Sunday, March 31, 2019

A Tip-of-the-hat...

A few years back someone on the Web asked what books people had especially enjoyed or found important in their path to mathematics. A book I was completely unfamiliar with, “Mathematics: A Human Endeavor” by Harold R. Jacobs, got several very favorable mentions. Having never heard of it I was curious about it, and figured if I ever ran into it I’d get a copy. About a year ago I stumbled upon it for a dollar :))) at a used book sale and snapped it up. It’s gone through several editions since the original 1970 volume which is what I got  — by then I was in college, easily explaining how I had missed this volume that is for a middle or high school level.
Anyway, it is indeed a wonderful text (with a Foreword, incidentally by Martin Gardner), especially for its time when most school volumes were pretty dry and boringly pedagogic; Jacobs’ love for his subject shines through, as well as his desire, way back then, to make math attractive to others. He wrote several other math volumes as well, all of which get high ratings on Amazon, and I think several may be used especially by the homeschooling crowd:

I briefly mentioned the volume on Twitter awhile back, and again some folks responded with  fond memories of it. So I'm surprised by how little biographical information I could find about Mr. Jacobs on the Web. He did at one time receive the ‘Most Outstanding High School Mathematics Teacher in Los Angeles’ award, and the Presidential Award for Excellence in Mathematics Teaching in 1988. I assume he is retired and still living (at least I found no obituary for him); at any rate he seemed like someone deserving a tip-of-the-hat; doing early on what so many are striving more visibly to do these days in the direction of improving and broadening math education.

If anyone can fill in a little more information about Mr. Jacobs, or simply has fond memories of his books or personal encounters with him to pass on, I’d be pleased to hear of such.

Friday, March 29, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Sunday, March 24, 2019

Matt Parker, Man of Mystery (…and damn good writer)


^ One of these is a headshot of a true national Australian gem; the other is Matt Parker.

Who’s the hottest mathematician going these days? ...probably debatable, but if we ask instead who’s the "hottest-mathematician-AND-stand-up-comic" going these days, then it’s more likely a slam-dunk for Matt Parker, newly-out with his second fab popular math volume, “Humble Pi.” So just a post today paying tribute to the jaunty numbers bloke who spreads math far-and-wide... including biographical information taken from public sources, private spies, and NSA operatives (…or, made up out of thin air):

Though he hails from Britain now, Matt’s originally from Australia, and I have a soft spot for anyone from the land of cockatoos and Crocodile Dundee. Somehow he seems to also sneak into the U.S. on occasion (…so much for Homeland Security!).
Born in 1980, Matt is ~39 years old (coincidentally, an age I was at one time). I couldn’t find his actual birthday, but he seems very Sagittarian to me; or perhaps Capricornish, though Aries and Gemini are definitely also in the running. ;)) God forbid if I’m wrong and he’s really a friggin’ Leo!

Matt was born 60 years after Ramanujan died, and began channeling the Indian prodigy at a very young age (OK, I made that up; I honestly don’t know who he channels; it could be Henny Youngman).
As best my crack genealogical team was able to uncover (without actual DNA samples) he is no relation whatsoever to Fess Parker of Davy Crockett fame. Nor will Sarah Jessica Parker or Trey Parker claim any connection to him at all. It’s even possible he was raised by a pack of wild dingoes in the Australian outback (…though admittedly, unlikely, and I don’t wish to start rumors here... but you, dear reader, can do what you wish). 
According to rough estimates, Matt stands somewhere between 4’9” and 8’7” tall, and has blue, brown, or green eyes (maybe). He's been known to appreciate an occasional beer, but ONLY on days of the week that include a vowel (perhaps the result of a strict Zoroastrian upbringing; just a guess).

In college he went from mechanical engineering to physics to mathematics, before finding his true love in whatever the heck you call what he’s doing nowadays.

Matt’s first book was a tour-de-force in book titling: “Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More.”
Luckily, he learned to control his incorrigible verbal impulses (or got a better editor) with his current delicious volume, “Humble Pi” (I haven’t read it yet, but some folks say it's palatable).
In fact recently, and remarkably (even allowing for massive bribes), the book hit #1 on the British nonfiction best-seller list, a fact, which, in his humility, Matt only managed to note on Twitter 1729 times (a number Ramanujan fans will appreciate).

