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.

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

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