br /></i></div><div class="separator" style="clear: both;"><a href="http://cameroncounts.wordpress.com/2014/03/26/generation-t-designs-and-other-mathematical-notation/" target="_blank">Peter Cameron reflects on mathematical notation and type-setting</a>.</div><div><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://flexmonkey.blogspot.co.uk/2014/03/nodality-node-based-calculator-for-ios.html" target="_blank">FlexMonkey has produced an app</a> offering<span style="font-family: inherit;"> "<span style="background-color: white; line-height: 18px;">a new type of calculator: numbers and operators appear as nodes in a network interface and the relationships between them define your calculations", which sounds intriguing. And another calculator app, Tydlig, is <a href="http://aperiodical.com/2014/04/app-review-tydlig-a-calculator-app-for-ios/#more-11769" target="_blank">reviewed by Aoife Hunt at the Aperiodical</a>.</span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px;"><br /></span></span></div><div class="separator" style="clear: both;">Max Tegmark's recent book <i>Our Mathematical Universe</i>, which argues that our universe <b>is</b> mathematics, has divided readers: some think it fascinating, others say its hypothesis is untestable and therefore scientifically meaningless. For a typically entertaining discussion (which, if I understand correctly, goes along with both these views) see <a href="http://www.scottaaronson.com/blog/?p=1753" target="_blank">Scott Aaronson's blog Shtetl-Optimized on the topic</a>.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">(Aaronson's post has the the wonderful title "This review of Max Tegmark's book also occurs infinitely often in the decimal expansion of pi". If you have a slow internet connection, feel free to save time by, instead of following all the links in this edition of the Carnival, simply going through the digits of pi till you find the content. You might find some other interesting stuff on the way: if so, submit it to the next Carnival!)</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">From deep or possibly meaningless ideas about whether the universe <b>is</b> simply mathematics to a simple question about multiplication: what exactly do we mean by "6 x 4"? Here are thoughts from <a href="http://reflectivemaths.wordpress.com/2014/03/11/6-x-4-means/" target="_blank">reflectivemaths</a> and <a href="http://www.flyingcoloursmaths.co.uk/6times4-4times6/" target="_blank">flyingcolours</a>.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">For those looking for serious maths, Terry Tao gives <a href="http://terrytao.wordpress.com/2014/03/28/the-cayley-salmon-theorem-via-classical-differential-geometry/#more-7395" target="_blank">a new proof of the Cayley-Salmon Theorem</a>. And of course <a href="http://www.theoremoftheday.org/" target="_blank">Theorem of the Day</a> features a different result every day - as I type this, today's theorem is <a href="http://www.theoremoftheday.org/CombinatorialTheory/Wagner/TotDWagner.pdf" target="_blank">Wagner's Theorem about planar graphs</a>.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">The wonderful lecture archive at Gresham College (disclosure: <a href="http://www.gresham.ac.uk/visiting-professor-of-computing-mathematics" target="_blank">I'm a visiting professor there</a>) contains <a href="http://www.gresham.ac.uk/gresham-professor-of-geometry" target="_blank">the lectures by Raymond Flood, the Gresham Professor of Geometry</a>: the video of the most recent lecture, on "Modelling the Spread of Infectious Diseases" should be available very soon. And don't miss <a href="http://www.gresham.ac.uk/lectures-and-events/notation-patterns-new-discoveries" target="_blank">Colin Wright's wonderful recent lecture on "Notations, Patterns and New Discoveries"</a>: the title in my opinion seriously undersells its entertainment value!</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px;">And finally, the <a href="http://gabrielecirulli.github.io/2048/" target="_blank">game at which everyone seems to be spending all their time this month - 2048</a>. (I also suggest you have a look at an old one - <a href="http://www.johnrausch.com/puzzleworld/app/lunar_lockout/lunar_lockout.htm" target="_blank">Lunar Lockout</a>). If you've discovered 2048, you'll appreciate <a href="http://xkcd.com/1344/" target="_blank">this from xkcd</a>.</span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px;"><br /></span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px;">If you have suggestions for Carnival 110, which will be hosted by <a href="http://www.flyingcoloursmaths.co.uk/blog/" target="_blank">Flying Colours Maths</a>, please <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank">submit them here</a>.</span></span></div>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com2