Hopefully, we can expect many more such books from him in the future, though he’s almost out of time for a Fields Medal (...but does still have πgoogol more chances for a Nobel Peace Prize than Donald Trump).
Last year he did deservedly share the “2018 Communications Award” of the Joint Policy Board for Mathematics, with Vi Hart, leading inevitably to the pressing question, if Matt & Vi ever had a child together would that offspring prove the Riemann Hypothesis? We may never know.

And so far as I could discover, Emmy Noether NEVER had a single good word to say about Matt, though other female mathematicians seem quite willing to associate with him (Katie, Jo, Holly, Hannah, etc. may need to explain themselves).

Nor do I know what Matt's Erdös Number is, but don't let him ever try to convince you that he's anywhere in this photo (ain't so; that's me on the right with my grandpa checking over some proof we found in the margin of a book... NOT):

Many mathematicians are known to indulge in puns. Unfortunately, Matt IS one of them (but I won't "pun"ish you here with any examples).
He is a bright, exuberant, engaging, creative, witty, sharp-as-a-tack, well-spoken nerd math communicator… except of course when compared to his colleague James Grime, who makes Matt seem like just another boring, humdrum, numbers dullard scrounging a living off of his far-more capable/employable physicist wife. (James, by the way, it's been rumored, goes to bed every night with a king-size clothes-hanger in his mouth.)

No chicken to controversy, Matt once famously and unsuccessfully tangled with the British government when he realized their geometric depiction of a football (on road signs) was inexplicably far off from its true appearance:
[If you spot Matt ripping down football signs anywhere on the streets of London feel free, indeed obliged, to call the nearest office of Interpol.]

Next, he may run for Prime Minister, since that would be a much easier improvement to make in Britain these days than road signage.

It's believed Matt spends most of his spare time obsessing over magic squares, listening to old Tom Lehrer records, and re-reading absurd blog posts about himself. Hi Matt!!

If you open your browser to the internet and randomly tap some buttons, you’ll likely come across Matt; I mean he’s everywhere! Has his own website here:

...but regularly appears on Numberphile (where a no-holds-barred, winner-take-all grudge match between he and Tadashi Tokieda could well be in the works):

and still more videos here:
…ohh, and, here:

(occasionally, the initial thought upon viewing one of his videos, like this one, is to question Matt's sanity, but, like most first impressions, that will pass... unless, of course, you're trained in psychiatry)

He’s also had several appearances on BBC’s “The Infinite Monkey Cage” with Brian Cox and Robin Ince (as well as several other science outlets):

of course he’s on Twitter, where 80,000 lads & lassies, who apparently have nothing better to do with their lives, follow him:

And, with others, he sells math gear here (…or as he lamely calls it in mangled English, “maths gear” -- those Brits constantly humour me with their warped English spelling):

Matt came in second in last year’s “Big Internet Math Off" contest, meaning he can only lay claim to being 'The World’s Second Most Interesting Mathematician' (well-ahead of Gauss, Euler, and Galois, but behind Nira Chamberlain who won the event).

And he is such a master of disguise, employing so many different aliases, that many don't even realize he once acted in an American sitcom:

I hear-tell that Benedict Cumberbatch is cast to play Parker when they do the film version of Matt's life (though maybe I have it backwards and Matt intends to play Benedict when they do the big screen version of Cumberbatch's life).

I hope this brief informative profile adds to your knowledge/appreciation of this standup guy, as we look forward to many more lemmas, conjectures, theorems, and jokes from Matt in our dystopian future.
To help you recognize him in that future I had my computer generate this algorithmically age-enhanced photo of him (...yeah, yeah, I know, it looks surprisingly like Keith Devlin):

I could tell you even more about Matt, much more… but if I did he’d have to kill me.

(...anyway... seriously... buy the book)

Friday, March 22, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

...and be sure to return here on Sunday for an in-depth ;) report on a mathy bloke and author you all know and love.

