tag:blogger.com,1999:blog-58111244408382835022024-03-05T05:11:06.023+00:00Tony's Maths BlogA blog about maths things which interest me.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.comBlogger98125tag:blogger.com,1999:blog-5811124440838283502.post-54856106050087850822024-03-01T20:50:00.004+00:002024-03-01T20:50:35.402+00:00My Wordle failure<p><span style="font-family: helvetica;"> Earlier this week, on 24 February, my Wordle streak of over 200 successes came to a sad end. (For those who don't know the game, each day one has to deduce a five-letter word in at most six guesses, learning after each attempt whether each letter is correct and in the correct place, or correct but out of position.) I play the hard version, which requires that each guess be consistent with the results of all previous attempts.</span></p><p><span style="font-family: helvetica;">After two guesses I knew the word had the form _I_ER. At this point I was in trouble - the hard version meant that every subsequent guess had to have this form, so I could try only two new letters each time and there were too many possibilities to guarantee success in the six permitted rounds. And I failed to find the right one.</span></p><p><span style="font-family: helvetica;">So I was interested to see <a href="https://www.youtube.com/watch?v=tVzggzivpq0" target="_blank">this puzzle tackled by Mark Goodliffe</a> of Cracking the Cryptic</span><span style="font-family: helvetica;">. He faced the same problem I did. Spoiler warning - next sentence is in white text so you don't have to read it. <span style="color: white;">Mark succeeded - with a little good fortune, which I feel he thoroughly deserved because he was aware of the possibility of repeated letters, which I had overlooked. </span> But (without indulging in schadenfreude) I was relieved to see from the comments that I was not the only person whose streak had ended with that puzzle - perhaps I shouldn't blame myself too much!</span></p><p><span style="font-family: helvetica;">So how could I have done better? It seems to me that as soon as I knew that the solution contained E and R I was in difficulty. My strategy is to choose an opening guess that is made up of common letters. But perhaps if I want to be sure of success in six guesses I can't afford to have both E and R in my initial guess? </span></p><p><span style="font-family: helvetica;">Of course, success in six attempts isn't the only possible objective. I am tyring to get my average number of attempts as far below 4 as I can, and so I may have to accept the occasional failure in order to achieve that goal - a strategy that guarantees success in six goes every time might have a higher average number of guesses. </span></p><p><span style="font-family: helvetica;">I</span><span style="font-family: helvetica;"> know there are people who have done much more analysis than I have who know exactly how to optimise their choices but my comments are based on intuition rather than hard evidence!</span></p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-71936544033124950142024-01-01T15:30:00.005+00:002024-01-01T15:30:42.127+00:00Laptop irritations<p><span style="font-family: arial;"> This isn't really maths (except in so far as all computing is maths) but I am going to vent a couple of frustrations regarding laptops. </span></p><p><span style="font-family: arial;">First, I am sure there is a good reason for this design feature and I would be delighted if someone could enlighten me. Here is part of the keyboard of this laptop:</span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXqPlEbdBf1tsXGNZYAjuWLHvVfMdZ0nZ9rhjLDk5776X1fwB8t9DaEyb4hzxSGkDPwDk-fkkarP1rYGT7AQEWCMrfGp9nwJI_47Brwzyxq_uR9YuPv63xjOw_Qc9WfHXzKXsrbzjYNjCcVpYQBpD8wiJf0vO_iiisE9Lr-SC-8NRqJwSTZmU6ca_VB6zn/s640/IMG_3052%20copy.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: arial;"><img alt="Laptop keyboard" border="0" data-original-height="480" data-original-width="640" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXqPlEbdBf1tsXGNZYAjuWLHvVfMdZ0nZ9rhjLDk5776X1fwB8t9DaEyb4hzxSGkDPwDk-fkkarP1rYGT7AQEWCMrfGp9nwJI_47Brwzyxq_uR9YuPv63xjOw_Qc9WfHXzKXsrbzjYNjCcVpYQBpD8wiJf0vO_iiisE9Lr-SC-8NRqJwSTZmU6ca_VB6zn/w320-h240/IMG_3052%20copy.jpg" title="Laptop keyboard" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;">Now, note the position of the on/off switch (lit up in the photo). It's in the middle of the top row of keys. Why? In other laptops I have used, the on/off switch is easy to find because it is placed apart from the other keys. With this keyboard, it's hard to find (with my eyesight the marking isn't very clear to me, especially in low light, and of course it isn't lit up when I want to switch the laptop on). Worse, it is next to the delete key, and on more than one occasion I have hit the off key instead of delete. (You have to hold it down for a time to switch the machine off, and so far I have only accidentally switched it off once when in the middle of a piece of work, but the possibility is now always in the back of my mind.)</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;">It seems to me utterly illogical to place the on/off switch where it is. But it is so illogical that there must have been a deliberate design decision to do so and there must be a good reason for it. Can anyone tell me?</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;">My second point is just venting at an irritation. I don't like my laptops making noises when I am working, so the first thing I do is switch off all notifications. But still some persist. (Why, when I save a Word document as a PDF and the file already exists, does it have to beep as well as displaying an "are you sure" box? I find myself swearing aloud at the machine when it does this, and I have spent more time than it is worth trying to find out how to switch it off.)</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: arial;">What is worse than that is that the machine makes random beeps at odd times. (These do not correlate to emails arriving or anything like that, as far as I can tell.) It even does it when I have locked the computer. Presumably these are important warnings, but I can't find out what they are. Nothing has popped up in another window and there is no indication that I can see as to the reason for the beep. What event is so important that the laptop has to interrupt me to tell me it has occurred, but isn't sufficiently important for me to be told what has happened?</span></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><p><br /></p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-2719168079830818702023-09-09T16:32:00.000+01:002023-09-09T16:32:06.609+01:00Is mathematics universal? An argument from sudoku<p><span style="font-family: helvetica;"> I have a horrible feeling that a long time ago I believed, and perhaps even tried ti persuade others, that mathematics is different from other forms of human knowledge and endeavour. The works of Shakespeare, Beethoven and Rembrandt, for example, are contingent on human attributes, such as our language and emtions and our senses of sight and hearing. Other species, however evolved, would be unlikely to appreciate these works as we do. And our science is contingent on the way the universe happens to work: creatures in our universe might appreciate our ideas, but aliens in a completely different universe with different physical laws would not.</span></p><p><span style="font-family: helvetica;">But mathematics, I once may have thought, is different. Mathematical truths, like the facts that there are infinitely many primes or exactly 26 sporadic finite simple groups, are (it would seem) true universally and don't depend on the way humans have evolved or on the physical laws that happen to hold in our universe. So it makes sense to send the pattern 2, 3, 5, 7, 11, 13, 17, 19, ... as a signal to outer space as a message to potential aliens (although whether it is wise to do so is a different question - our own relationships with other creatures don't suggest that engagement with more powerful species is likely to end well.)</span></p><p><span style="font-family: helvetica;">I link to think I was always a little uneasy about this arrogant claim and that as I have grown older and perhaps wiser, I am increasingly aware that mathematics is a cultural construct. But what prompted this current ramble is, of all things, an absolutely beautiful sudoku presented on the Cracking the Cryptic Youtube channel - <a href="https://www.youtube.com/watch?v=9LEFdNinbdg" target="_blank"><i>x'clusion</i> by Florian Wortmann</a>. The break-in (which I didn't see for myself, though I should have) is the most wonderful I have seen. (Spoiler alert: the rest of this parenthesis uses white text - <span style="color: white;">to appreciate it you probably need to know a couple of sudoku theorems</span>.)