Anyone out there know if this is an original observation, or is it something that has been known and explored previously? Or, is it perhaps something that is actually in some way obvious within number theory that I’m just not familiar with?

2) Have mentioned this sort of thing in the distant past… one of our local radio stations plays 60’s/70’s pop hits all day (MY kinda music ;) and am always amazed at how, when I’m spinning the dial in the car, I can catch just 2-3 notes of an old hit and know immediately what it is (and who sang it)! It seems an incredible feat, yet easily and automatically done — and I imagine there’s even some mathematics involved, given all the different note combinations that are possible, and all that have been heard over a lifetime, yet with the notes, instruments, cadence, rhythm, etc. a single song can be recognized so quickly.

Anyway, anyone know of research papers/work that have looked at this specific identification of musical pieces from a few individual notes?; not general papers about memory for long ago events or auditory patterns, but specifically pertaining to musical notes? Surely, somewhere in cognitive psychology this has been studied? I'd be curious.

3) On Twitter, Jeet Heer linked to this piece reviewing what professions are sending the most contributions to major Democratic presidential candidates. The surprising bit (or at least one of them), if you scroll down toward the bottom, is Elizabeth Warren getting her strongest support from… drumrrrroll... mathematicians (29% of them)!

Meanwhile, Andrew Yang’s strongest support came from “Pizza delivery drivers,” 18% of whom back him, and on a side-note, 54.8% of fork-lift operators support Bernie Sanders.

(…not sure I believe any of these figures, but 'tis the season ;)

5) And lastly, another favorite tweet from the week:

My advisor's advisor, advisor, advisor, advisor, advisor was Leonhard Euler. I tell my calculus students, this is a good example of a monotone decreasing sequence.

Start the week with a little 'reverse' puzzle-thinking... There are lots of variations of the Monty Hall “paradox,” but as probably all readers here know in its standard form it is statistically better (indeed twice as good) for the contestant to ‘switch’ doors in the usual game as to stay with their initially chosen door. But one can make a small change in the set-up to make it better for the contestant to stick with their chosen door.

Instead of the usual premise of say a car behind one door, and billy goats behind the other two, consider the case of a goat behind one door and cars behind the other two (continue to assume the contestant prefers a car... even though billy goats can be downright appealing… and host Monty still knows what’s behind all 3 doors).

Now, contestant Sheldon picks say Door #1, and Monty in turn opens Door #3 revealing a car… should Sheldon switch to Door #2 or remain with #1 (or, does it make any difference)?

The quick math shows that at the outset Sheldon had a 2/3 chance of selecting a car by random chance, better than the chance that either one of the remaining doors hide a car, and thus Sheldon is better off staying with his 2/3 first pick in this instance (essentially, exactly the reverse argument of the normal set-up).

(Though, IF, as a prior, we know that Sheldon is a telepathic extraterrestrial, or one with X-ray vision, than that could alter the odds considerably…;)

1) I listen to several weekly podcasts, but oddly very few math podcasts. Still, am surprised (embarrassed?) I was unaware of the “Breaking Math” podcast that apparently has been around for awhile. Caught my attention because of the current session done with Ben Orlin (he's this cartoony guy who thinks he can write math books that blogger-dupes will fall in love with and recommend non-stop ;), but so many of their past episodes look great as well:

3) I linked earlier (on Twitter) to this short inquiry from Ken Abbott’s blog (quoted verbatim below), related to ever-fascinating prime number pattern/distribution, but never saw any response to it (Ken doesn’t seem to have a “comments” section on his blog and I don’t know if anyone responded to him via email?):

*********************

“About the distribution of Twin Primes.

Let p1 be the first prime of a twin prime pair and let p2 be the first prime of the next consecutive twin prime pair. Then, I'm quite surprised how small p2-p1 stays. And when it does increase it will suddenly drop back to a very small number such as 12.

Anybody have any input on this?”

*********************

I also asked, but never got a response, if there was some table readily available on the Web of such p2-p1 values? Or, more generally, any papers out there that have studied this particular set of values? (surely someone has?)