Wednesday, March 20, 2019

Mid-week Mindbenders...

Another trip-down-memory lane with this old mind-bender about a few cards, that I’ve posted before and copied verbatim from Futility Closet:

Given these premises, what can you infer?
  1. If there is a king in the hand then there is an ace, or if there isn’t a king in the hand then there is an ace, but not both.
  2. There is a king in the hand.
What is your answer???
Now, go read the Futility Closet post:
As you will see, the post claims that “almost no one sees” the correct answer, and "practically everyone" infers (wrongly, with "cognitive illusion"... "but not both," being the key phrase) instead that “there is an ace in the hand.I think the correct answer (that you can’t infer anything) may be more obvious to word mavens or to mathematicians (given their familiarity with logic and conditionals), but I don’t know for sure. Again it’s an example of the ambiguity of verbal cues misleading people... language is rarely as precisely interpretable as individuals assume it to be. Indeed, Gödel used to say that it was almost inexplicable that humans could converse effectively with one another at all given how ambiguous most words are.

Reminds me a bit of a lovely old classic math conundrum that throws most people off (if not familiar with it), which in one version (from Wikipedia) runs like this:
"Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 as a tip for himself. Each guest got $1 back, so now each guest only paid $9, bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?"

Sunday, March 17, 2019

Followup Videos...

Last Monday I posted several videos, and now a couple of followups to those, worth mentioning. Jim Simons’ second delightful 90-min. discussion at MIT, this time on finance (the first specifically on mathematics), is now up:

I assume the third meeting with the remarkable Simons, on his philanthropic activity, will be up within a few days [it is now up HERE ]. There are also many more (usually shorter) interviews with Simons on YouTube:

For any who don’t already know, Jim is THE Simons (with his wife) behind the Simons Foundation which brings us the excellent Quanta Magazine among other public offerings. Learn more about it here.

…In another followup, this time to Andrew Booker’s finding of a solution to the sum of 3 cubes equalling 33, Numberphile posted this timely piece:

...and finally, Michael Harris has provided links to what look to be wonderful videos from last year's Celebration of Barry Mazur Conference:

While filling out your brackets try to find time to watch some of these!

Friday, March 15, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Thursday, March 14, 2019

Let The Madness Begin...

In honor of Pi Day the formula for the volume of a sphere is:   


OK, now that that's out of the way, what's more timely than a post about pi this week, but of course a post bout ‘March Madness’! — ‘cuz there are a lot more basketball enthusiasts around than math enthusiasts!… And shortly, illegal gamblers ;)) all over the country will, with wild abandon, be filling out their bets/brackets for the upcoming NCAA tournament.
Mathematically, there are 263 ways to complete a 64-team bracket form, but of course realistically that means little -- there are far fewer ‘realistic’ ways that one with knowledge of the sport would ever fill out their forms (but still a very large number indeed, i.e. billions).

Basketball fan, occasional investor Warren Buffett publicly offered a billion-dollar prize (yup, you read that right) through his Berkshire Hathaway group for a perfect bracket back in 2014. He’s altered the offer since then (now guaranteeing some prize money is given away, but restricting entrants), but the odds remain firmly in his (risk-averse) favor; heck, most bracketologists are out of the running for a perfect sheet after the first round (as some note, the bigger fear/danger is not of someone ever scoring a perfect bracket finish, but someone motivated by huge pay-offs hacking the whole, well-secured system).
Anyway, news of Buffett's 2014 offer:
…and his 2019 contest: 

I believe ESPN runs the largest public-participation bracketology competition:

Every year I try a slightly different method for making my own picks. This year I’m choosing an insightful combination of astrology, homeopathy, and phrenology. Wish me luck.

Finally, here's Davidson's Tim Chartier, from last year, explaining some of the conclusions from his long-time mathematical study of the tournament:

Wednesday, March 13, 2019

The Shrinking Dollar...