</span></p><p><span style="font-family: helvetica;">But would sudoku-solvers from an alien species appreciate it, or does it depend on the structure of the human brain? I can imagine aliens with a different brain structure, with much larger memory. Such a species could hold all possible sudoku grids in their working memory, and solving a sudoku for them would be quickly achieved by finding the one grid compatible with the puzzle by a brute force search - not using the fascinating logic which our brains require us to apply.</span></p><p><span style="font-family: helvetica;">I think this example suggests that mathematics is not as universal as I might once have thought, and that aliens whose brain happened to be structured differently might well have no interest in our mathematics.</span></p><p><span style="font-family: helvetica;">(In thinking about this I have also been influenced by <a href="https://www.youtube.com/watch?v=wL3uWO-KLUE" target="_blank">this video about algorithms</a>, which shows that a hypothetical computer with a huge amount of fast-access memory could solve by brute force problems far faster than a more traditional conventional computer - the two examples given being solution of a scrambled Rubik's cube, and breaking the Double-DEC encryption system whose 112-bit key might wrongly be assumed to be immune to brute-force attack in a reasonable time.</span></p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-23489248063311390272023-08-05T07:32:00.003+01:002023-08-05T07:32:49.248+01:00Carnival of Mathematics 218<p><span style="font-family: verdana;"> I'm delighted to host for the second time the monthly Carnival of Mathematics for August 2023. Information about the Carnival and links to previous issues can be found at <a href="https://aperiodical.com/carnival-of-mathematics/" target="_blank">The Aperiodical</a>.</span></p><div style="text-align: left;">
<span style="font-family: verdana;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiifC5T4SCI7D5wchbvFYBqvXCyBxLfwXXDKjyDPlfw_3ByWliG33d7nIWdLt4EqeUEX2tpqeMlQyNkrkDYHhRO9SqhllKamA7bz-6frCaBgaX7jsJl8gf155VWI9S-ByKCf7w1P7LThIzOmbKfmHI01HZFeLdb8ynuyICFWO7VSIFEh6kNA6azXbmjb3Cr/s1000/218%20bus.webp" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Picture of London bus No 218. Image from https://bus-routes-in-london.fandom.com/wiki/London_Buses_route_218, CC-BY-SA" border="0" data-original-height="563" data-original-width="1000" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiifC5T4SCI7D5wchbvFYBqvXCyBxLfwXXDKjyDPlfw_3ByWliG33d7nIWdLt4EqeUEX2tpqeMlQyNkrkDYHhRO9SqhllKamA7bz-6frCaBgaX7jsJl8gf155VWI9S-ByKCf7w1P7LThIzOmbKfmHI01HZFeLdb8ynuyICFWO7VSIFEh6kNA6azXbmjb3Cr/w320-h180/218%20bus.webp" title="Number 218" width="320" /></a><br />
</span><p class="MsoNormal"><span style="font-family: verdana;">This is Carnival 218, and it is customary to begin with some
facts about the issue number. </span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Following my immediate predecessor Vaibhav at <a href="https://www.blogger.com/blog/post/edit/5811124440838283502/2348924806331139027" target="_blank">DoubleRoot</a>,
and using the approach currently being adopted by some ill-advised students, I
asked ChatGPT for some cool facts about the number 218. Here is a sample
of its responses:</span></p><p class="MsoNormal"><span style="font-family: verdana;">1.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">218 is an even number.</span></p><p class="MsoNormal"><span style="font-family: verdana;">2.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">It has three digits: 218 is a three-digit
number, with the hundreds digit being 2, the tens digit being 1, and the units
digit being 8.</span></p><p class="MsoNormal"><span style="font-family: verdana;">3.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">It is a Harshad number: 218 is a Harshad number,
which means it is divisible by the sum of its digits (2 + 1 + 8 = 11, and 218 ÷
11 = 19).</span></p><p class="MsoNormal"><span style="font-family: verdana;">4.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">The number 218 can be expressed as a sum of
positive integers in multiple ways. One possible partition is 218 = 109 + 109.</span></p><p class="MsoNormal"><span style="font-family: verdana;">5.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">A year on the planet Mars is approximately 687
Earth days. Interestingly, 218 days on Mars is approximately equal to 1/3 of a
Martian year.</span></p><p class="MsoNormal"><span style="font-family: verdana;">6.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">The number 218 is a numeric palindrome, meaning
it reads the same forwards and backward.</span></p><p class="MsoNormal"><span style="font-family: verdana;">7.<span style="font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 7pt; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal;">
</span></span><span style="font-family: verdana;">218 is sometimes used in scientific research as
a numerical identifier for various purposes, such as labeling experimental
groups or samples.</span></p><!--[if !supportLists]--><p></p>
<p class="MsoNormal"><span style="font-family: verdana;">What do you say?<span style="mso-spacerun: yes;">
</span>These seem to be either trivial (1, 2, 4, 7), plain wrong (3, 6), or stretching
the meaning of “Interestingly” and “approximately” (5).<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>Well,
what did you expect from ChatGPT?<span style="mso-spacerun: yes;"> </span>If you
want something that is interesting and true, much better to consult Wikipedia
on “218 (number)”, which tells me<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-family: verdana;">“218 is the number of
inequivalent ways to color the 12 edges of a cube using at most 2 colors, where
two colorings are equivalent if they differ only by a rotation of the cube.”<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">But on to the content you’ve come for.<span style="mso-spacerun: yes;"> </span>What’s new in the maths online world? (Due to
a technical problems some suggestions for the July Carnival were temporarily
lost in the ether so “new” includes some links that might have appeared last
month.)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Well, July saw the premieres of two blockbuster movies, one
of which is of obvious mathematical interest.<span style="mso-spacerun: yes;">
</span>So here from Cambridge Mathematics is <a href="https://www.cambridgemaths.org/blogs/a-brief-history-of-barbie-and-mathematics/" target="_blank">A brief history of Barbie and mathematics</a>. (Apart from the infamous “Math
class is hard” which a version of Barbie in 1992 would “say”, it also mentions
the dreadful Barbie book “I Can Be a Computer Engineer” of 2013 (which was
subsequently retracted), but also contains interesting historical analysis of
Barbie matters over the last 64 years). <span style="mso-spacerun: yes;"> </span>And if you are more interested in the other
film, <a href="https://www.maa.org/press/periodicals/convergence/quotations-in-context-oppenheimer" target="_blank">Michael Molinsky in his “Quotations in Context” column</a> on the Mathematical
Association of America website has explored a talk by J. Robert Oppenheimer –
“Today, it is not only that our kings do not know mathematics, but our
philosophers do not know mathematics and – to go a step further – our
mathematicians do not know mathematics.”<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">For another mathematician of the middle of the last century,
the <a href="https://infinitelyirrational.podbean.com/e/29-the-godelfather-a-mathematical-masterpiece/" target="_blank">Infinitely Irrational podcast explored The Gödelfather: A Mathematical Masterpiece</a> with special guest Ioanna Georgiou.<span style="mso-spacerun: yes;"> </span>And, on the topic of logic, my own <a href="https://www.youtube.com/watch?v=VY78Te3eK3g" target="_blank">short talk about an earlier philosopher, the legendary John Buridan</a><span class="MsoHyperlink">, and his mathematical
paradoxes</span>, was published by G4G Celebration.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">The hot mathematical topic recently has been tiling – since the Carnival is organised by The Aperiodical,
it is entirely appropriate that we continue to cover the continuing excitement about
aperiodic tilings which followed the discovery of the “Ein Stein” single tile which
aperiodically tiles the plane.<span style="mso-spacerun: yes;"> </span>Ayliean
MacDonald went (for some reason) to the village of Newtyle in Scotland (which
turns out to be not far from my father’s hometown of Forfar) to record an
update to her previous video which featured in last month’s Carnival.<span style="mso-spacerun: yes;"> </span>Her new one is <a href="https://www.youtube.com/watch?v=ArADlJx7SlU" target="_blank">a Numberphile video about the New Tile</a> (and the topic is moving so fast that Future Ayliean had to
