Been reading a review copy of a wonderful new book (due out in March) by Matt Cook from MIT Press, entitled “Sleight of Mind,” all about one of my favorite topics… paradoxes. It’s chockfull of them, and I’ll do a fuller review closer to the publication date, but will entice you with just a few fun, simple one-sentence logic puzzles (variations of the “liar paradox”) that appear near the start of one section of the volume:
“Answer truthfully, yes or no: Is ‘no’ the correct answer to this question?"
“Now answer this one truthfully: Is ‘yes’ the correct answer to this question?”
“What about, ‘Does this question have an answer?”
…and I’ll leave you to contemplate these on your own. (but essentially, for one of them neither answer works, for one of them both answers work, and for one of them only one answer fits).
One tidbit I found especially interesting/surprising (and had never heard about) from David Spiegelhalter’s current “The Art of Statistics” volume, is that Bayes' Theorem or Bayesian inference/reasoning has been banned in British courts… yeah, you read that right Nate Silver, BANNED in court! ;) As if statistics weren’t confused and misused enough by lawyers, one of the most common and vaunted modern approaches to probability isn’t even allowed, and this decision goes back at least 8 years.
Here are a few of the links that talk about the decision:
If any of our friends from across the pond can tell us more about how much controversy or debate this prohibition has produced in UK (or is it 'settled law' at this point?) I’d be curious to hear.
With the House impeachment vote looming in a few days perhaps I should squeeze in another relaxing ASMR clip before the never-ending Republican bullsh*t hits the proverbial fan & airwaves... Did an earlier ASMR with a feline, so here's one with Simba, a big, gorgeous Shetland Sheepdog... enjoy:
Just a little clash of thoughts in my mind recently….
By chance I’ve read several books/articles lately where Ed Witten’s name pops up; always of course in glowing terms, i.e., ‘the most brilliant scientist of our time,’ ‘the most important thinker alive,’ ‘the greatest living theorist,’ etc. Indeed, many of the other leading thinkers of our time admit that Witten is a person with ideas they can barely comprehend or keep up with. He is a mathematical physicist (and a Fields Medalist, among many awards), and I won’t even try to offer a bio or synopsis of him here (other than to say he and I were born in the same year, after which our lives diverged considerably ;) — most readers no doubt know basically who he is.
But what struck me in seeing his name so much recently, is that most Americans (outside the astute readers of this blog of course ;)) likely have no idea who Ed Witten is — his name would be largely unknown I bet to 90++% of all literate Americans.
I suspect, in his day, and certainly now, the name “Albert Einstein” was known to most all Americans. Even names like Heisenberg, Bohr, and others probably are somewhat recognizable to a majority of again literate Americans. Late in life, and especially following his death, “Richard Feynman” became very well-recognized among masses of American citizens. And of course there's Stephen Hawking too. Yet this current ‘most brilliant man alive’ exists somewhat in virtual obscurity outside of the scientific community that reveres him.
I’m not someone who thinks we should place individuals on pedestals (in fact I dislike the practice of building monuments/statues to individuals, or even naming buildings/streets after them, etc.), but I do wish our education system and press made citizens more aware of such major contributors to our knowledge and progress. I can’t help but think what a splendidly better country this would be if more of the electorate knew the name “Ed Witten” than knew the name “Kim Kardashian.” We might even then be a nation that would not elect (more tragically than laughably) an incompetent, authoritarian demagogue to the highest post of the land… and take 3 damn shattering years to impeach his autocratic ass. ...ehh, just some idle thoughts for this Tuesday. ;)
OK, for your Sunday entertainment and perspicacious pondering I’ll pass along this optical illusion that seems to be making the rounds — it’s somewhat cool, and I didn’t recall seeing it before:
It works better for some folks than for others, and there are actually multiple ways to make it work besides the “breathing” routine. Despite being passed around a lot right now, when I tried to learn more about it I found it went back to at least 2015, but still haven’t found the originator or precise explanation for how it works. In fact, for something that’s been out there for at least 4 years I’m surprised how little I could find about it on the Web. Maybe the best explanation (and still not all that incisive or neurologically revealing) is this comment placed on one site:
“The dots aren't moving. There are two sets of dots in alternating points around the circle, and they're flashing between the two sets. The illusion of movement comes from the suggestion to the viewer that the dots are moving, resulting in subconsciously following in the chosen direction.”
