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Thursday, February 28, 2013

A Puzzle Re-run

One of the advantages of math blogging is that so much of mathematics never gets old! If you're doing political blogging than what you wrote about a month ago is likely already aged, boring news. But in math so much material is timeless (people still do posts on the Pythagorean theorem).
So being in a puzzle mood today, I've reached back to a post from over a year ago for another of my favorites (and of course the puzzle is older than that). I'll quote largely from that post with a few changes (and apologies to all who are familiar and bored with this classic, but for newbies who've never seen it....):


Verbatim from Richard Wiseman's blog:
"Imagine there is a country with a lot of people. These people do not die, the people consists of monogamous families only, and there is no limit to the maximum amount of children each family can have. With every birth there is a 50% chance its a boy and a 50% chance it is a girl.  Every family wants to have one son: they get children until they give birth to a son, then they stop having children. This means that every family eventually has one father, one mother, one son and a variable number of daughters.  What percent of the children in that country are male?"
Wiseman's Friday puzzles are frequently devious… but, often once the answer is given and explained, one feels impelled to slap one's forehead and exclaim "DOH!, well, of course!" So perhaps his best offerings are those that, even once explained, are still not totally clear, and generate much ongoing discussion...


This is one such effort, which often generates dissenters when presented, once again demonstrating how tricky/misleading, probabilities can be. I confess to originally requiring extra time to convince myself that 50% was the correct answer. It's one of those quirky puzzles that is patently obvious to many, yet thorny for others (one of the keys, I think, is to remain tightly focused on strict statistical probability, and not let your brain get distracted by what could theoretically happen). Read all about it for yourself at Richard's original post:


(...and peruse as many of the 270+ comments as you care to!)

One of the simplest explanations from his comments (for anyone having trouble seeing it) is just to imagine the statistics for a sample that begins with 128 families (assuming strict 50% chance of a boy or girl at each point):

128 starting families produce 64 boys and 64 girls
next round, the 64 families with girls now produce 32 boys and 32 girls
next round, the 32 families with girls produce 16 boys and 16 girls
16 families with girls produce 8 boys and 8 girls
8 families with girls produce 4 boys and 4 girls
4 families with girls produce 2 boys and 2 girls
2 families with girls produce 1 boy and 1 girl

Total at conclusion: 50% boys, 50% girls


Tuesday, February 26, 2013

Prime Savants… Fascinating

If you are an Oliver Sacks fan you may well be familiar with the story of twin male autistic savants from his best-selling "The Man Who Mistook His Wife for a Hat" volume. Anyone captivated by prime numbers and savantism can't help but love the account, and George Johnson explores it in this current blog post for Discover Magazine:


A fascinating read. There are further excerpts from the Sacks' chapter here:


The story tells of twin males who demonstrate great pleasure and joy in recognizing prime numbers… LARGE prime numbers, of the sort that normally only a computer might verify as prime in a short order of time, but that the twins appeared able to discern upon brief reflection. It is as if they had personal access to the so-called "Platonic" realm of numbers… the same might be said for other savant-like math knowledge or computational skills that are well-documented, but this is an even rarer facility.

There are dissenters who question Sacks' credibility or interpretation of this particular story which he only related about 20 years after its actual occurrence:


The above rebuttal also includes a YouTube video of the actual twins demonstrating their calendar-calculation skills (not their prime number skills).

Sacks' account is anecdotal and so the skepticism is understandable (by the time Sacks drew attention to the twins they had been separated and lost their special numerical skills)… but still... I admit… I want to believe it true ;-)

Here's a further followup to Sacks' story by another researcher:


And this research abstract indicates that the prime number recognition skill has indeed been reported for other savants:


Mind-blowing stuff!

