Saturday 9 September 2023

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.

Saturday 5 August 2023

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.

Picture of London bus No 218. Image from, CC-BY-SA

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 50th 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

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 

Monday 24 October 2022

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.)

Monday 18 April 2022

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!

Sunday 12 September 2021

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.  

Sunday 4 July 2021

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.

Saturday 12 June 2021

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!