Anyway, if you know more about it feel free to elucidate in the comments below! ;)
Last Sunday the new monthly ranking of Twitter math feeds (from Kelly Truelove) came out with polymath Eric Weinstein leading the pack,** which rather surprised me (though it seems based solely on total number of followers) -- I don't recall if he'd been in the listings in recent prior months. Eric is trained in math, economics, physics, but more recently is known best as a central member of the so-called "Intellectual Dark Web," and doesn't actually tweet much about mathematics.
Moreover, while I generally enjoy Eric in long form (where he often stresses depth and nuance and clarity, in addition to being provocative), I find his Twitter feed far more annoying/irritating (and lacking in depth/nuance/clarity!).
Meanwhile, he appears on many podcasts these days, including his own relatively new one, "The Portal," and in a bit of synchronicity within hours of noticing the Twitter rankings, I discovered his latest (~hour-long) Portal episode focuses on some esoteric aspects of math (and physics and art), not often talked about, with mathematical artist London Tsai (it starts off a bit slow, but begins building at around the 5 minute point):
(this is just audio; video version will likely be up on YouTube at some point)
Here's a (completely separate) clip I've used before of Weinstein talking to Joe Rogan about the octonions (and more):
...Speaking of math podcasts, Numberphile has been doing a great irregular series of them (interviewing various individuals), that you should check out if you haven't already done so: https://player.fm/series/the-numberphile-podcast
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** interestingly, on a separate page, the listings show an individual's "public" vs. peer group ranking, and while Eric was #1 in the public sphere he was only #95 among "peers," which makes a bit more sense.
And who, you might wonder ranked #1 among math Twitter peers....
Evelyn Lamb! (no. 10 on list)
Popular math volumes continued to flourish in 2019 (or 'Year 3 of the Trumpocalypse' as it is otherwise known), and I'll just mention a few faves in this year-end wrap-up for ye merry holiday shopping....
My actual favorite read of 2019 isn’t even eligible to be my “book-of-the-year” pick… because it was published in 2006! I’ve mentioned briefly before that I thoroughly enjoyed John Derbyshire’s 2006 account of algebra, “Unknown Quantity” that I picked up perchance for a buck at a used book sale; perhaps the best, most accessible introduction to a lot of modern algebra that I’ve ever seen.
OK, onward though to the 2019 volume-of-the-year:
Despite my better judgment, my intentions not to let it happen, and my visceral internal sense of justice in the Universe… Ben Orlin has friggin’ done it again! For the second year in a row Ben takes the prize, this time for his “Change Is the Only Constant.” Ben must now promise NOT to write a book in 2020 just to give some piker like Keith Devlin, Ian Stewart, or Marcus du Sautoy a fighting chance next year. So if you’re reading this Ben, just slowly, gently put that pen down, amble off, slink into hiding for one year (maybe play some pickleball), and there need be no trouble, OK? Hope to see you again in 2021!
No one book shot to the top of my list as in some prior years, but in the end Ben won out for a couple of reasons:
1) the uniqueness and consistency of his writing/humor/wit… something I’ve not encountered in 50+ years of reading popular math (I mean Jordan Ellenberg is funny, but not funny every 23 seconds like Ben — and now I want to know if Ben is this funny when he’s changing his daughter’s diapers... or is he, at that point, merely mentally calculating fluid dynamics?).