Monday, February 25, 2013

EdX, MOOCs, the Road Ahead

I suspect in response to my latest post over at MathTango, and/or recent one here, Sol Lederman emailed me a link to another blog that has a recent interview with Anant Agarwal, President of edX, another major MOOC endeavor that competes with Coursera and other online education efforts (edX is a non-profit backed by Harvard and MIT, two educational interests you may have heard of).  It is a great, and positive, discussion of the future of the MOOC model (over 7 pages though, so it requires some reading commitment, but worth it); it also delineates some of the differences between Coursera and edX:


Too much to try and summarize, but I think this quote from the interviewer, Sramana Mitra, actually captures much of what Agarwal is suggesting:
"This is the paradigm where education is going. I truly believe online education is the driver. The old model of education was the ‘sage on stage’ model, where a professor stood in the front of a class to guide learning. That is fine if you have excellent professors, but there are really smart professors who do not have the charisma to present a topic in a way that students can understand. When quality content is delivered from an online source, there is a consistency in the content. The course itself can be benchmarked and the results tracked, allowing the course content to mature to ensure students are learning the material they are supposed to learn. The paradigm shift is toward the blended model. Even if you have a mediocre or substandard professor, the course material itself is excellent. The professors can still serve a  worthy function as they are there managing the classroom and guiding discussions. They provide structure and administration. The students learn from world-class content. That is a highly scalable education model."
I've already expressed my belief that MOOCs ARE a wave of the future (in some form)... but IF, with the rapidly-growing time, energy, money, and talent going into them, they indeed fail, it won't be for lack of creative effort... and it will say a great deal about the nature of human learning.

Saturday, February 23, 2013

Weekend Potpourri

So many mathy reads to choose from this weekend, if you've missed them (really enough for a couple of weekends!)....

1) Brand new from Sol Lederman, a wonderful podcast with mathematician/writer Erica Klarreich:


I've previously mentioned my belief that interviewing lesser-known math figures is almost more enticing than interviewing the titans in the field because readers/listeners already know so much about the 'big names' out there that much of what they say may seem repetitious (even if important) of things they've voiced before. Lesser-known folks have a fresher appeal as there is so much new to learn about them, and I think this podcast demonstrates my point. I was fascinated hearing Erica's views and experiences on a wide range of topics. See if you don't agree.
And below, some of Erica's prior writings for the Simons Foundation:


2) MIT physicist Max Tegmark is famous for his belief that the Universe is "built" of mathematics. He expresses his viewpoint in this straightforward interview from the ScienceNow site… and attracts a lot of comments in the process (including cynical ones):


Julie Rehmeyer, who I just posted about a few days ago, has a new piece on Fermat's Last Theorem and its axiomatic basis, in ScienceNews here:


4) Patrick Honner again laments a question from a NY State Regents Math exam that entails unstated assumptions and in so doing short-circuits deep mathematical thinking:


5) Some bloke named Keith Devlin has a fantastic longread on math games in American Scientist, leading up to release of his own company's new animated game for math learning. He sets forth the criteria or qualities a video needs to possess to be successful as a math instruction tool (and explains why MOST games FAIL):


From the description, it sounds to me as if the prospective player (young person) of these new games will learn math in a manner reminiscent of the original Karate Kid (the movie) learning karate without ever knowing it from his master Mr. Miyagi. Keith's discussion of the "symbol barrier" and symbol manipulation in math is especially enlightening, but the entire piece is GREAT. He employs a music (piano) metaphor to explain what a successful math game should be like.

6) The brilliant Barry Mazur, recent recipient of the National Medal of Science, gives us a rich (and philosophical) piece called "Shadows of Evidence" on what constitutes "evidence" in the realm of mathematics. The essay ends with a quote from Chris Anderson essentially arguing that "modeling," as traditionally used in science, is becoming obsolete (because ALL models are, technically, flawed), and that with the advent of computer number-crunching ability, only "correlation" derived from huge data sets will be needed. To which Mazur responds that, "correlation alone will never replace the explanatory power of mathematics.":


This essay in turn leads to an even longer Mazur read entitled "What Is Plausible" here (pdf):