interrupt the video with news of developments since it was recorded).<span style="mso-spacerun: yes;"> </span>The discovery by Craig S. Kaplan, David
Smith, Joseph Samuel Myers, and Chaim Goodman-Strauss of the Ein Stein, and subsequently
the Spectre Tile which tiles aperiodically without reflections, also featured
in <a href="https://www.youtube.com/watch?v=OImGgciDZ_A" target="_blank">a new G4G Celebration video</a>. <span style="mso-spacerun: yes;"> </span>Meanwhile <span style="mso-spacerun: yes;"> </span><a href="https://fractalkitty.com/2023/06/07/spectre-tiles/" target="_blank">Fractal Kitty provides translucent pngs of the Spectre tile.</a><u><span style="color: #0563c1; mso-themecolor: hyperlink;"><o:p></o:p></span></u></span></p>
<p class="MsoNormal"><a name="_Hlk141700059"><span style="font-family: verdana;">Here now is my random selection of further miscellaneous
maths things which have recently appeared (with many thanks to those who
emailed me their suggestions).<o:p></o:p></span></a></p>
<p class="MsoNormal"><span style="font-family: verdana;"><span style="mso-bookmark: _Hlk141700059;"></span><a href="https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html" target="_blank"><span style="mso-bookmark: _Hlk141700059;">The New York Times marked the 50<sup>th</sup>
anniversary of The Encyclopedia of Integer Sequences</span></a><span style="mso-bookmark: _Hlk141700059;">. (</span>Behind a paywall but with limited
free access.)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Given the “facts” put forward by ChatGPT when I asked about
the number 218, here is a timely article on </span><a href="https://www.understandingai.org/p/large-language-models-explained-with" style="font-family: verdana;" target="_blank">Understanding AI: how large models work</a><span style="font-family: verdana;"> by Timothy B Lee
and Sean Trott.</span><span style="font-family: verdana;"> </span><span style="font-family: verdana;">Meanwhile </span><a href="https://medium.com/@neil.j.saunders/ai-is-staggeringly-competent-but-it-doesnt-comprehend-a-thing-as-google-s-lamda-shows-f7e0fad21c36" style="font-family: verdana;">Neil
Saunders argues at Medium.Com</a><span style="font-family: verdana;"> that while generative AI is “staggeringly
competent” (the examples above don’t entirely convince me of that) it nevertheless
doesn’t have any understanding.</span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Here is an account by Robert Smith about <a href="https://www.stylewarning.com/posts/brute-force-rubiks-cube/" target="_blank">creating an algorithm for a computer to solve Rubik’s cube</a>.<span style="mso-spacerun: yes;"> </span>And here is an older (but I only recently
found it) insightful <a name="_Hlk141699964"></a><a href="https://www.youtube.com/watch?v=wL3uWO-KLUE" target="_blank"><span style="mso-bookmark: _Hlk141699964;">video account of an algorithmic subtlety</span></a><span style="mso-bookmark: _Hlk141699964;"> </span>on the polylog Youtube channel,
which starts with Rubik’s cube but gets into cryptographic security.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">You’ll enjoy <a href="https://www.youtube.com/watch?v=B1J6Ou4q8vE" target="_blank">Alan Becker’s charming Animation vs. Math</a> - and <a href="https://www.youtube.com/watch?v=2VQDqzT4SOM" target="_blank">Dr Tom Crawford (@TomRocksMaths) has made a reaction video</a>.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">As always, the Cracking the Cryptic youtube channel
presented examples of its expert sudoku solvers thinking mathematically in
solving variant sudokus.<span style="mso-spacerun: yes;"> </span><a href="https://www.youtube.com/watch?v=9LEFdNinbdg" target="_blank">In solving this wonderful puzzle by Florian Wortmann</a>, Simon Anthony finds an astonishingly beautiful
break-in (which I have to admit eluded me when I tried the puzzle, although, being
familiar with the two sudoku theorems required, I felt that I should have seen it for myself).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Colin Wright happened across </span><a href="https://www.solipsys.co.uk/new/PythagorasByIncircle.html" style="font-family: verdana;" target="_blank">a proof of Pythagoras’s Theorem using the Incircle</a><span style="font-family: verdana;">.</span></p><p class="MsoNormal"><span style="font-family: verdana;">James Propp presents </span><a href="https://mathenchant.wordpress.com/2023/07/16/the-triumphs-of-sisyphus/" style="font-family: verdana;" target="_blank">a wide-ranging discussion about mistakes in calculations</a>.</p><p class="MsoNormal"><span style="font-family: verdana;">Karen Campe, whose blog will host the September Carnival of
Mathematics, has provided <a href="https://karendcampe.wordpress.com/2023/07/13/shout-out-for-squares/" target="_blank">A Shoutout for Squares</a>.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;"><a href="https://aperiodical.com/2023/07/interview-kyle-evans-on-his-2023-fringe-show-maths-at-the-museum/" target="_blank">Kyle Evans was interviewed by the Aperiodical</a> about his forthcoming Edinburgh
Fringe maths show, Maths at the Museum.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Matt Parker took advantage of the recent publication by David Cushing and David I. Stewart to <a href="https://www.youtube.com/watch?v=zYkmIxS4ksA" target="_blank">buy just enough UK national lottery tickets to guarantee a win</a> (but "a win" doesn't mean "a profit"!) (For a link to
the paper see underneath Matt’s video.)<o:p></o:p></span></p>
<p class="MsoNormal"><a href="https://www.youtube.com/watch?v=LXZIQ51W3Ns" style="font-family: verdana;" target="_blank">Sam Hartburn presents a song about a Knight’s Tour</a>.</p><p class="MsoNormal"><span style="font-family: verdana;">Snezana Lawrence is the guest of Mathematical Association President <a href="https://www.youtube.com/watch?v=t2G4t2bDjZk" target="_blank">Professor Nira Chamberlain OBE in his VLOG</a>.</span></p><p class="MsoNormal"><a href="https://twitter.com/TaliaRinger/status/1681410191278080000?s=20" style="font-family: verdana;" target="_blank">Here's a Twitter (as it once was) thread on diversity in understanding mathematics</a><span style="font-family: verdana;">, started by Talia Ringer.</span></p>
<p class="MsoNormal"><span style="font-family: verdana;">Want to play the mathematical pattern-spotting game Set but
would prefer a Non-Abelian or Projective version?<span style="mso-spacerun: yes;"> </span>Check out <a href="https://www.youtube.com/watch?v=EkFX9jUJPKk" target="_blank">Catherine Hsu’s Numberphile video</a>.<o:p></o:p></span></p><p class="MsoNormal"><a href="https://padlet.com/missradders/maths-on-quiz-shows-37vlr8kntwqsezgg" style="font-family: verdana;" target="_blank">Here is a Padlet roundup of maths questions on TV quiz shows</a><span style="font-family: verdana;"> (by missradders).</span></p>
<p class="MsoNormal"><span style="font-family: verdana;">For those seeking an alternative to whatever the former Twitter is now called, </span><a href="https://samjshah.com/2023/07/01/mastodon-mathstodon-join-us/" style="font-family: verdana;" target="_blank">here from the Continuous Everywhere but Differentiable Nowhere blog is an invitation to join mathstodon.xyz</a></p><p class="MsoNormal"><o:p><span style="font-family: verdana;">And that's the end of this month's Carnival of Mathematics. Enjoy! And when the time comes, check out the September Carnival</span></o:p><span style="font-family: verdana;"> via </span><a href="https://aperiodical.com/carnival-of-mathematics/" style="font-family: verdana;" target="_blank">The Aperiodical</a><span style="font-family: verdana;">.</span></p><p class="MsoNormal"><o:p><span style="font-family: verdana;">Image of London bus from <a href="https://bus-routes-in-london.fandom.com/wiki/London_Buses_route_218" rel="nofollow" target="_blank">https://bus-routes-in-london.fandom.com/wiki/London_Buses_route_218</a>, <a href="https://www.fandom.com/licensing" rel="nofollow" target="_blank">CC-BY-SA</a> </span></o:p></p><p class="MsoNormal"><o:p><span style="font-family: verdana;"><br /></span></o:p></p><p class="MsoNormal"><o:p><span style="font-family: verdana;"><br /></span></o:p></p><span style="font-family: verdana;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: verdana;"><br /></span></div><br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-56024427485402222792022-10-24T18:03:00.004+01:002022-10-24T18:03:35.927+01:00The 24 Hour Maths Game Show<p class="MsoNormal"><span style="font-family: arial;"><span style="font-size: 12pt; line-height: 107%;">Following the 24 Hour
Maths Magic Show two years ago, this coming weekend brings the 24 Hour Maths
Game Show! Starting at 7pm on Friday 28 October there will be 24 hours of
mathematicians presenting mathematical games, or games about mathematics.