==> ADDENDUM: It occurred to me that the wonderful OEIS site would likely have something on this, and typing in the first several p2-p1 values, sure enough it is there:

I think I can safely say that one of the few and best things to come out of America’s ill-fated war in Viet Nam was.… Fawn Nguyen (who came here from there in 1976, when I was just a toddler — well, a toddler with a beard and mustache and two college degrees).

There’s an awful lot of wonderful, helpful cross-talk among primary/secondary teachers on internet social media, especially on Twitter. Often when you see a first name (Bob, Carol, Ted, Alice...) you have to check further to see exactly who it is…. BUT when you see the name “Fawn” EVERYone KNOWS who it is!! (ya know kinda like when you see the name Beyonce, Cher, Bono, Euclid, or Quasimodo). Because if you don’t know Fawn Nguyen, you’re probably not a real teacher to begin with, but just some planted Russian spy… who, in your best-case scenario, is hoping to marry a filthy-rich, power-hungry American real estate doofus, and eventually become First Lady (or Escort) of these here not-so-United States… but, I digress (again).

Fawn’s early life story, making her way to the U.S. as a youngster from Viet Nam, is inspiring (especially when compared with Ivanka Trump's childhood stories of Saturday Gucci shopping sprees), as she told it when I interviewed her back in 2014:

She eventually made her way from the Midwest (where she first arrived) to the West Coast of the U.S., where she has taught math since the French Revolution, first in Oregon and then in California. She’s still a huge fan of the Oregon Ducks (though frankly, I suspect that’s ONLY because “duck” rhymes with her favorite word). After teaching in the classroom for 30+ years (but who’s counting) they’ve now made her a ‘Teacher On Special Assignment’ (perhaps because too many of her students were deliberately flunking just so they could spend another year with her). Anyway, it all has a sort of clandestine ring to it -- psssst… Netflix, I think it would make a great TV mini-series: “Teacher On Special Assignment, starring Fawn Nguyen” (or, Angelina Jolie playing Fawn)… but then, my best ideas are always ignored. :(

I’d recommend Fawn for a teaching position at Pomona College, but with California either burning to the ground or falling into the Pacific Ocean in the next decade, I hesitate to do that. Better perhaps that she move back to Minnesota (where she first arrived in the U.S. lo those many years ago), hang out with Ben Orlin, and discuss the never-ending intricacies of the Pythagorean theorem. Maybe they could even write a book together — she could do the drawings and it would be entitled “Math With F*cking Good Drawings, At Long Last Dammit” (...or whatever Simon & Schuster preferred). I also think Fawn would make a great Secretary of Education; I mean after 3 years of Betsy DeVos it would be nice (even earth-shattering) to have someone in charge who understands teaching, and education, and learning, and middle school boys.

I can’t do justice to Fawn here... Like Sean Carroll who I profiled earlier and who can’t possibly be summarized in a blogpost either, there is just too much stuff to tell — but at least I can comprehend Fawn when she talks; whereas Sean is caught up in that crazy cosmology cacophony that leaves me confounded... especially the part where I'm both dead and alive at the same time until someone observes me.

Fawn’s blog “Finding Ways” is one of the more distinctive math blogs out there; a mix of teaching points, personal observation, uncommon wisdom, and Fawn’s helpful household hints (like the hysterically absurdist, "Make your bed every morning;" she's such a kidder!). You never know just what she might write about next, or when a four-letter word will pop up without warning (while you're reading pleasantly to your 7-year-old). I can’t even employ Fawn’s favorite words here on a nice family blog like this (…well, except that is, when I’m talking about Donald T. or any member of his dysfunctional family… and I use the word “family” advisedly in the same loose, sketchy sense that people speak of the ‘Manson family’). By the way, "Visual Patterns" and "MathTalks" are a couple of other sites for math teachers, that Fawn started.

Fawn learned English as a second language, but writes better than most of us who learned it first (or, right after pig-Latin); and certainly better than anyone in Mitch McConnell's Senate Office. In fact, her writing has the ability to make grown men cry (but don’t ask me how I know that), and I’ve lost track of how many times I’ve linked to this 2012 blogpost of hers.

But seriously if some prescient publisher doesn’t snap her up one day soon it will be a miscarriage of book publishing (…I may be lookin’ at you Princeton University Press).