Have been re-running some previous puzzles/paradoxes since they seem to be popular and several current followers of the blog weren’t following back when they were initially posted, so another one today.  When posted a few years back I essentially headed it:
Gladly paying $1:10 for $1:00 bill… or why rational choices ain't always so rational

I saw it first at Greg Ross's "Futility Closet,”  and it's known as the "dollar auction" paradox (created by economist Martin Schubik). This setup I've adapted from Wikipedia:

An auctioneer is to auction off a single dollar bill with the following rule: the bill goes to the highest bidder, AND the second-highest bidder LOSES the amount that they bid (to the auctioneer). The winner could gain a dollar for say 20 cents, for example, but only if no one else bids higher. The second-highest bidder is the biggest loser since they pay out their bid and get nothing in return.
The opening, minimum bid is 5 cents (with 5-cent increments thereafter) from one player, who would make a 95-cent profit if no one else bid. But it's sensible for another player to bid, say 10 cents, and still make a 90-cent profit. Then similarly, another bidder may now bid 15 cents, making 85-cents profit.
Whoever is the second-highest bidder at any point in time will wish to convert his potential loss to a gain by bidding higher than the highest-bidder, and so on. Obviously, if this keeps up, at some point, the dollar will COST someone a dollar to purchase -- but at least they will suffer no loss, while the 2nd highest bidder will lose 95 cents, giving them an incentive to bid $1.05 and thus decrease their loss to a nickel... at which point, the other bidder loses a whole dollar... and on and on. Bids beyond $1.00 mean that both top bidders lose money, thus minimizing the amount of loss then becomes the focus. A series of rational bids will reach and ultimately surpass the one dollar point, as the bidders seek to minimize their losses. Thus, "rational" bidding leads inevitably to both the two highest bidders losing money (while the auctioneer makes out well).

No wonder some call economics "the dismal science." ;-)

Monday, March 11, 2019


So many great math-related videos coming out all the time. Passing a few along in the event you’ve missed them and have time to catch up:
From Numberphile this is actually an older video on the 'sum of three cubes' equation n = x3 + y3 + z3
...The video is suddenly very pertinent because a solution for the integer 33 has now been discovered, and here it is in all its glory!:
  33 = 88661289752875283 + (-8778405442862239)3  + (-2736111468807040)3 

From Matt Parker another pedagogical video with a mind-blowing ending:

Eugenia Cheng here in a TED talk applying pure, abstract math to our view of society:

If you can possibly find time for it (~90 mins.), phenomenal investor Jim Simons recounts his mesmerizing life in mathematics (and this is just the first of 3 discussions at MIT with Dr. Simons; I assume the others will also be uploaded):

...Lastly, I’ll just re-mention this fun note I tweeted out a couple days back: About a year ago I discovered a retired local math professor who gives free math talks that I attend, had previously worked with Tom Lehrer — I thought that was pretty cooool! Now just this weekend I learned he also previously worked with both Grothendieck and Ted Kaczynski. Wow, THAT'S quite an array of math folks! (and sets me thinking about my own  'six degrees of separation'). ;)

Sunday, March 10, 2019

Of Human Ability...

Often when there is discussion of so-called “geniuses” or “savants” there is also debate over whether their heightened skills are innately-derived, or more the result of concentrated practice and intense focus.

I'll sidestep that fray momentarily, but it does cause my mind to drift elsewhere:
One of the most incredible human feats I’ve ever heard about is human echolocation (using mouth clicks and subsequent echoes to recognize, or “see” your environment, when blind), made famous with Daniel Kish’s TED talk here:

When I first heard of it I thought it might be some sort of hoax, so implausible did it seem… the sightless maneuvering around the world by means of echolocating the size, shape, and position of objects… but, it is not.

Another pioneer of this phenomenal skill was Ben Underwood a young California blind man who sadly died at age 16 from the same cancer that took his eyesight. A longer, wonderful documentary on him here:

Visioneers” is the non-profit organization that seeks to teach blind people the echolocation techniques that Kish and others promote.