For details of the stellar line-up and the exciting fun in prospect see </span><span style="font-size: 12pt; line-height: 107%;"><a href="https://24hourmaths.com/" target="_blank">the event website</a>. The event is raising money for four excellent
charities - <span style="background: white;">Beat Eating Disorders, the Rheumatoid Arthritis Foundation, the Disasters
Emergency Committee, and the Malala Fund: donations to one or more on </span><a href="https://www.justgiving.com/team/24hourmathsgameshow" style="font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; text-decoration-line: none; transition: text-shadow 0.5s ease 0s;"><span style="background: white; border: none windowtext 1.0pt; color: #0f79d0; mso-border-alt: none windowtext 0cm; padding: 0cm;">the event Just
Giving page</span></a><span style="background: white;">. <o:p></o:p></span></span></span></p><p>
</p><p class="MsoNormal"><span style="background: white; font-size: 12pt; line-height: 107%;"><span style="font-family: arial;">I have
the coveted Saturday morning 8:30-9am slot UK time (which is prime time for
viewers in some part of the world) and I’ll be talking about the Hypergame
paradox and how I trapped two students into starting a game which may go on for ever. (There is a happy ending.)</span></span><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><o:p></o:p></span></p><p><br /></p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-11736345839596369932022-04-18T17:10:00.001+01:002022-04-18T17:13:00.436+01:00Mathematicians' diaries in the time of Covid<p> How did mathematicians cope with the Covid lockdowns? <i><a href="https://covidiarymath.com/" target="_blank">COVIDIARY of Mathematicians</a> </i>is a new book published by the Mathematical Society Archimedes of Belgrade. It presents diaries from April 2020 of seven mathematicians in different parts of the world - Tiago Hirth (Lisbon), Guido Ramellini (Barcelona), James Tanton (Phoenix), Jovan Knezevic (Belgrade), Kiran Bacche (Bangalore), Sergio Belmonte (Altafulla), and Tijana Marković (Belgrade). Their diaries feature mathematics and (excellent) puzzles, mountainlettes, cooking and shopping, in the strangest of times. The book is beautifully produced, with copious colour pictures, and QR-codes taking the smartphone-equipped reader further afield. The editors Aleksandra Ravas and Dragana Stošić Milijković have done a wonderful job, providing full explanations in footnotes of any reference which might be unfamiliar to some readers. Solutions are provided to the puzzles! If you have the opportunity, I strongly recommend that you explore this book!</p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-33067006904147226802021-09-12T20:33:00.002+01:002021-09-12T20:44:49.297+01:00Did studying maths help Emma Raducanu win the US Open?<p> Emma Raducanu's feat of winning the US Open as a qualifier, without dropping a set in her ten matches, was astonishing. In feature articles this summer, during and after Wimbledon, her A-level achievements this summer, and especially her A* grade in A-level Mathematics, have been reported, Is there a connection between her tennis accomplishments and her study of maths?</p><p>Now, mathematics implicitly arises in tennis tactics. I've discussed toy examples in public lectures on game theory, which (it seems to me) is relevant to choices players make - whether to serve to the forehand or backhand, and where to expect for your opponent to serve, for example. I very much doubt if players ever analyse in these terms, but they are intuitively doing game theory when making their tactical decisions.</p><p>But I think in the case of Raducanu there is a more general point. Several times I have heard knowledgeable commentators - most recently Tim Henman immediately after Raducanu's victory over Leylah Fernandez in the US Open final last night - talk about her qualities as a problem solver. She thinks deeply about her tactics, adapting to opponents and match situations. Now problem-solving is a quality which is developed in studying mathematics. (Fernandez, who showed in her victories over three top-five opponents in New York remarkable abilities to turn around matches in which she was behind, apparently enjoys solving Rubik's cube - more mathematical problem-solving!0</p><p>Obviously her study of mathematics is not why Raducanu is a great tennis player. But it seems to me that the problem-solving skills which she displays on the tennis court are the same skills which make her good at mathematics. </p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-66567356218190830102021-07-04T17:25:00.006+01:002021-07-04T17:30:07.219+01:00Another dubious probabilistic argument<p> Here's a variation on my previous post - another piece of hand-waving probabilistic reasoning, which I think is basically correct, but I suspect many will disagree. </p><p>A long time ago I used to do the challenging weekly cryptic crosswords in <i>The Listener </i>magazine. (<i>The Listener </i>has been defunct for many years, though I think the crossword continues in <i>The Times</i>.) The story was that if no-one solved the crossword, it was too hard, and if more than one person solved it, it was too easy. While that was an exaggeration, it wasn't easy, and I judged it worthwhile, if I completed the puzzle, to submit my entry for the prize draw.</p><p>I had completed about fifteen puzzles in a row, and submitted my answers each time, but hadn't won the book token. Then there was an unusual puzzle - it was mathematical rather than word-based. Two mathematician colleagues and I worked on it - we didn't find it at all easy - and we eventually solved it. I submitted our answer, and this time, we won the book token.</p><p>So - my conclusion was that (probably) more people solved the word-based puzzles than the mathematical one, and that therefore I was more likely to win the prize for that puzzle (as I had done) than for the others.</p><p><b>Is this conclusion valid?</b></p><p>As it happens, at the end of the year statistics for all <i>The Listener</i> crossword entries were published, so I was able to see if this was indeed the case. It turned out that the mathematical crossword had attracted about three times as many correct entries as any of the others. So my conclusion was in fact false, but I still think the reasoning was sound.</p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-91481171148032106452021-06-12T18:24:00.002+01:002021-06-12T20:52:06.712+01:00Is this analysis correct?<p>When I was an undergraduate, I was walking through the town centre one morning when a journalist from the student newspaper asked my views on the Rag Mag that had just appeared. When the report came out, six students' views were quoted. Three of the six happened to be from the same cohort of six maths undergraduates at my college. My Director of Studies concluded (jokingly) that this disproportionate representation of his tutees showed that we were spending too much time out and about when we should have been studying.</p><p>Here is a similar scenario, based on a recent experience. Suppose that I enter a lottery. It is open to, say, 10,000 people, but I do not know how many choose to take part. As it happens, I win the lottery. It seems to me that I can conclude that (with a high degree of probability) only a small proportion of those who were eligible chose to enter the lottery. If all 10,000 people entered then my chance of winning was only ,00001, whereas if 5 people entered it was 0.02. Since I did win, my estimate of the likely number of entrants will be at the lower end.</p><p>Now, you are my friend and you are equally friendly with all 10,000 potential participants. I tell you about it, I think you should agree that I am right in my conclusion. But you cannot come to the same conclusion: whichever of the 10,000 people had won the lottery would have told you of their success, so you can't deduce anything from what I have said to you about the likely number of participants.</p><p>So - I have good reason to come to a conclusion (with a high degree of probability). You agree that I am right to come to this conclusion, but also you have no reason to suppose that the conclusion is valid. How can this be?</p><p>Apologies if I am missing something but I think this analysis is sound. Have I read about this somewhere?</p><p>However, I haven't yet managed to persuaded anyone else to agree with me (which is perhaps ironic, or perhaps I am just wrong). Comments welcome!</p><p><br /></p><p>. </p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com2tag:blogger.com,1999:blog-5811124440838283502.post-77932711887971749092021-03-10T10:23:00.003+00:002021-03-10T10:23:53.396+00:00Tomorrow's Mathematicians Today<p>I spent last weekend at the online conference <a href="https://sites.google.com/view/imatmt2021" target="_blank">Tomorrow's Mathematicians Today</a>, hosted by the University of Greenwich and the Institute of Mathematics and its Applications. This is a conference for undergraduate mathematicians to present to their peers on mathematical topics that excite them - this might be their own project work, or something they have come across in the curriculum or outside it that has fascinated them. The conference was created in 2010 by Noel-Ann Bradshaw when she taught at the University of Greenwich, and has been held in various venues since.</p><p>We hosted it physically at Greenwich in 2010, 2013, 2016 and 2019, and before Saturday I was apprehensive that the conference would not work online. At the physical conferences what made the event so exciting was the atmosphere - friendly and supportive of the speakers, with great enthusiasm from audience and presenters. Would this work online? </p><p>Well, it was obviously different, but the enthusiasm was certainly present, and the online networking sessions were well attended and worked far better than I had imagined. Here are the words the attendees chose to describe their feelings:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxSac-Iciyt_ZXCuU85UrqZvn4DcbZ6snliB7ZzhSgF02oGxjGBfkgC3IrYEmROEEr8zGlDMlpcv4H-PgikX7so7CQgUjggwMLTli3iQRXpJ5jf-XZ7b4csI9Enl5B28IwQIBcWxm94RYs/s2048/3-what-three-words-would-you-use-to-describe-tmt2021.