Carnival of Mathematics #249

Welcome to Carnival of Mathematics issue 249! Before we dive into our monthly assortment of maths, it's customary to provide some fascinating information about the issue number.

So I asked ChatGPT for interesting nuggets abut the number 249. ChatGPT tells me that 249 is not prime, that it is 15^2+24 and 24 is close to a square number, and that its digital root is 6.  None of that excited me, so I asked Copilot, which tells me that Asteroid 249 Ilse is the 249th catalogued asteroid, that Eglinton Island is the 249th largest island n the world, and that 249 is the international calling code for Sudan.

Giving up on AI, I asked Wikipedia about 249. It tells me that it is the natural number between 248 and 250.

So let's move quickly on to this month's links.

• Number Builder is a new online game from the tireless Christian Lawson-Perfect.
• John Cook argues that eigenvalue homework problems are backward.
• Ethan N. Epperly investigates if a polynomial function is trapped in a box, how much can it wiggle?
  
• John Cook (again) gives us Wagon's algorithm in three easy steps, with interesting stops along the way
• Lance Fortnow advises a reader worried about the future of mathematics and mathematicians
  
• Peter Cameron muses on whether mathematics is discovered or invented
  
• Kit Yates tells us about Middle Bias
  
• Skewray Research offers insights into an ancient stringed instrument through animated GuQin harmonics
  
• Tanya Khovanova proves a longstanding conjecture of Alexander Karabegov  
• Dave Richeson presents a 3-D printing idea of Colm Mulcahy's
• Colin Beveridge's Double Maths First Thing reflects on good and evil in the Links section of issue 4D
• Grant Sanderson and Peter Winkler present the Lattice Bacteria Puzzle, the solution to which I find truly astonishing.  Think about it before the answer appears on the 3blue1brown YouTube channel!
• Cracking the Cryptic tackled a sudoku by Olima celebrating the first day of the Gathering4Gardner meeting G4G16

And that is it for this month.  My grateful thanks to everyone who suggested links.

Links to previous Carnivals of Mathematics (and in due course to future ones) can be found at https://aperiodical.com/category/columns/carnival-of-mathematics/.

Bayes at the Bus Stop

 I make no claims to be a statistician (though I recently read a textbook* on Bayesian probability from cover to cover) and I apologise for any errors in this post: my excuse, in the words of Dr Johnson, is "Ignorance, madam, pure ignorance".

But having made that caveat, I feel that I think like a Bayesian when I am waiting for my bus home.   If there are a lot of people waiting, I reduce my estimate of my likely wait.  If there are a lot of people waiting, but then the other bus that the stop serves arrives and they all get on it, I raise my estimate.  If I see my bus leaving just before I arrive at the stop, I expect a longer wait.  And so on.

Now, I have always known that there are pitfalls in taking averages.  If I ask random people how long was their last wait for a bus and take the average, assuming they answer accurately I will possibly get a meaningful estimate of waiting times.  But if I ask random people who play golf what was the result of the last shot they played, almost all of them will have holed that shot.  That doesn't mean that almost all golf shots are holed!  Similarly if I ask cricketers what happened to the last ball they faced, most will have got out to it, but that doesn't mean that most cricket balls take a wicket.

But here is a point I have only recently fully appreciated.  What is the average class size in a school?  If a school has 60 students and two classes it might seem that the average is 30 students per class.  Suppose they are divided so that one class contains 35 students and the other 25.  The average still seems to be 30 students per class.  But if I ask every student how many are in their class, 25 will say "25" and 35 will say "35", so that average is actually higher - 30.833 by my calculation.  And arguably this is a much more meaningful figure since it better represents the experience of the students.

OK.  So now suppose that sometimes when I am waiting for my bus a friend meets me by chance at the bus stop, and that, whenever that happens, they afterwards ask me how long my total wait for the bus was on that occasion.  Let's make a few assumptions - buses come every 20 minutes, I arrive randomly at the bus stop so my average wait is uniformly distributed between 0 and 20, and I don't see buses coming or going: they materialise and dematerialise instantly giving me no clue as to how recently the last bus came.  My friend similarly arrives entirely randomly.  And of course we assume that the bus stop and my phone don't display live information about how far away the bus is, which takes all the fun out of waiting for a bus.** 

How long does my friend think my average wait is based on the data they gather?  I should be able to calculate this directly but out of laziness I simulated it in Excel.  Excel tells me (in one typical simulation of 1000 waits) that my average wait was 10.0904 minutes, while my friend's estimate was 13.3478 minutes.  Of course my friend's estimate is larger than my experience because they are more likely to meet me during a long wait than a short one.

So - here's the question.  I am waiting at the bus stop when I see my friend coming to join me.  Should I revise upwards my estimate of my likely wait, since on the occasions I meet my friend my wait is longer?  And how about if I see my friend on the other side of the street, but they don't see me?

I think I know the answer but I will leave you to think about it if you are so inclined.

* Chris Ferris, Bayesian Probability for Babies

** Not that there is any fun in waiting for a bus

A puzzle from my childhood

I've reconstructed a puzzle which I remember being amused by when I came across it in a school textbook.

