Carnival of Mathematics 198

The Carnival revisits an old home

Friday, October 1, 2021 · 4 min read

Hello and welcome to the 198th Carnival of Mathematics, a roving monthly roundup of mathy blog posts from around the Internet! Longtime Comfortably Numbered readers should be no stranger to the Carnival: I hosted #134, #148, and #159 in the past.

As is traditional, I want to start by thinking a bit about the number 198. It is actually a very dear number to me — I even own a “198” t-shirt! Why, you ask? The number 198 is the emblem of the wonderful CS198 program at Stanford, which hires a huge team of undergraduates to help teach the introductory computer science courses every year. (As far back as 1995, the faculty running the program published a retrospective on its impact on campus.)

Here is another fun fact about the number 198: suppose I start looking at all the integers ($1, 2, 3, 4, 5, \dots$), keeping only the ones that are perfect powers ($1, 4=2^2, 8=2^3, 9=3^2, 16=2^4, \dots$), and then taking their cumulative averages ($1$, $2.5=(1+4)/2$, $4.33=(1+4+8)/3$, $5.5=(1+4+8+9)/4$, $7.6=(1+4+8+9+16)/5, \dots$). You might wonder, are there any other integers in this sequence? It turns out, yes! And I bet you can guess the first one…

Now, time for some links! I’ve provided some “teaser” text from each submission to get you interested.

The BRSR blog asks: can you find a Euclidean triangle on a non-Euclidean surface?

A question that came up in a math chatroom (yes, I’m the kind of nerd who spends time in math chatrooms): find a “Euclidean” triangle on a non-Euclidean surface. More exactly, find a geodesic triangle on a surface with non-constant Gaussian curvature having the same sum of interior angles as an Euclidean triangle.

On his blog bit-player, Brian Hayes asks: why do pandemics peak in distinct waves as the months go by?

I’m puzzled by all this structural embellishment. Is it mere noise—a product of random fluctuations—or is there some driving mechanism we ought to know about, some switch or dial that’s turning the infection process on and off every few months?

I have a few ideas about possible explanations, but I’m not so keen on any of them that I would try to persuade you they’re correct. However, I do hope to persuade you there’s something here that needs explaining.

On Twitter, Thien An asks: would you rather have a bishop and a knight, or two bishops?

To get this figure, it suffices to grab 100k games and only keep the positions with BB+pawns v.s. BN+pawns and see who eventually wins. That’s very simplistic since there are indeed many other important positional features, but I was actually expecting less convincing results…

Looking at a video of a pomegranate sorting mechanism, Nisar Khan asks: how much do the pomegranate sizes vary between sorted buckets?

After watching the below video, thought of finding the approximate size differences of pomegranates sorted in different boxes…

On his math blog, Tony asks: did studying maths help Emma Raducanu win the US Open?

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…

On Risen Crypto, Trajesh asks: how does the quadratic sieve factoring algorithm really work?

The Quadratic Sieve is the second fastest algorithm for factoring large semiprimes. It’s the fastest for factoring ones which are lesser than 100 digits long.

On The Universe of Discourse, Mark Dominus asks: why is the “S” combinator named “S”? He also posted a nice puzzle that is easily solved if you happen to know Dilworth’s theorem; separately, Jim Fowler built a musical PICO-8 game where you race to find the chains guaranteed by Dilworth’s theorem.

I thought about this for a while but couldn’t make any progress. But OP had said “I know I have to construct a partially ordered set and possibly use Dilworth’s Theorem…” so I looked up Dilworth’s theorem.

On their blog “Gödel’s Last Letter and P=NP,” Ken Regan and Dick Lipton discuss “baby steps” in math, in the context of some exciting recent results.

Reckoned against a later paper by Cohn and Elkies, [Viazovska’s] improvement was 0.0000…000001. The recent post where we discussed the phrase “the proof is in the pudding” involves a number with six more zeroes than that. These are not what we mean by “baby steps.”

And finally, something more about sequences— on her math blog, Tanya Khovanova explores the top 50 largest numbers to start OEIS sequences.

My son, Alexey Radul, wrote a program that finds the largest numbers to start a sequence in the Online Encyclopedia of Integer Sequences (OEIS). To my surprise, the top ten are all numbers consisting of ones only.

That’s all for this month’s Carnival! What a joy to see — once more — the math blog world overflowing with curiosity. Hosting the Carnival of Mathematics truly is a delight.

Want even more math? The previous Carnival, the 197th, was hosted by Jeremy at Math $\cap$ Programming, and the next — the 199th, to which you can contribute here, starting now! — will be hosted by Vaibhav at DoubleRoot. See you there!

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