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Wordcloud picture" border="0" data-original-height="1216" data-original-width="2048" height="190" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxSac-Iciyt_ZXCuU85UrqZvn4DcbZ6snliB7ZzhSgF02oGxjGBfkgC3IrYEmROEEr8zGlDMlpcv4H-PgikX7so7CQgUjggwMLTli3iQRXpJ5jf-XZ7b4csI9Enl5B28IwQIBcWxm94RYs/w320-h190/3-what-three-words-would-you-use-to-describe-tmt2021.jpg" width="320" /></a></div><p>The quality of student presentations was outstanding: the judges of the <a href="https://sites.google.com/view/imatmt2021/gchq-prize" target="_blank">GCHQ Prize for the Best Presentation</a> had an extremely difficult task deciding the winner. The winning paper by Yanqi Cheng (UCL) was remarkable not only for its content but for Yanqi's sangfroid in managing a seamless transition to live delivery when the computer playing her pre-recorded video crashed early in the presentation. "Honourable Mentions" were awarded to Yousra Idichchou (Greenwich) and Oscar Holroyd (Warwick), and the other shortlisted papers by Finley Wilde (Bristol), Kaiynat Mirza (Keele), Muhiyud-Dean Mirza (Warwick) and Sheeru Shamsi (Keele) were all excellent, as indeed was every single student talk.</p><p>The conference also benefited from two contrasting keynotes by established mathematicians - Colva Roney-Dougal on random games with groups and Kit Yates on the mathematics of epidemics. It was surprising to hear from Colva how the Riemann Hypothesis turns up in group theory!</p><p>Like its predecessors the conference was inspiring in showcasing the enjoyment today's students are taking from mathematics. It totally proved the point made by IMA President Nira Chmberlain in opening the conference when he said that the presenters were not "tomorrow's mathematicians" but are already fully qualified for the title of mathematician.</p><p>I obviously hope future TMT conferences will be held physically again, but the online conference was as friendly and inspiring as we could have hoped!</p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-36510937468247192622020-12-28T18:11:00.000+00:002020-12-28T18:11:01.373+00:00Reading "The Queen's Gambit"<p> I haven't watched the Netflix series <i>The Queen's Gambit</i>, but (prompted by a review in <i>Private Eye</i>) I've just read Walter Tevis's 1983 novel on which it is based. (I'm a reader rather than a viewer by inclination.) It's an interesting book, which I thought well worth reading.</p><p>Now, I know I am missing the point, but there were a couple of things I found irritating. first of all I found the accounts of the chess unconvincing. I haven't played seriously for many years, and the author was undoubtedly a more serious chess player than I ever was, but the games he describes don't seem plausible to me - the heroine wins too much material too quickly. I am probably wrong, because the author took expert advice on the chess positions. I'd be interested to know whether better chess players than me felt the same, or am I out of touch?</p><p>Of course, the bit that really annoyed me was the description of a tournament that just didn't add up. After four rounds three players are in the lead with 4 out of 4: Beth wins her fifth one while the other two with 100% play each other. In the last round she loses to the winner of that match, and only finishes sixth after two other players on 5/5 agree a quick draw in the last round. This just doesn't work!</p><p>The chess novel which I loved as a teenager (and still do) was Anthony Glyn's <i>The Dragon Variation</i>, and the games in that book seemed authentic. The descriptions of Beth Harmon's games in <i>The Queen's Gambit </i>didn't work for me.</p><p>So if you are looking for an entertaining novel about chess, I still recommend <i>The Dragon Variation.</i></p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com2tag:blogger.com,1999:blog-5811124440838283502.post-70053177925993739802020-12-20T12:51:00.003+00:002020-12-20T13:00:04.717+00:00Memories of Peter Neumann<p> </p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoGgCU96IYWW0S0ybFqMzAOuf9Na9aFV3vwhZ4uZvgJx2aeCQkfZYE3qWddO2ofY4VsaxdHXPELlPQ92aJAyBusoUZ4idAm7rP6UzFMep2hcJvYveEUzo4lRmQ1-00FMtaqUhY9dzDGPsC/" style="margin-left: auto; margin-right: auto;"><img alt="" data-original-height="314" data-original-width="220" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoGgCU96IYWW0S0ybFqMzAOuf9Na9aFV3vwhZ4uZvgJx2aeCQkfZYE3qWddO2ofY4VsaxdHXPELlPQ92aJAyBusoUZ4idAm7rP6UzFMep2hcJvYveEUzo4lRmQ1-00FMtaqUhY9dzDGPsC/" width="168" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Peter M. Neumann (1940-2020) <br />(Photo by Bert Seghers, <br />Wikimedia Commons, Public Domain)</td></tr></tbody></table><br />A very sad post - the mathematician Peter M. Neumann died on Friday 18 December of Covid-19/ Peter was my moral tutor when I was a postgraduate student at The Queen's College, Oxford, and like many others I was enormously influenced by him. I was very lucky as a student to have wonderful support from several great mathematicians - Ray Lickorish and Michael Vaughan-Lee in particular - but I feel myself especially privileged to have known Peter. Indeed, in many ways I consciously tried to model myself on him, and the best things I have done in my career in teaching owe much to Peter.<p></p><p>In this post I'm going to share some rather rambling and inconsequential memories of him (with the warning that my memory is fallible).</p><p>I first met him when I went for interview at Queen's - which turned out to be a conversation in the Fellow's Garden with Peter and Graham Higman. I had been reading my notes from my course on Combinatorial Group Theory on the train journey, which was probably a mistake because I confused myself and attributed to Higman what was actually the Baumslag-Solitar Group. Peter questioned it and my heart sank as I realised my blunder, but before I could say anything Higman jumped in to say that in fact that group had been his original idea and Baumslag and Solitar acknowledged him in their paper. </p><p>Anyway, despite such blunders, I was accepted. I was very impressed that, when I turned up at Oxford six months later and knocked on Peter's door, he remembered who I was. But then when I introduced myself to my office-mates in the Mathematical Institute the next day, one of them said, 'Yes, I know your name. When I knocked on Peter Neumann's door last week, he said, "Hello, you're Tony Mann, aren't you?"' So perhaps Peter's strategy was to guess one of the names of the new DPhil students so that he would be right at least once!<br /></p><p>I attended Peter's famous <i>Kinderseminar</i> - a Wednesday morning gathering of doctoral students and visiting professors in Peter's rooms, with coffee and friendly conversation before a presentation on someone's research. I remember my first presentation. It was dreadful. (To be fair to myself, I had never given a presentation before - it wasn't part of the Cambridge mathematics curriculum - and so my experience of mathematics talks consisted entirely lectures which were primarily dictation of notes and research seminars which were generally over my head.) Knowing the expertise of the three visiting professors, I naively assumed they knew everything there was to know about my research topic, so I cased through the background material to avoid boring them. After five minutes of rushed, garbled talking, I looked at the audience and realised that none of them had followed at all. If I were in that position now, I would stop and restart the presentation at a sensible pace, but at the time I panicked and continued with the presentation for another hour, completely wasting everyone's time. When Peter gave feedback, he began by saying "That was terrible, wasn't it" (I guess it's to my credit that I was already aware of that) and, although that's all I remember of his feedback, it must have been very generous because I felt encouraged rather than dismayed.</p><p>I also remember that Peter particularly disliked the works of C.P.Snow, a favourite writer of my father's whose novels I had also liked. I recall Peter saying that good novelists "show" and leave readers to form their own judgments while bad ones like Snow tell the reader what they should think. (While there is a lot in this, I don't entirely agree with Peter: there are different but valid ways of story-telling.) Years later I discovered that Peter's antipathy to Snow may have been due to his strong feelings about Snow's introduction to G.H. Hardy's <i>A Mathematician's Apology</i>: a much-praised account of Hardy which Peter felt misrepresented the great man by exaggerating his supposed unhappiness.</p><p>Peter himself should have written much more - but he always said "There are already too many books". While this is true, more books by Peter would have benefitted us all.</p><p>Other old memories of Peter - when he sent one a message it was always on an interesting picture postcard rather than just a scrap of paper. This is something I copied, building up a stock of postcards to use when I had to write a short note to somebody (and I have retained my postcard-buying habit even though electronic communication means I never send anybody any notes any more!)</p><p>I remember going to a barbecue at Peter's house at which several of us we watched the England-Argentina World Cup match, and regretting my enthusiastic response to Maradona's wonder goal when others were upset that it put England out of the tournament.</p><p>Peter had a cautionary tale for those of us who have to write many of references for students. A prospective student applied to Queen's and his teacher's reference said, "Without a doubt this student is by far the best mathematician this school has ever produced." Since the school in question was Peter's own school, this reference did not have the positive effect the teacher intended!</p><p>I remember Peter's stories about his house number - 403. He used to go into schools to talk about maths and he would refer to prime numbers "like, 2, 3, 5 or 403, for example". Only once, a hand immediately went up in the audience, "But, sir, 403 isn't prime, it's 13 times 31". Peter told how, when the house was built, he went to the Post Office to be allotted a number for it, and was told, "There are no numbers left - you'll have to give it a name instead." After offering to find some more numbers for them, he and Sylvia called the house "Burnside" after the great mathematician. I remember him saying that he had thought of calling the house "Burnside Hall" after his two favourite mathematicians.