2) I loved the last 6 chapters (out of 27 chapters) of this book. The first 2/3 of the volume were fun, but also a bit disjointed or choppy, and I didn’t initially think it would be book-of-the-year... until he totally won me over with those 6 final chapters, touching on some of my favorite topics in his inimitable way. By the way, a fine recent transcribed interview with Ben about his book, here:
So much for numero uno; plenty of other good books this year, including these:
Infinite Powers -- Steven Strogatz’s deservedly-lauded volume bringing the realm of calculus to the masses
Mind and Matter — John Urschel’s quite different autobiography of his unusual career combining mathematics and football
Tales of Impossibility — Dave Richeson’s creative, distinctive compendium of interesting math history centered on classic impossible problems The Art of Statistics — David Spiegelhalter ...good introduction to statistics for a general audience from a leading British practitioner (this may well be my 2nd-place fave book of the year, though it does require some initial interest in stats to enjoy)
The Best Writing In Mathematics 2019 — Mircea Piticianother great, diverse volume from Pitici with a lot on problems, puzzles, and proofs this year, but with philosophy and history thrown in as well
[also, worth remembering that this book lists MANY other worthwhile volumes and readings from 2018/2019]
The Doomsday Calculation — William Poundstone’s quirky volume largely on Bayesian probability, but touching upon various other interesting topics as well
The 9 Pitfalls of Data Science — Gary Smith ...from Oxford University Press and not well-distributed, but one of many recent, entertaining popular volumes on the problems with much-heralded ‘big data'
The Universe Speaks In Numbers — Graham Farmelo ...I wasn’t as enamored of this volume as many others were, but certainly worth a mention
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Several other books look interesting, but don’t know if/when I’ll find time to read (the Parker & Eastaway volumes are British-published, and for reasons that escape me, take their lazy, %#(@&!!ing, convolutedly-sweet time to even show up in the U.S.):
Calculus Reordered — David Bressoud Calculus Simplified — Oscar Fernandez (...at least the third calculus-focused book for a general audience from Fernandez in recent years)
Do Dice Play God? — Ian Stewart
Humble Pi — Matt Parker
Limitless Mind — Jo Boaler Mathematics For Human Flourishing -- Francis Su
Math Recess — Sunil Singh
Maths On the Back of an Envelope — Rob Eastaway
Proof — Amir Alexander
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Feel free to mention your own favorites from the year in the comments (especially ones I've left out), including more technical works you also think deserve attention.
Additionally, there were several books aimed specifically at math teachers that I haven't had the chance/time to see, but are probably worth knowing about if you are a teacher.
Things are slow at Math-Frolic, as I bide time watching the drip, drip, drip of daily news and those who be twisting slowly, slowly in the wind. ;) Sometime in the next week will probably post my usual year-end compendium of books (and 'math book-of-the-year') for your Holiday shopping; but today just another off-topic post for the few who may be interested (ye "boomers" ;) .… Over a year ago I wrote HERE about my personal experience with PSAPs (personal sound amplifying products), cheap substitutes for hearing aids. With a year+ added experience I’ll update that info:
I’ve now owned 4 of these devices (for a single ear), 2 of which were the same $20 model I initially talked about. For my ear that $20 model was the most comfortable (and slimmest) model, and while I wear this item limited hours on any given day, the longer you do wear them the more important comfort becomes as a consideration. And again I was surprised by how much benefit this cheapie product did afford (especially in non-crowded situations). On the downside, though initially a single charge of this (re-chargeable) unit would last close to 2 days, soon the product was only lasting about 6 hrs. on one charge — this becomes quite a nuisance to need re-charging before even getting through a day. Also, the product was somewhat flimsy and needed to be handled with care (let's just say I had to repair it with super glue a couple of times). I wouldn't much expect it to last more than a year even with proper care (but again you could buy one of these every 6 months for years on end and still spend less than the cost of a single real hearing aid).