I have no idea how many people are capable of learning the technique (or maybe everyone is if they start young enough)?
Perhaps, not surprisingly, echolocation has probably been studied most in bats, though certainly sonar (principally underwater echolocation) has been widely studied as well.
There’s no doubt some interesting math going on in human echolocation, though I’m not bringing it up for that aspect, but solely for the inspiration I find in this odd attunement of a modality rarely activated in people! If you do want to scan some more technical and mathematical research on it check out this interesting 2017 piece:
…and here’s a simpler layman version of the above study:

At any rate, people popularly say that we only use 10% of our brain (or some such) and this sort of capability almost make it seem true!

Before his untimely death in a car accident, I encountered Marty Ravellette multiple times. Marty was born without arms but learned early on to use his legs as most of us employ our arms (others have done the same). It was remarkable to view him living a relatively normal life, sipping coffee and smoking a cigarette (with his feet) in a restaurant, driving a vehicle, and even running his own one-man business, believe-it-or-not called “Hands-On Landscaping” (yeah, he had a sense of humor too). He gained national news attention once when he saved a lady from a burning car on a highway, and won various awards throughout his life. If you didn’t witness it with your own eyes it would be hard to imagine how well he managed in life without arms.
Just another inspirational figure; making do, quite successfully, with what he had. What abilities do the rest of us have that we never develop because we're never pushed to?

Even the adroit and rapid manipulation of abstract symbols, both in mathematics and language, is an incredible talent, we take for granted, that, in a Wignerian sense, is largely inexplicable -- linguists and psycholinguists have tried for decades to account for the phenomena (of language learning and speech processing-and-production), with little real progress methinks! At a very young age we learn to both voice and decipher high-speed phonemic sounds (such that, to this day, I really have no good idea how I write/compose these blog posts, nor read those of others!). The human animal is marvelous... so too others:

There's much that humans seem to commonly share in the way of capabilities... but surely not everything. I often encounter individuals questioning the notion of 'innate' talents — but I honestly don’t understand how anyone, given our knowledge of genetics and twin studies and the like, keep putting forth blank slate arguments and doubting innate differences in skill levels. We all enter this world with a range of talents, but those ranges clearly differ from individual to individual (despite overlap), just as we all have a range of height we will reach — one person may grow to somewhere between say 5’2” and 5’10” depending on their environment, and another somewhere between perhaps 5’4” and 6’2”… and no one, no matter how rich the environment, will reach 18’7” (genes won’t permit it). Individual skills operate within ranges as well.
People espousing the more open-ended view fear that young people will be harmed if it’s implied that they lack certain innate skills or aptitudes… I’m GLAD to attribute my weaknesses to a lack of innate skills; the alternative is to believe I just didn’t have the discipline, the work ethic, the perseverance to master certain fields. Essentially, telling kids who don’t do well in math that they did have the necessary talent all along, but didn't cultivate it, is to tell them they are lazy… and that to me is more harmful. We can't all be good writers, spellers, mathematicians, musicians, or even finger-painters (and, moreover, we ought be exalting in our skill differences, instead of stressing our sameness -- if anything, I'm more interested in our 'sameness' to other primates, mammals, and vertebrates, than our sameness to other people).

Dr. Jo Boaler takes a somewhat different view of matters in this article (and her general work at YouCubed) on advances in neuroscience:

I agree with the basic view that specific neuro skills are not "fixed" from birth, but still believe  that skill-ranges (as well as many other aspects of psychology and personality) are fixed and vary widely from person-to-person, which is a subtly different point. Nonetheless, her multifold approach to math-teaching may well produce the most good for the most (not all) students until the day comes when truly individualized teaching is possible.

Anyway, I'm fascinated by the minds of towering mathematicians…
Thomas Lin’s (editor) “The Prime Number Conspiracy” is a darn near riveting compendium of essays from Quanta Magazine.  Section 4, “How Do the Best Mathematical Minds Work,” is one of my favorites, wherein each of 8 essays profiles a different modern premier mathematician. These are people whose brains are definitely wired differently from mine, or most people’s. Anyone who tries to tell me that, no, these individuals simply concentrated more, practiced more, focused more, in certain areas than did I and others, I hardly have patience to respond to. All human brains are different going right back to the womb, as are their fingerprints, but the kinds of minds outlined in section 7 of Lin's volume see the world, and the patterns of the world, and the patterns of the patterns of the world, and... so on, differently than do I. You may as well tell me that I and Pablo Picasso and Evel Knievel and Ted Bundy were the same at birth, as tell me that I share much with these penetrating mathematical minds. And I don't care if we call them "geniuses," or "savants," or "prodigies," or "gifted," or whatever. These are just short-hand labels we use for convenience; terms that need not be over-stressed or dwelled upon, but nor should they be abandoned for fear of harming others who are set apart from them.
Meanwhile, excuse me now while I attempt to go and echolocate my car keys.