</p><p>I was privileged to work with Peter on a couple of book projects: it was a joy particularly to collaborate with him and Julia Tompson (I was very much the junior partner) in producing Burnside's <i>Collected Papers</i>. One of Peter's qualities was that he had very high standards and I feel that his book reviews could be rather ruthless. So it was a rather good move on my part that both books I have edited involved working with Peter so he wouldn't be able to review either of them!</p><p>I was lucky enough to attend the conference celebrating Peter's 60th birthday, twenty years ago next month. Peter's birthday was at the end of December and so the birthday took place at the beginning of January 2001 - giving Peter the possibly unique achievement that his 60th birthday was marked by a conference in a different millennium from the event it was celebrating!</p><p>I owe so much to Peter - and so do many many others. His contribution to mathematics goes far beyond his own mathematical discoveries, significant though these are, He taught, inspired and encouraged so many others, and was much loved as well as much admired.</p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com2tag:blogger.com,1999:blog-5811124440838283502.post-68607221179615271632020-12-07T19:04:00.001+00:002020-12-07T19:04:14.885+00:00Two mathematical magic tricks<p> So over the last few weeks I have performed a couple of mathematical magic tricks which I have put on my Youtube channel.</p><p>As part of the <a href="https://24hourmaths.com/" target="_blank">24 Hour Maths Magic Show</a>, when I was hosted by the amazing Chris Smith, I performed a trick based on the curious phrase "David Lovel in yon abbey". You can <a href="https://www.youtube.com/watch?v=s11e_NpVewg&t" target="_blank">watch the trick</a> and find out how it relates to the phrase.</p><p>At the virtual <a href="https://mathsjam.com/gathering/" target="_blank">MathsJam 2020 Gathering</a>, with the help of Ruth, I performed my favourite Martin Gardner card trick - you can <a href="https://www.youtube.com/watch?v=e7fk6w7APAw" target="_blank">watch this one here</a>.</p><p>While I love both these tricks, I feel that the real magic may not be apparent to the viewer. The first trick relies on the kind of linguistic / mathematical properties that appeal to many mathematicians (see for example Alex Bellos's wonderful new book <i>The Language Lover's Puzzle Book</i>) while the second one depends on the magician dictating the sequence of play in a game of knots and crosses. Are these tricks are more interesting to perform than to watch? </p>Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-13691380086379890052020-04-20T20:22:00.004+01:002020-04-20T20:22:59.380+01:00Memories of John ConwayThe mathematician John Horton Conway died just over a week ago. (I recommend Siobhan Roberts's biography <i>Genius at Play</i>.) I' intend to write a post later about what his book <i>On Numbers and Games</i> meant to me, but in this post I will share a couple of memories of Conway. I emphasise - these are memories and their accuracy is not guaranteed.<br />
<br />
First,a personal story of how I was caught out by his exam question. When I studied Part III of the Mathematical Tripos at Cambridge I took Conway's course on Sporadic Simple Groups (this at a time when the Classification of the Finite Simple Groups was not complete and indeed before it had been proved that the Monster group actually existed). In the exam we had to do any three questions out of six (or similar). The last question was something like "Write an essay on anything in the course you haven't covered in your previous answers." Now, I had three good topics I knew pretty well, and they all came up in other questions. So I thought to myself, "Well, I'll do the essay question first, while I think about which two of the other three questions to do." As a result I wrote the essay on my fourth-best topic, and didn't use in the exam one of my three strongest topics. Not good exam technique! (To be honest, I feel that's why it wasn't a very fair question to set in an exam!) I have sometimes wondered: if I had answered that question first, and written about my best topic, and then afterwards answered the question on that topic, since that wouldn't have been a previous answer, could I have got credit for the same material twice?)<br />
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My other memories are of an email mailing list for those interested in the history of mathematics, which ran for some years from the late 1990s before splitting and dissolving in acrimony as these things tended to do at the time. Anyone could discuss or ask any questions, and Conway was one of the regulars.<br />
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One one occasion two schoolboys posted an email saying they were in primary school and they wanted to know more about some mathematical topic. Conway sent a lovely long reply, at an appropriate mathematical level, saying probably how nice it was to hear from such enthusiastic students. The boys sent a thank-you letter, adding at the end that they weren't actually at all interested in mathematics, but their teacher had told them to ask a question.<br />
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On another occasion a university student (I think a PhD student) asked about a result of Lagrange (I think), noting that there were several proofs of the result and asking if anyone knew which was Lagrange's original proof. Conway wrote a long reply, saying that he didn't know, and going on to explain why he didn't think it mattered: there were several simple proofs and it would have been a matter of chance which one Lagrange came up with first, since the proof of the theorem was trivial. Immediately someone jumped in, accusing Conway of appalling rudeness in describing the question as "trivial" (he hadn't) and suggesting that such treatment from an established mathematician would likely deter the student from further mathematical studies. Others argued about this (not all members of the list were native English speakers and the word "trivial" did seem to have very negative connotations for some: which surprised me as I was very familiar with its use in this mathematical context and didn't think it at all offensive). Conway said he hadn't intended to be rude or to suggest the question was trivial - indeed, by writing a long and detailed answer, he had shown his respect for the questioner - and he apologised profusely. But the criticism continued until the student who had asked the original question replied. He said that far from feeling upset by Conway's response, not only did he not interpret it as critical, but he was absolutely thrilled that the great Conway himself had replied to his question: he had been walking on air ever since!<br />
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I admired Conway's willingness to engage with anybody on that mailing list, regardless of their age or experience. I hope I have learned from that.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-32453893695698923812020-03-29T20:30:00.000+01:002020-03-29T20:30:27.981+01:00Self-referential humourI' am one of these annoying people who loves self-reference. I had a sign saying "Why can't anyone these days do anything properyl" in my student room: now I wear a T-shirt labelled "Prefectionist", so my sense of humour hasn't developed. Books which I loved as a student (and still do) are Patrick Hughes and George Brecht's <i>Vicious Circles and Infinity</i> and Douglas Hofstadter's <i>Gödel, Escher, Bach: An Eternal Golden Braid</i>. The first of these books pushed my interest in art towards Fluxus, and then later I discovered Hughes's brain-baffling perspective paintings. And through Hughes I found the wonderful, amaxingly modern logical paradoxes of the fourteenth-century philosopher Jean Buridan.<br />
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And so I'm one of these annoying people who think it's witty to include "Self-reference" in the index of a software user-manual, referring to the page number for that index page. (Yes, I did that, but it was a long time ago.)<br />
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When did this enjoyment of self-reference start? Possibly with my father, although I can't immediately think of any examples. Certainly when I was a teenager. For me, the funniest Monty Python sketch was the one where we see a group of intrepid explorers reaching their destination, never before visited by man, and duly celebrate with handshakes all round. But while they are celebrating, one of them wonders, "Who is filming us?" and we see them looking in all directions before pointing, rushing straight towards the camera and then introducing themselves to the camera crew, who obviously got there before them. And as we cut to another angle, they realise that there must be another camera crew too ... (This is my memory of the sketch and it may not be very accurate.)<br />
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I had wonderful maths teachers at school - Jimmy Cowan and Ivan Wells - but I don't remember them particularly encouraging this kind of humour. But recently (ie when I first planned writing this blog post, so not very recently) I was amused by this gag from Moray Hunter's radio comedy "Alone": .<br />
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<i>"Remind me, Morris, never to ask you to do anything … at
all … ever"<o:p></o:p></i></div>
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<i>"Does that include reminding you never to ask me to do
anything at all ever?"</i></div>
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<i><o:p></o:p></i></div>
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<i>"Probably – I lost interest halfway through the question"</i></div>
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<o:p></o:p></div>
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(The first speaker is the Angus Deayton character, and the concluding line is not only ideal for his world-weary delivery, but really perfects the joke.)</div>
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Why do I mention this? Because Moray Hunter was at my old school and shared the same maths teachers as I had. So perhaps there is a connection...</div>
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<o:p></o:p></div>
Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-31662780213155463912020-02-22T10:50:00.000+00:002020-02-22T10:50:05.926+00:00A brilliant solution to the Tower of HanoiI have always underestimated the interest value of the Tower of Hanoi problem. If you don;t remember the puzzle, there are three positions for piles of discs of different sizes, starting with all discs piled from smallest to largest on one pile (in this case the leftmost).<br />