The second product I bought (~$75) had slightly better sound-filtering quality, and was sturdier than the $20 product, but this was counterbalanced by the same need for frequent re-charging and being much less comfortable (due to being slightly thicker). Overall, the two products were almost a toss-up; one with slightly better hearing quality and sturdiness, the other being cheaper and more comfortable (though comfort will depend on the size/shape of your individual ear).
The last device I bought, again ~$70, was battery operated (instead of re-chargeable) — some of these PSAPs require 2 very small batteries (which I wished to avoid), but the one I chose required a single, larger, easier-to-handle and maneuver battery — the hype said a battery would last 500 hrs., but I find it to be more like maybe 200 hrs. — still better than having to re-charge each-and-every day (and the batteries aren’t too expensive). Also, its hearing quality is at least on par with the 2nd device I owned, if not slightly better, and it’s thinner and more comfortable (than the 2nd device), so overall (price/comfort/sound-quality) I regard it as the best of those I’ve tried (though they're all somewhat close), and for the time being will probably stick with single-battery-run options in the future, until more progress is made with re-chargeables (I also may look more seriously at German, Japanese, and American made models at some point; the ones I’ve had thus far are all Chinese made, and I suspect subject to lower quality control).
Even buying 4 of these devices in one year’s time, I’ve spent less than $200 (and attained some significant satisfaction) versus the $2000++ a real hearing aid would cost (and perhaps need replacement within 5 years time). Your mileage may vary (and there are certainly many instances/conditions where a true hearing aid may be necessary; I would just caution that real hearing aids, despite the increased cost, can still be difficult to maintain, fit, adjust or operate properly, and be easy to lose!). Obviously, if money is no object for you, or you have insurance covering hearing aids, you may not want to mess with PSAPs. And again, prices, quality, and availability of all these products ought continue to improve notably as the market for them continues to widen. (Walmart is increasingly expanding into the actual hearing aid market, and Costco has been there for some time).
Oh man, what a blood-curdling week if you've been viewing the impeachment hearings and watching Republicans criminally pretend to represent Americans' interests (with their clearly oligarchic Russian values)... and it ain't even close to over yet -- the current brilliant, rocket-science Trump strategy seems to be to commit endless impeachable acts each-and-every week, in hopes such hearings may never be brought to a close. ;) So, time already for another ASMR clip to temporarily change all that seething to soothing:
Alex Bellos poses one of the great puzzle 'paradoxes' of all time (well, actually a set of them) in his Guardian column today. I don't doubt he'll offer his usual lucid explanation later on... but also don't doubt that sparks will fly in the comments section... bring it on!: https://www.theguardian.com/science/2019/nov/18/can-you-solve-it-the-two-child-problem
Frank Ryan (NFL quarterback for Cleveland Browns I well remember)
Alexander Solzhenitsyn
Ed Thorpe (stock trader)
Lawrence Tribe (lawyer)
Virginia Wade (tennis star) They all deeply studied or majored in math in college, but mostly not going on for the PhD. (though Frank Ryan did), nor pursue the subject much career-wise. I stumbled upon this somewhat fun page of “Famous Nonmathematicians” awhile back -- the above is just a small sampling from it… a couple of these folks I knew about, but most on the list were new to me.
Mathematics… it takes all kinds! (check out the whole list)
A little filler post to end the weekend… not exactly math so much as an odd, quirky thing that once happened to me involving numbers:
Every month I scan (as everyone should) my monthly bank statement when it arrives, for inaccuracies. It’s almost always fine, but one month a few years back I noticed two entries for my local electric utility that does automatic withdrawal from my account. Early in the month was a figure that looked like my usual amount of dollars being withdrawn, but then later in the month was a second significantly larger withdrawal. Checking my records I knew the first withdrawal was correct, but the second one made no sense to me. Had someone hacked my account, stolen my identity… to pay a utility bill??? Or just some computer glitch? Going to the bank and spending an hour they couldn’t figure out what happened either, but said they’d keep working on it and call me if they discovered something. I went home and sure enough a couple hours later the call came with the oddball explanation:
A lady had recently moved into a new house and requested automatic withdrawal to pay her utility bill. They gave her a form to fill out, but where she was supposed to write in her 11-digitbanking account number she recorded instead a 12-digitcredit card number (wrongly thinking she could pay by credit card). When she turned the form in, a computer (looking for a bank acct. no.) only read the first 11 digits she’d written down (of her credit card number)… which turned out to be identical to MY bank account number! What are the chances!