Fawn admits to having once been too judgmental, something that having 3 kids and more than 3 doughnuts remedied (still, I'd be a tad cautious about crossing Fawn whenever she's holding a fork):

Fawn loves food, and if you tell her she makes the best Pho on the planet you’ll likely be moved to the #1 slot on her speed-dial, AND be invited over in a few weeks for Thanksgiving dinner. She’s also been known to down a beer on days of the week beginning with a consonant... and, perhaps also a Bloody Mary on occasion: https://twitter.com/fawnpnguyen/status/1184089101655396353

Fawn is on TwitterHERE in case you’re not already one of her 20,000 ardent followers (more than triple what Donald Trump has, after you eliminate Russian bots).

I get the feeling that “Nguyen” in Viet Nam must be like “Smith” in America — I mean there’s a LOT of them. So if you look Fawn up on the internet DON’T get confused by any similarly-named felons, strippers, embezzlers, ne’er-do-wells, real estate agents, etc. that you may run across there (at least I think they’re different people). You want only the reeeeal Fawn (the one who answers to “Fawnzie” because of her affection for the Fonzie character of old “Happy Days” episodes that she credits with helping her learn English). All the others are lame imitations.

Needless to say Fawn has a lot of fans. Here is one member of her cult extolling her virtues in a blogpost from a year ago:

I don’t know how much Fawn normally pays to have such posts planted around the blogosphere, but I want to assure readers that I’m NOT receiving a single penny for today’s blog entry! (…though I may check my mailbox a little more often over the next few days).

Fawn has talked to teacher groups and conferences more times than I've flossed my back molars (not that that’s saying much). Here is a brief inimitable example of her in action:

With the money she’s amassed from teaching, Fawn will no doubt be able to retire in just a few more decades, and all her past students (who have become Wall Street bankers or professional poker players), can then fondly visit her at one of Los Angeles’ finest homeless shelters.

No math today, and musical 'Chill Fridays' are over, but thinking of doing occasional ASMR Chill Days in the future. Most folks by now probably know what ASMR is all about so I shan't explain/describe it (plenty on the Web if you need to look it up). I have a couple hundred bookmarked ASMR videos that I enjoy/find relaxing (but peoples' tastes for these things vary widely to say the least!). So from time-to-time will share one -- usually one that may be less familiar (many of the more popular ones have millions of views). Shoe-shining is one favorite category of ASMR and here's a fellow who has his own distinctive technique, but hasn't garnered as many views as several others:

...and, believe-it-or-not, yet another Math-Frolic 'profile' this coming Sunday! Hmmm... who from the West Coast might it be?

"Physicists can predict what atoms will do. But gather enough atoms, and the calculations grow unwieldy. We need new, emergent laws -- chemical laws. Then, gather enough chemicals, and complexity overwhelms us again. We need biology to step in with new theories and rules. And so on down the line. At each tipping point, the role of math evolves: from certain to tentative, from deterministic to statistical, from consensus to controversy. Simple phenomena (like quarks) follow mathematical rules with slavish fidelity. Complex phenomena (like toddlers) less so." -- Ben Orlin from "Change Is the Only Constant" (incredibly here, Ben almost succeeds in stringing together 9 sentences without a joke being involved... but, then he had to throw in "toddlers" at the end ;)

Readers have been clamoring non-stop for another scintillating mathematician “profile” (well, not really, that's more like a KellyAnne Conway alternative factoid), so without further ado….

The hottest thing going in mathematics these days is statistics/probability (I mean other than the 613 books out this year on calculus), so figured a profile of a statistician was in order (just a short while back I mentioned several that I follow on Twitter)… BUT, I rarely do what I ought… so instead, will briefly profile a former law student...

In case you think that makes no sense, well not so fast Stephen Hawking wannabes! In recent years this particular law student had one of the most popular stat-related websites of anyone on the entire Web (in fact, he turned it into a little book). If you’ve followed math much at all on the internet over that time you likely already know of Tyler Vigen and his “Spurious Correlations” site. IF you’re NOT familiar with Tyler’s work, he regularly posts graphs of variables that seem to correlate VERY strongly, even though they have no logical reason for doing so (he's generated ~85,000 graphs!). So for example, here’s one of his graphs:

Or, for something less pleasant to think about:

Perhaps that even makes sense to you!? Anyway, you get the idea…. (statistics that are ludicrous... with or without fat tails).