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<tr><td class="tr-caption" style="text-align: center;">The Tower of Hanoi</td></tr>
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The problem is to move disks so that the whole pile is now in the rightmost position, but one can only move one disc at a time and one can never place a larger disc on top of a smaller one.<br />
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The surprise is how many moves are necessary - to move a pile of <i>n</i> discs takes 2^(<i>n</i>-1) operations. (it's a good example showing various ideas about algorithms, such as recursion.) There is a legend that monks in India are engaged in moving a tower of 64 discs, one operation per day, and that when the move is complete, the world will end. How long would that take?<br />
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Although I won a T-shirt by solving the puzzle in Lewisham Shopping Centre on a Saturday morning in Maths Year 2000, I had never felt that the puzzle has mass appeal: the procedure for solution is too laborious. But recently I took the puzzle in the photograph to the Green STEM Fest for children at the University of Greenwich (along with things which I thought were more likely to attract interest) and this was the exhibit the kids enjoyed most. A few nine-year-olds (or so) started racing each other against the clock, recording times of around 35 seconds, which seemed pretty quick to me.<br />
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Then one of them produced the brilliant solution in the video below (which is a reconstruction)! This is the work of a real mathematician who has recognised the symmetry in the situation.<br />
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<br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-19651641693188266252020-01-12T18:18:00.000+00:002020-01-12T18:18:08.570+00:00Thoughts on Hannah Fry's Royal Institution Christmas LecturesAnother New Year, another resolution to write more blog posts. We'll see.<br />
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A mathematician delivering the Royal Institution Christmas Lectures is always something special. Hannah Fry's three lectures this year were rather different from Christopher Zeeman's classics. Rather than one person directly talking to the audience for the full hour each time, she brought in guests for short interviews, showed activities outside the lecture room, and presented a huge range of activities and apparatus with volunteers from the audience (and in one case, a plant, when the audience member asked to solve Rubik's cube turned out to be the nation's champion speedcuber). This was a team effort and everyone who took part performed splendidly: Matt Parker's many contributions deserve special mention. <br />
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And it was wonderful. The excitement was palpable. The enthusiasm of the audience, the rush to put hands up whenever a volunteer was wanted - even allowing for possibly selective editing, it was clear that all the students were having a whale of a time. Did one ever expect to see young people so excited by a maths lecture? (Sure, Zeeman was also exciting, but in a very different way.)<br />
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I have seen some comments to the effect that there wasn't very much maths in the lectures. I think that is misguided. There was plenty of maths, with the applications shown but without the technical details. I don't have any problem with that. As a kid I was always motivated by the abstract mathematics rather than the applications, but I'm in a minority. And today, a kid wanting to know the details of anything Hannah talked about can just get out their phone. And what a wonderful panorama Hannah presented of the power of mathematics in today's world of data and machine intelligence. (It was nice to see MENACE, the match-box game player, taking its rightful place in the show!)<br />
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(The one unfortunate thing was that the first lecture included an upbeat segment about using maths to judge when it was safe to explore volcanoes - although it must have been filmed before the tragedy in New Zealand which could not have been foreseen, that bit should have been edited out or reshot for the broadcast.)<br />
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So - Hannah Fry's lectures have inspired schoolkids to take maths seriously. Hopefully some of them will be motivated to study maths at University. And what will happen when they attend their first lecture? If these Christmas Lectures are their first experience of mathematics lectures, they will be expecting wildly interactive sessions with guest speakers introduced every few minutes, lots of demonstrations and fast-moving material. How will they react to a lecturer spending an hour going through a complex pure mathematical proof line by line?<br />
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Have the Royal Institution misled their audience by presenting as a lecture something so far removed from a traditional lecture? Possibly, but the Christmas Lectures is their brand, so they cannot be blamed for doing that.<br />
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If we as university lecturers are to avoid disappointing our future students, perhaps we need to rethink our lectures. Rather than go through detailed mathematics at a pace which cannot be right for everyone in the room, perhaps we should try to emulate Hannah's RI Lectures. We could (as I'm sure some already do) present many voices (on video if not live), lots of ideas, and as much interactivity as we can manage to keep the audience enthusiastic, leaving the technical details for students to study in their own time. We can provide lecture notes (or, better, screencasts) that they can go through at their own pace, pausing when their brain is full and returning to them later, and going to Youtube or similar when they get stuck, just as we ourselves study from books and papers. We can use our large-class time to build enthusiasm and give the big picture rather than getting lost in detail.<br />
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If Hannah's "lectures" help speed up the move to more useful use of students' time than the traditional lecture, that will be another benefit from these remarkable Christmas Lectures.<br />
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(As always, I am presenting my own personal views - that is what a blog is for! I don't expect everyone to agree with me.)<br />
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<br />Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com1tag:blogger.com,1999:blog-5811124440838283502.post-68951670168985004862019-06-15T15:40:00.000+01:002019-06-15T15:47:19.594+01:00Looking forward to the Festival of MathematicsSo in ten days time the University is hosting Greenwich Maths Time, the 2019 IMA Festival of Mathematics and its Applications. The Festival takes place on Tuesday 25 and Wednesday 26 June. Everything is free!<br />
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The weather forecast is currently fine and we are looking forward to a feast of mathematics. The programme and booking details can be found at <a href="http://www.tinyurl.com/imafest19" target="_blank">the Festival website</a> www.tinyurl.com/imafest19and the Twitter hashtag is <a href="https://twitter.com/search?q=%23IMAFest19" target="_blank">#IMAFest19</a>.<br />
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The Festival has attracted a wonderful collections of performers and activities. Visitors will have the opportunity to learn about noon-Newtonian fluids by walking on custard! We're grateful to everyone who is taking part and who has worked to make the Festival possible, and especially to the Festival sponsors, the University of Greenwich, the Institute of Mathematics and its Applications, the Advanced Mathematics Support Programme, The OR Society, FDM, the Mathematical Association, and an anonymous individual donor.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-15509958160245455682018-07-08T13:32:00.000+01:002018-07-08T13:32:07.517+01:00The Big Internet MathOff<a href="https://aperiodical.com/2018/07/the-big-internet-math-off-round-1-jo-morgan-v-tony-mann/#more-18568" target="_blank">The Big Internet MathOff</a> is now taking place. Sixteen mathematicians are competing in a knock-out tournament to present the most interesting piece of mathematics. With the final two matches of the first round now taking place, there has already been a wealth of fascinating mathematics presented. Christian Lawson-Perfect deserves huge thanks for creating this wonderful showcase of exciting maths.<br />
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But not only are the contestants' pitches fascinating. There is also interesting maths involved in the background.<br />
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First, Each match runs for 48 hours. As a contestant, my most pressing question today is whether I will win my first round match against Jo Morgan, in which case I will need to work urgently on my pitch for round 2. So the question is, how soon can I extrapolate from the early votes to be reasonably sure of the outcome?<br />
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I am sure that this has been studied in the context of political elections. But this is harder because the electorate is everybody with access to the internet - several hundred million people (though not all will actually vote). <br />
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Now, when I first looked at the score ten minutes after the match started, I was leading by 2 votes to 1, but I think that was a bit too early to be confident of the result. As I write this, four hours into the match, Jo is leading by 58 votes to 29, and my share of the vote has consistently been around 30% for some time. Although time differences may mean that America has yet to vote, my intuition is that I can safely abandon my pitch for round 2 and turn to other things like writing blog posts.<br />