Just a peculiar happenstance, and a reminder to always check your financial statements… in the age of computers weird things still happen. I'm glad I caught it the very first month it occurred -- even though I'm used to seeing XYZ Utility on my statement (and thus usually ignore it), the second entry was big enough to get my attention.
Anyone else got a quirky story to tell of how computers and/or numbers threw some strangeness or mystery into your life?
Just an ASMR clip to end the week -- this one for all the cat lovers:
(...it's looking like next week will likely be a verrrrrry stormy, tempestuous news week, so you may wish to keep this handy for purposes of remaining calm) ;)
New month, so a new math “profile” today (#15 in the series)… all you 65+ year-old mathematicians will already know this individual well, but all you baby-faced, Instagram-addicted 30-year-old-and-younger whippersnappers, dabbling in category theory and elliptic curves, listen up now… AND, respect your elders (despite the damn mess we've left you with)!
The first amazing fact is… Tom Lehrer is still alive!!! Glory be (rampant rumors of his demise the last couple decades have been consistently exaggerated). He’s in his nineties and therefore no longer making appearances on Ed Sullivan, but you can still find him on YouTube(thank you Sergey and Larry). "Thomas Andrew Lehrer" (named after those bulwarks of American comedy, Thomas Jefferson and Andrew Johnson), fittingly now lives in Massachusetts, the wise, sensible state that voted for George McGovern, Walter Mondale, and Michael Dukakis (while mindless, Zombie-like Americans were casting ballots for, if you can believe this, Nixon, Reagan, and Bush/Cheney, LOL). Tom's own political hero was Adlai Stevenson, who famously, and timelessly, said of Republicans back in the 50's that "if they will stop telling lies about the Democrats, we will stop telling the truth about them."
Tom Lehrer may have been the funniest mathematician of all time (…yeah, I know, Matt Parker is trying to give him a run for his money, but if Tom had had the Internet at his disposal, he might have left Matt in his Cantor dust).
Anyway, let’s jump right into it; here, for any Lehrer virgins out there, one of his all-time favorites, that he wrote for me when I was two-years old:
Now, compose yourself, and we’ll continue on…
Tom was born on April 9, 1928 in Manhattan, NY.; the same year that Noam Chomsky was born and also (like Tom) realized by age 2 that the United States was an imperialist corporatocracy. Lehrer was considered a prodigy early on and after graduating prep school, entered Harvard at age 15 (something I personally refused to do, waiting instead until Pomona College made room for me). There he studied mathematics while also writing comic songs for friends (sort of like we all did on college Saturday nights… except, HE was good at it!). He earned both a BA and MA in math from Harvard, magna cum laude, which I believe is Latin for ‘funny as Hell’. And he was inducted into Phi Beta Kappa, which I believe is Greek for ‘magna cum laude’.
Following that classroom stint at Hahhhvahd he taught math classes at MIT, Wellesley, Harvard, and University of California, never getting a dream job at the Claremont Colleges, but oh well. Lehrer continued off-and-on with classes at Harvard pursuing a math PhD. at the same time that his musical parody proclivities drew national attention. He spent time as a researcher at Los Alamos Laboratory as well. He was drafted into the Army and served from 1955-7 working at NSA, no doubt cracking up the code-crackers every time he burst into cryptic song (meanwhile, I was exploring the intrinsic structure of vintage humor on TV under the tutelage of Captain Kangaroo and Dr. Soupy Sales). Lehrer eventually gave up on the PhD. once it was clear there was more money to be made making nerdy Americans guffaw. (Of course all the laughing came to a screeching halt for America in November of 2016 when happy-go-lucky socialism was replaced with creeping, unfunny, and lunatic kleptocracy… but, I digress.)