Friday, March 8, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

(...if nothing delays it, I'll be back here Sunday with a bit of a ramble)

Wednesday, March 6, 2019

Midweek Probability

Another re-run today of a posting I did previously....
Marilyn vos Savant famously introduced the "Monty Hall problem" to the public and gave the correct answer even when many professional mathematicians initially labelled her "wrong." 

Here’s another fun question she later posed for her readers: 

Say you plan to roll a die 20 times. Which result is more likely: 
(a) 11111111111111111111 or 
(b) 66234441536125563152 

Marilyn answered that both sequences were equally likely as outcomes from such a procedure… and there is little controversy over that answer… from a frequentist view of rolling a fair die, all sequence-outcomes being equally likely (that likelihood being very small, BTW). BUT, then Marilyn went on to note, “But let’s say you rolled a die out of my view and then said the results were one of those series. Which is more likely? It’s (b) because the roll has already occurred. It was far more likely to have been that mix than a series of ones.

At least one mathematician again took her to task, claiming the answer is "wrong" and the probabilities are still equal… that "rolling the die out of view" has no consequence. But clearly there is a difference between anticipating in advance a resultant sequence out of ALL the possible sequences that a procedure might produce, versus addressing just two given sequences AFTER a procedure has already taken place. Vos Savant has essentially altered the original question (in order to make an interesting/worthwhile point).

[One way to think about it is simply to make the sequence more ridiculously long: suppose I roll a FAIR die a million times; I record the results and tell you that the outcome was either a million ones, OR, some more-random-looking list of figures… prior to rolling the die both sequences would be equally likely, but with the task already completed, and ONE of the TWO given choices GUARANTEED to be the actual sequence, the second one is clearly more probable.]
What might be a more interesting question to explore is at what point along "randomization" would two given sequences approach equal probability? i.e., suppose I throw the dice a million times and show a sequence of 225,000 ones, followed by 225,000 twos, followed by 225,000 threes, followed by 225,000 fours, versus a more genuinely-random-looking sequence -- still the second would be more probable... but I could keep altering the first sequence slowly step-by-step and at some point its probability would 'tip-over' to being the same as the second sequence. But when does it happen?

Sunday, March 3, 2019

Rebels With a Cause

I’m long fascinated by the varying lives and minds of mathematicians. Thusly, one of my favorite sections of Thomas Lin’s (editor) fabulous volume, “The Prime Number Conspiracy,” is Part 4 “How Do the Best Mathematical Minds Work?,” offering fascinating profiles of several individual prominent mathematicians. One of them, Freeman Dyson held a well-known disdain for PhD. degrees, and I’ll just pass along these lines (that just maybe will inspire some):

…I’m very proud of not having a Ph.D. I think the Ph.D. system is an abomination… It’s good for a very small number of people who are going to spend their lives being professors. But it has become now a kind of union card that you have to have in order to have a job, whether it’s being a professor or other things, and it’s quite inappropriate for that… The Ph.D takes far too long and discourages women from becoming scientists, which I consider a great tragedy. So I have opposed it all my life without any success at all…

…So I’m very proud that I don’t have a Ph.D. and I raised six children and none of them has a Ph.D., so that’s my contribution.

[Dyson is still going strong at 95, contemplating unsolved math problems, and 5 of his 6 children, by the way, are women.]

...and then, on a separate note, a bit ago I came across this wonderful Paul Lockhart quote (from "A Mathematician's Lament") in a piece by Sunil Singh:

Friday, March 1, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]