7) Finally, perhaps appropriately after all of the above, I'll end with 50 varied quotations just put up by Guillermo Bautista on what mathematics is:


a few of my favorites:
"Mathematics is no more computation than typing is literature." – John Allen Paulos

"Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guide to this growth is not calculation and formulas, but an open ended search for pattern."  -- Lynn A. Steen

"Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them." — Joseph Fourier 

.... ADDENDUMExperimenting with shorter time segments, Sol Lederman has already put up another podcast interview (22 mins.), this time with Jason Ermer, creator of the "Collaborative Mathematics" project (who I referenced a bit ago):


Mathematicians have been at the forefront of bringing productive Web collaboration to academic subjects, and now Jason is attempting it at lower levels. Check it out!

Wednesday, February 20, 2013

The Future of MOOClear Power?

I'm fascinated watching Keith Devlin twist and squirm a bit as he works his way through MOOCs, wavering slightly back-and-forth between optimism and uncertainty (or at least caution) about their future. His latest upbeat post, as he embarks on his second go-at-it (beginning March 4), is here:


Devlin notes that MOOCs are still very young, barely out of the starting gate (indeed he compares the effort to "running a marathon"), and that 'missteps' are to be expected. I totally agree, and find it remarkable that some folks already pan them as a failed experiment, or, as Keith writes, "Anyone who views such outcomes as failures has clearly never tried to do anything new and challenging, where you have to make up some of the rules as you go on."
My own ultimate optimism for MOOCs (long-term) stems from one simple reason… supply-and-demand (...it's worked well for capitalism free enterprise in the past). There are a limited number of truly excellent programs and instructors in any given academic field, but there are often 100s of thousands of prospective students around a shrinking globe who would love access to them. The demand for such access, once it is viewed as possible, will drive those in charge to find an effective way of making it work -- we may not know that precise "way" yet, but 'necessity (or demand) is the mother of invention'… in short, as others have noted, MOOCs are the avenue to a scalability that is desirable but was previously unfeasible. Tangentially, after medicine, higher education is probably the next most inflationary cost in the U.S… a cost that must be reigned in if education is to remain broad-based and not the luxury of an elite. MOOCs are a clear egalitarian antidote for that cost-control.
Anyway, here's wishing Keith great success with his second MOOC endeavor... and his 3rd, and 4th, and 5th and....

(Again, if any reader here is taking Keith's course, I'd love to hear reports back about how it goes).

Tuesday, February 19, 2013

Rehmeyer On Tuesday

Sol Lederman faces the same difficulty with his podcasts that I've encountered with my transcribed interviews… finding more female mathematicians to participate. But he did snag the exuberant Julie Rehmeyer, mathematician/writer/columnist, for his latest effort. Listen here:


You can peruse Julie's old columns (called "Math Trek") for "Science News" here:


And if you're in the mood to bang your brain around a good bit I recommend you start with her 2010 piece on the so-called "boy born on Tuesday" problem. This is one of the successors to the Monty Hall puzzle as one of the most highly-debated probability conundrums in recent times (…and I'd call it considerably more complicated than Monty Hall). Her take below (be sure to read the 70+ comments that follow... and perhaps have some Advil handy ;-)):


(I'd just note that much of the debate over the 'correct' answer for this puzzle, and many others, stems from the different unspoken assumptions made with any given wording of the problem. Words and language almost always contain ambiguity, sometimes very subtly so, and ambiguity is the enemy of mathematics/probability... or consensus!).

Sunday, February 17, 2013

The Shape of Things To Come

"There are changes lyin' ahead in every road 
And there are new thoughts ready and waiting to explode 
When tomorrow is today the bells may toll for some 
But nothing can change the shape of things to come."

Pardon my stubborn connection to the 60's....