We expect that if you add a quantity to itself, and then take half of the result, you get the origal quantity.  That doesn't happen here.

Background: Imperial measurements. One rod (or perch) is 5.5 yards. 1 chain is 4 perches or 22 yards.  1 furlong is 10 chains or 220 yards  One mile is 8 furlongs or 1760 yards.   

Now. let's start with a distance of 1 mile, 7 furlongs, 9 chains, 3 rods and 7 yards,  Let's add this to itself.  

        

            1    7    9    3    7

    +    1    7    9     3    7

    =    4    0    0    0    3   which when we divide by 2 gives

          2    0    0    0    1.5

So the sum is four miles and 3 yards, and if we divide this by two, we get 2 miles 0 furlongs 0 chains 0 rods and 1.5 yards - not what we started with!  Can you explain?

(The addition, in detail, from the right)

7 yards plus 7 yards is 14 yards which is 2 rods plus 3 yards,

3 rods plus 3 rods plus 2 rods (carried) is 8 rods which is 2 chains plus 0 rods,

9 chains + 9 chains plus 2 chains(carried) is 20 chains = 2 furlongs 0 chains,

7 furlongs + 7 furlongs plus 2 furlongs (carried) is 16 furlongs = 2 miles 0 furlongs

1 mile + 1 mile plus 2 miles (carried) is 4 miles.

My Wordle failure

 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.

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.

So I was interested to see this puzzle tackled by Mark Goodliffe of Cracking the Cryptic.   He faced the same problem I did.  Spoiler warning - next sentence is in white text so you don't have to read it.  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.  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!

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?  

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.  

I 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!

Laptop irritations

 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.  

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:

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.)

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?

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.)

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?

Is mathematics universal? An argument from sudoku

 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.

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.)

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 - x'clusion by Florian Wortmann.  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 -  to appreciate it you probably need to know a couple of sudoku theorems.)

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.

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.

(In thinking about this I have also been influenced by this video about algorithms, 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.

Carnival of Mathematics 218

 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 The Aperiodical.

This is Carnival 218, and it is customary to begin with some facts about the issue number. 

Following my immediate predecessor Vaibhav at DoubleRoot, 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:

1.       218 is an even number.

2.       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.

3.       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).

4.       The number 218 can be expressed as a sum of positive integers in multiple ways. One possible partition is 218 = 109 + 109.

5.       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.

6.       The number 218 is a numeric palindrome, meaning it reads the same forwards and backward.

7.       218 is sometimes used in scientific research as a numerical identifier for various purposes, such as labeling experimental groups or samples.

What do you say?  These seem to be either trivial (1, 2, 4, 7), plain wrong (3, 6), or stretching the meaning of “Interestingly” and “approximately” (5).   Well, what did you expect from ChatGPT?  If you want something that is interesting and true, much better to consult Wikipedia on “218 (number)”, which tells me

“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.”

But on to the content you’ve come for.  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.)

Well, July saw the premieres of two blockbuster movies, one of which is of obvious mathematical interest.  So here from Cambridge Mathematics is A brief history of Barbie and mathematics. (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).  And if you are more interested in the other film, Michael Molinsky in his “Quotations in Context” column 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.”

For another mathematician of the middle of the last century, the Infinitely Irrational podcast explored The Gödelfather: A Mathematical Masterpiece with special guest Ioanna Georgiou.  And, on the topic of logic, my own short talk about an earlier philosopher, the legendary John Buridan, and his mathematical paradoxes, was published by G4G Celebration.

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.  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.  Her new one is a Numberphile video about the New Tile (and the topic is moving so fast that Future Ayliean had to interrupt the video with news of developments since it was recorded).   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 new G4G Celebration video.  Meanwhile  Fractal Kitty provides translucent pngs of the Spectre tile.

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).

The New York Times marked the 50^th anniversary of The Encyclopedia of Integer Sequences. (Behind a paywall but with limited free access.)

Given the “facts” put forward by ChatGPT when I asked about the number 218, here is a timely article on Understanding AI: how large models work by Timothy B Lee and Sean Trott.  Meanwhile Neil Saunders argues at Medium.Com that while generative AI is “staggeringly competent” (the examples above don’t entirely convince me of that) it nevertheless doesn’t have any understanding.

Here is an account by Robert Smith about creating an algorithm for a computer to solve Rubik’s cube.  And here is an older (but I only recently found it) insightful video account of an algorithmic subtlety on the polylog Youtube channel, which starts with Rubik’s cube but gets into cryptographic security.

You’ll enjoy Alan Becker’s charming Animation vs. Math - and Dr Tom Crawford (@TomRocksMaths) has made a reaction video.

As always, the Cracking the Cryptic youtube channel presented examples of its expert sudoku solvers thinking mathematically in solving variant sudokus.  In solving this wonderful puzzle by Florian Wortmann, 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).

Colin Wright happened across a proof of Pythagoras’s Theorem using the Incircle.

James Propp presents a wide-ranging discussion about mistakes in calculations.