But of course who says these correlations aren’t real!? “Correlation is not causation” is not so simple, my friend. As every third-grader knows by now, a butterfly flying over the Indian Ocean in April may cause a hurricane in Miami in October (or, the election of neo-fascists in two of the world’s oldest democracies later on… but I digress). Perhaps Adoes not directly cause B, but perhaps Adoes cause Q which causes R which causes X which causes Y which causes Z, which, causes... B! Well then!… (Hey, it could happen... and does!).

In any event, here's Tyler explaining his site in his own words:

So back to our main focus…. the profile of Tyler Vigen, known anagrammatically as “Ever Tingly”:

Ever, I mean Tyler, was born in Burnsville, Minnesota on Feb. 13, 1991 (the same year that "Silence of the Lambs" came out and fava beans with a fine chianti became a thing). That makes him an Aquarium, just like myself, and accounting for the constant ’sink or swim’ feeling I've had throughout life. No, no, truth-be-told, the 'Age of Aquarius' ended long ago, and ever since Reagan we’ve been floundering deplorably in the Age of Rush Limbaugh (not completely unlike what 1930s Germany experienced... in fact the correlation gets closer and closer over time; some sort of regression to the mean... and reprehensible).

Tylerearned a bachelor’s degree in criminal justice from Metropolitan State University in Minnesota and worked in the National Guard in areas of geospatial analysis and satellite imagery and other data. From there he went on toHarvard Law School, thus missing out on a real education at Pomona College, but his technical interests and computer programming continued. He created his “Spurious Correlations” website while at Harvard, writing a program to search out misbegotten statistical flukes (each one of which Republicans ponder super-seriously). He also wrote other programs to assist students along their way.

He graduated Harvard in 2016 but was not interested in pursuing a career in law (which is a shame because I may need a good lawyer before the current Administration, known colloquially as the 4th Reich, comes ignominiously to a grinding halt, just maybe around Christmas!). But I can't really blame Tyler; I mean if there are two things the world is awash in, it's lawyers and people who write calculus books.

Tyler, by the way, pronounces his last name “VEE-Gun,” just like that cult of San Francisco hippies who, unlike the rest of incorrigible human riff-raff, actually takes ethics seriously. Tyler got married earlier this year and now lives and works in Chicago as a management consultant for Boston Consulting Group,where, if you can believe this, John Legend, Mitt Romney, and Bibi Netanyahu all previously worked (probably Carl Yastrzemski too, but I haven't verified that). There, Tyler makes use of his programming skills, and, I assume if Government contracts are involved, is eligible to become a whistleblower.

Tyler seems to have been inactive on Twitter for about a year, but he does have a Facebook page where Mark Zuckerberg violates his rights each-and-every day while stealing his identity to put up for sale on the Dark Web, right alongside Jared Kushner's (no, I’m kidding of course, Mark would never do that... but Jack Dorsey, not so sure).

I hope that while he was at Harvard, Tyler taught those lawyers-to-be some statistics — we need more lawyers who understand statistics… heck, we need more mathematicians... and, more 12th graders for that matter... who understand statistics, if America is to be saved!

That's about all my crack investigative team could dig up on Tyler, but we wish him well in all future endeavors, and caution him to moderate his margarine consumption. Now, if you'll excuse me, I'm in the middle of exploring any budding correlation between presidents committing treason and impeachment... and I don't mean spurious.

I started the “Chill Friday” posts more than a year ago, essentially pre-loading over a year’s-worth of music-interlude posts. Those front-loaded posts have now concluded, but I’ll close out the series by re-running the very first selection:

(meanwhile, for Sunday, there is another Math-Frolic "profile" scheduled)

If you live anywhere in MathLand you likely know that today is the day that a new Ben Orlin book hits the streets (creating similar anticipation to the first Tesla!).

I’ve written so much and often about the creative mind, writing, humor, of Ben Orlin that it's time to STOP it!… stop sounding like a broken record (and as we all know broken records can only be right twice a day… er wait, nevermind, that’s clocks).