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I've already had to find my answer to my second problem. Each contestant has to provide a different piece of fun mathematics for each round so we all had to submit four topics before the tournament started. It's unlikely that each set of four topics were all equally good. So what order do you choose for your topics? Do you save your best one for the final? In that case, you might not get to the final and your best idea is wasted. Or do you use your best ideas in the early rounds, to improve your chances of progress, but risk putting your weakest idea forward in the final?<br />
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There may be a similarity with World Cup penalty shoot-outs. With five penalties to take, it's clear that a team should select their best five penalty takers. But in which order should they take the penalties? One suggestion is that you should save your best penalty-takers for the end, when the pressure will be greatest. But that could mean that your team has lost 5-3 and your best taker hasn't taken one of the penalties. The great Ally McCoist, commenting on last night's shoot-out between Russia and Croatia, argued for putting your best takers first, guaranteeing that at least they will take penalties. I'm not convinced that's a sound argument, though if you score your early penalties then the pressure on your opponents will increase.<br />
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So what did I do for the MathOff? Did I save my best idea for the final, or bring it out for the first round? Well, there is a further complication. I have no idea how to judge which of my topics would win most votes in an internet poll. Ranking them in order would have been pure guesswork. So, shamefully, I didn't use any game theory or simulation to decide the order of my pitches - I just entered them in the order in which I thought of them. Which may be as good a strategy as any!Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag:blogger.com,1999:blog-5811124440838283502.post-92084517561792796352018-05-20T16:53:00.001+01:002018-05-20T16:53:44.729+01:00Mathematical discoveriesI was lucky enough to attend a meeting organised by the <a href="http://www.bshm.ac.uk/" target="_blank">British Society for the History of Mathematics</a> on "The History of Cryptography and Coding". It was a quite exceptional meeting - six excellent talks. As one of the other audience members said, I learned something from every talk, and a lot from several talks. (Anyone who goes to these events will know that this isn't always the case.)<br />
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The final talk, by Clifford Cocks on the discoveries of the public key cryptography, was fascinating in many respects. (Cocks was one of the people at GCHQ who discovered both the Diffie-Hellman key exchange method and the Rivest-Shamir-Adleman (RSA) algorithm before those after whom the ideas are named, but this wasn't known until GCHQ made it public over 25 years after the event.) Cocks told us about the (different) reasons why the British and American discoverers were looking for these methods. I was particularly struck by his insights into the creative processes that led to the discoveries.<br />
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In 1970 James Ellis at GCHQ had the idea of public-key cryptography. Many people at GCHQ tried to find a way to implement it, without success. Cocks suggested that this was because of "tunnel vision" - because Ellis's paper suggested using look-up tables, everyone was focused on that idea. Cocks had just arrived at GCHQ from university, and his mentor mentioned Ellis's problem to him, but described it in general terms without mentioning look-up tables. Without having been led in a wrong direction, Cocks quickly came up with the idea of using factorisation, and the problem was solved. (When Cocks told his colleague and housemate Malcolm Williamson about his paper, Williamson overnight worked up the idea of using the discrete logarithm problem, anticipating Diffie and Hellman.)<br />
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Cocks also told us about how Diffie was working on these discoveries having left his academic job, supporting himself on his savings - something which I don't recall knowing.<br />
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Then Cocks told us about Rivest, Shamir and Adleman's discovery of RSA. They had tried about 30 ideas, none of which worked. Then after a Passover meal at which alcohol flowed freely, Rivest had the big idea, wrote it down, and checked the next morning to see if it still worked.<br />
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I think these stories shed some light on mathematical creativity. It needs hard work, of course, but it also needs flexibility. Cocks (by his modest account) had the advantage over his colleagues that his mind wasn't conditioned by an unproductive idea. Rivest's solution came after a break from thinking about it. Of course, there are many other examples - Poincaré's inspiration as he was getting on a bus is the standard one - but it is always interesting to hear how great mathematical discoveries came about, and to hear this story from Cocks himself was a wonderful privilege.Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com2tag:blogger.com,1999:blog-5811124440838283502.post-30479674922345607152018-01-28T20:05:00.002+00:002018-01-28T20:08:33.681+00:00London buses, and the use of the mean as an estimateA couple of weeks ago, I got onto my morning bus, climbed the stairs, holding on as the bus lurched forward, and sat down, to hear for the first time the new announcement "Please hold on: the bus is about to move", introduced by TfL (Transport for London). Over the next few days this announcement was widely ridiculed. It was broadcast after every stop, but often - in my experience almost always - AFTER the bus had started moving, and sometimes when it was slowing down for the next stop, making the announcement appear ridiculous. Occasionally, at busy stops like the railway station, it was broadcast while large numbers of people were still waiting to board, presumably causing consternation to prospective passengers who took it seriously. And on one occasion, while the bus was stationary, I heard "The bus is about to move" followed immediately by the announcement "The driver has been instructed to wait here for a few minutes", flatly contradicting the previous words.<br />
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What was happening? TfL explained that they were piloting the announcement for four weeks, to try to reduce the number of injuries sustained by passengers on moving buses - apparently of the order of 5000 each year. The timing of the announcement was based on the average time buses spent at each stop - I suspect by "average" they meant the mean.</div>
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The intention is laudable. But the problem with using a mean in situations like this is that it doesn't really tell you how long a particular bus will wait at a given stop. My bus home probably spends longer stopped at the railway station than at all the other stops put together. Just as most people earn less than the mean national salary, which is heavily influenced by the very small number of people earning millions each year, so I imagine most of the time a bus spends less time at a stop than the mean. So a system based on the mean time spent at a stop will result in the announcement usually being played after the us has left the stop, leading to ridicule.</div>
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Now, TfL are pretty good at maths - their planning of the transport around London during the 2012 Olympics was a very successful example of operational research in action. So did they really get this wrong? After all, one would think that a few tests would have shown the problem. </div>
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Certainly one result of the announcements was a great deal of publicity, which perhaps has made people more aware of the need for care when standing and moving on a bus. The announcements themselves may have a short-term effect, but in fact one very quickly ceases to notice them (or at least I have found that they very rarely impinged on my attention, after the first few instances on the first day). But perhaps the press coverage, and people talking about the announcements, had more impact than the announcements themselves.</div>
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But if the announcements are to continue, how can TfL avoid the absurdity of an announcement that the bus is about to move being broadcast after it has moved? The solution TfL have adopted (as well as apparently changing the timing) is simple. The wording of the announcement is now "Please hold on while the bus is moving". The timing no longer offers the possibility of absurdity. The solution to this problem was not mathematical modelling, but thoughtful use of language.</div>
Tonyhttp://www.blogger.com/profile/08832715837375830128noreply@blogger.com0tag: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 />
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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 />
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<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>
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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 />
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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 />
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<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 />
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<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>
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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 />
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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 />
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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 />
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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 />
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<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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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.com0