Tom also wrote songs for PBS’s “The Electric Company” back in the day, so some of you are familiar with his work even without knowing it.
Here’s a couple more Lehrer classics, before we get too far along:
His take on “the New Math” from decades ago makes you wonder what he might have to say about Common Core today:
...and then branching out to chemistry, "The Element Song"... with lyrics even harder to memorize than Billy Joel's "We Didn't Start the Fire":
Tom was by no means just a writer of math/science hijinks, but a general satirist poking fun... sometimes very controversial/irreverent fun... at stodgy society. Indeed, when he toured parts of the world in the 1960's some of his ‘radical,’ liberal offerings were censored or banned in various locales — because, hey, you know you can’t laugh at the Establishment without consequences, or the next thing you know, you'll be asking them to pay taxes for gosh sakes! (Or, as Tom says at one point, "You can't be satirical and not be offensive to someone.") Of course societal standards were quite different back then — unlike now, you couldn’t even have nude pics of the First Lady of the U.S. splattered across the internet back in those Puritanistic days! Also, long before Mel Brooks took full credit, Tom voiced the idea of doing a musical based on Hitler. Despite his leftward leanings Lehrer was uncomfortable with antics of the 60's so-called "New Left" movement, as well as with some of the 'political correctness' that would follow, recognizing that 'free speech' is threatened by both the far left and far right.
Tom’s last main public performance was in 1972 on behalf of the presidential campaign of George McGovern (who I proudly voted for, and you younguns don't remember). 25 years later he did do one London performance for a new generation of fans. Through the 1970s, bored with performing the same material (he wrote 37 songs) over and over again, he mostly returned to teaching, and was beloved by students. His multitude of fans have kept his name and material alive and available, though a restrained Lehrer repeatedly eschewed the fame, fortune, and limelight headed his way, maintaining a surprising degree of privacy (making him, according to one writer, "the J.D. Salinger" of musical comedy).
Tom was quite an original, who influenced many future satirists like Harry Shearer and Mark Russell, who is close to Lehrer’s age, but who’s career took off about the time that Lehrer was withdrawing. Russell in turn likely influenced the Capitol Steps who gained national attention starting in 1981, and who will probably use their upcoming annual year-end review to tell Donald Trump (if, incredibly, he's still in office) where to shove it (just my educated hunch). On-the-other-hand, Lehrer once noted that much of the satire that followed him had been "de-fanged" and lost its edge.
Despite his elusiveness over the years, the Web bears a great many pieces on/about him; here are just a few...
This fabulous 2014 Buzzfeed piece on Lehrer has a lot more details than I've given here: https://www.buzzfeed.com/bensmith/tom-lehrer
(In this piece he’s quoted as once saying his entire repertoire was “part of a huge scientific project to which I have devoted my entire life. Namely, the attempt to prolong adolescence beyond all previous limits.”)
It’s kinda a shame Tom isn’t active today given all the material the current “Administration” (and obviously, I use that word loosely) hands out on a platter to work with — but then who wants an 80-hour work week. Still, I'm in the mood for perhaps a timely "Impeachment Rag":
Now I don't want to be accused of starting conspiracy theories, but one pressing question that still remains is, did ANYone ever see Tom Lehrer and Buddy Holly in the same room at the same time... EVER!? huhhhh?:
B.H.
T.L.
Perhaps we should end with one of Tom's prescient and uplifting ditties:
Happy dreams everyone; I hope you've enjoyed this trip down memory lane… and remember sometimes the only alternative to laughing at life… is to incessantly wail.
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.