Anyway, some smart-aleck genealogist will probably discover that Ben and I are 17th cousins thrice removed, and claim that that alone explains my incessant, redundant praise. So I shan't be telling you anything about the marvelous fun and insights of Ben’s newest fantabulous, beautifully-produced, sparkling opus, “Change Is the Only Constant,” the 713th book this year (so it seems) on the subject of calculus... BUT the the only one with page-after-page of cartoons (conferring that rare privilege of laughing at the term “Weierstrass’s function”). The book is not a novel, though it IS novel; it’s not a textbook, but it will explain some calculus, even to the phobic — it’s almost what you might expect if Dave Barry ever wrote a volume about calculus… but no, not really (zero mention of "boogers" for example). My only review of this, and all future Orlin volumes, therefore will be, “BUY and laugh along, and be eternally thankful that his wife and first epsilon are willing to share Ben so generously with the rest of the public!" (also, do eat, sleep, and go to the bathroom on occasion).

So, sorry, I WILL NOT describe to you any of the joyous, meticulous bits this volume contains from beginning to end! (or even in the middle). Why spoil your fun.

Just buy, read, learn, laugh… and, wait patiently for Ben’s third book to come out, the one where, at long last, I expect round-faced buddy #2 will prove the Riemann Hypothesis -- I feel certain Ben secretly has a proof, but it's too long for this blog post.

p.s…. be sure to read all the book's interesting endnotes as well: Class Notes, Bibliography, Acknowledgements (where, interestingly, Ben divulges that the book “collapsed” at one point before he salvaged the “wreckage”).

Meanwhile, I'm absolutely committed to leaving the waaay-overused phrase** "Ben Orlin" out of all future posts (uhhh, wish me luck). ** needless to say, this promise doesn't hold for any other commonly over-used phrases, like "shit weasel"*** for Donald Trump *** or, "Putin Puppet"

Just biding time lately, watching the current pretend Administration/regime implode, crash, self-immolate, and perhaps destroy forever Vladimir Putin’s sweet, sweet dream of defeating America without ever firing a shot (…but perhaps I’ve said too much).

Anyway, just a filler post today, again recommending books for the popular math crowd. Over the 9+ years of this blog I’ve recommended a LOT of books and will just re-list here my eclectic favorites, through 2018,among 100s, for any who may have missed some (in alphabetical order by author):

Here’s Looking At You and

The Grapes of Math — both by Alex Bellos

Single Digits — Marc Chamberland

How Not To Be Wrong — Jordan Ellenberg

Mathematics Without Apologies — Michael Harris

Foolproof — Brian Hayes

When Einstein Walked With GÃ¶del — Jim Holt

The Prime Number Conspiracy — ed. by Thomas Lin

Weapons of Math Destruction — Cathy O’Neil

Math With Bad Drawings — Ben Orlin

Things To Make and Do In the Fourth Dimension— Matt Parker

Genius At Play — Siobhan Roberts

The Mathematics Lover’s Companion — Edward Scheinerman

Exact Thinking In Demented Times — Karl Sigmund

The Simpsons and Their Mathematical Secrets" — Simon Singh

The Joy of X — Steven Strogatz

The Outer Limits of Reason — Noson Yanofsky

Will also toss in Judea Pearl’s The Book of Why which I’m just now reading (originally published in 2018, but newly out now in paperback).

———————————

Oddly, the above list does not include a single volume from any of my 3 FAVORITE math popularizers: Keith Devlin, Ian Stewart, and Richard Elwes. But I certainly recommend anything from those three writers (all of whom, interestingly, are of British origin).

Lastly, for sheer perseverance and a unique contribution to popular math, I’ll cite Mircea Pitici’s anthology series, “The Best Writing In Mathematics” with annual volumes from 2010 to the present. It is a monumental effort (especially since Mircea has his own life to attend to!) to sort through the countless math offerings available each year and thoughtfully choose the selections for these collections. Always a wide range, and always some surprises. I think almost all of the above are now available in paperback and/or might be found in a well-supplied local used bookshop, as well as certainly online or through your local bookstore. So read away....

This has been getting passed around fairly well, but in case you missed it, another wonderful enlightening problem from James Tanton (suitable and interesting for most grade levels):