Karen Campe, whose blog will host the September Carnival of Mathematics, has provided A Shoutout for Squares.

Kyle Evans was interviewed by the Aperiodical about his forthcoming Edinburgh Fringe maths show, Maths at the Museum.

Matt Parker took advantage of the recent publication by David Cushing and David I. Stewart to buy just enough UK national lottery tickets to guarantee a win (but "a win" doesn't mean "a profit"!) (For a link to the paper see underneath Matt’s video.)

Sam Hartburn presents a song about a Knight’s Tour.

Snezana Lawrence is the guest of Mathematical Association President Professor Nira Chamberlain OBE in his VLOG.

Here's a Twitter (as it once was) thread on diversity in understanding mathematics, started by Talia Ringer.

Want to play the mathematical pattern-spotting game Set but would prefer a Non-Abelian or Projective version?  Check out Catherine Hsu’s Numberphile video.

Here is a Padlet roundup of maths questions on TV quiz shows (by missradders).

For those seeking an alternative to whatever the former Twitter is now called, here from the Continuous Everywhere but Differentiable Nowhere blog is an invitation to join mathstodon.xyz

And that's the end of this month's Carnival of Mathematics.  Enjoy!  And when the time comes, check out the September Carnival via The Aperiodical.

Image of London bus from https://bus-routes-in-london.fandom.com/wiki/London_Buses_route_218, CC-BY-SA 

The 24 Hour Maths Game Show

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 the event website.  The event is raising money for four excellent charities - Beat Eating Disorders, the Rheumatoid Arthritis Foundation, the Disasters Emergency Committee, and the Malala Fund: donations to one or more on the event Just Giving page. 

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.)

Mathematicians' diaries in the time of Covid

 How did mathematicians cope with the Covid lockdowns?  COVIDIARY of Mathematicians 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!

Did studying maths help Emma Raducanu win the US Open?

 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?

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.

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

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.  

Another dubious probabilistic argument

 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.  

A long time ago I used to do the challenging weekly cryptic crosswords in The Listener magazine.  (The Listener has been defunct for many years, though I think the crossword continues in The Times.)  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.

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.

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.

Is this conclusion valid?

As it happens, at the end of the year statistics for all The Listener 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.

Is this analysis correct?

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.

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.

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.

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?

Apologies if I am missing something but I think this analysis is sound. Have I read about this somewhere?

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!

.  

Tomorrow's Mathematicians Today

I spent last weekend at the online conference Tomorrow's Mathematicians Today, 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.

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? 

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:

The quality of student presentations was outstanding: the judges of the GCHQ Prize for the Best Presentation 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.

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!

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.

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!

Reading "The Queen's Gambit"

 I haven't watched the Netflix series The Queen's Gambit, but (prompted by a review in Private Eye) 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.

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?

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!

The chess novel which I loved as a teenager (and still do) was Anthony Glyn's The Dragon Variation, and the games in that book seemed authentic.  The descriptions of Beth Harmon's games in The Queen's Gambit didn't work for me.

So if you are looking for an entertaining novel about chess, I still recommend The Dragon Variation.

Memories of Peter Neumann

 

Peter M. Neumann (1940-2020)
(Photo by Bert Seghers,
Wikimedia Commons, Public Domain)

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.

In this post I'm going to share some rather rambling and inconsequential memories of him (with the warning that my memory is fallible).

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.  

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!

I attended Peter's famous Kinderseminar - 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.

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 A Mathematician's Apology: a much-praised account of Hardy which Peter felt misrepresented the great man by exaggerating his supposed unhappiness.

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.

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!)

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.

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!

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.

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 Collected Papers.  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!

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!

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.

Two mathematical magic tricks

 So over the last few weeks I have performed a couple of mathematical magic tricks which I have put on my Youtube channel.

As part of the 24 Hour Maths Magic Show,  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 watch the trick and find out how it relates to the phrase.

At the virtual MathsJam 2020 Gathering, with the help of Ruth, I performed my favourite Martin Gardner card trick - you can watch this one here.

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 The Language Lover's Puzzle Book) 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?  

Memories of John Conway

The mathematician John Horton Conway died just over a week ago. (I recommend Siobhan Roberts's biography Genius at Play.)  I' intend to write a post later about what his book On Numbers and Games 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.

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?)

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.

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.

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!

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.

Self-referential humour

I' 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 Vicious Circles and Infinity and Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid.  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.

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.)

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.)

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":  .

"Remind me, Morris, never to ask you to do anything … at all  … ever"

"Does that include reminding you never to ask me to do anything at all ever?"

"Probably – I lost interest halfway through the question"

(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.)

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...

A brilliant solution to the Tower of Hanoi

I 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).

The Tower of Hanoi

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.

The surprise is how many moves are necessary - to move a pile of n discs takes 2^(n-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?

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.

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.

Thoughts on Hannah Fry's Royal Institution Christmas Lectures

Another New Year, another resolution to write more blog posts.  We'll see.

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. 

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.)

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!)

(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.)

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?

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.

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.

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.

(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.)