Am always a tad nervous about heading down the rabbit-hole of math education… the topic is so huge (but also so vital), with a multitude of views, from so many intelligent, passionate people… but seems impossible not to venture there on occasion:
I'm familiar with Conrad Wolfram's views for reforming math education from some of his prior interviews, but somehow I missed this 2+ year-old TEDTalk (17 mins.) that summarizes his thoughts nicely (emphasizing programming/coding over calculating/computation, computer work over manual, pad-and-pencil work):

Speaking of math education, any day now Keith Devlin should be announcing his own personal company's animated approach to young math education (from innertubegames.net), so, anxious to see what that looks like.

And speaking of animation, Sol Lederman's newest podcast was with the producers of the popular computer-generated film "Dimensions" who are now out with "Chaos."
Unfortunately, the podcast audio was unsatisfactory so Sol had to transcribe the interview.  His post below links to the pdf transcription of that interview AND to a link for viewing "Chaos" (in 9 parts):


And finally, on a somewhat different note about the future of mathematics, this potent piece from John Baez on what is being deemed "green mathematics" and its role in our lives to come as we approach significant planetary problems ahead:


Baez writes that "mathematics will become increasingly driven by our need to understand the biosphere and our role within it" and notes that interdisciplinary "network theory" will be a major tool for solutions going forward.

Saturday, February 16, 2013

Bogomolny's Insights...

One of the most expansive, fun, brimming math websites on the internet is Russian-born Alexander Bogomolny's "CTK Insights" or related "Cut the Knot" site. A recent interview with Alexander here:


He explains where the name of the site comes from, and has interesting things to say about math education in the U.S. as well (some of which reflects thoughts I put forth in my latest, longish MathTango post).

On a side-note, I just recently discovered that Bogomolny also has a nice page (with lots of links) devoted to one of my favorite topics, "self-reference" -- I discovered this while I was researching a classic problematic (self-contradictory) sentence: "Their are three misteaks in this sentence" (...think about it):


Friday, February 15, 2013

Appreciating Feynman...

Today marks the 25th anniversary of the death of Richard Feynman. Can't let that pass without celebrating him a bit, with this (non-physics) clip of he and his buddy Richard Leighton playing to an appreciative crowd:

Thursday, February 14, 2013

A Valentine's Tradition...?

(via Dave1185 at en.wikipedia)

It's Valentine's Day again and hey, I guess everybody loves a little warm-fuzzy…
Rather unexpectedly, the most highly-trafficked post I've ever done here was a Valentine's Day entry posted one year ago, simply linking to an older piece by English-major-turned-science-writer Jennifer Ouellette recounting her betrothal to Caltech physicist Sean Carroll.
So, I won't argue with success, and will link to it once again this Feb. 14. With any luck, and cooperation from Sean and Jennifer, maybe this will become a Math-Frolic tradition :-):


To get you in the mood, once again, an excerpt:
"It turns out that the world is filled with hidden connections, recurring patterns, and intricate details that can only be seen through math-colored glasses. Those abstract symbols hold meaning.  How could I ever have thought it was irrelevant?
"This is what I have learned from loving a physicist. Real math isn’t some cold, dead set of rules to be memorized and blindly followed. The act of devising a calculus problem from your observations of the world around you – and then solving it – is as much a creative endeavor as writing a novel or composing a symphony....
"As with mathematics, so with love. There are no hard and fast rules to be blindly followed, no matter what the self-help gurus may tell you. Sometimes you just need to take a Fourier transform of yourself, shatter the walls and break everything down into the component parts. Once you’ve analyzed the full spectrum, you can rebuild, this time with just the right mix of ingredients that will enable you finally to combine your waveform with that of another person."
Now doesn't THAT warm the cockles of your heart! So pass the Belgian chocolates...
But in case warm-fuzzy just isn't your style, I'll also re-run this short (15-min.) independent film, "The Calculus of Love," I've posted on the blog before:

THE CALCULUS OF LOVE from Dan Clifton on Vimeo.

Wednesday, February 13, 2013

More Stats… & a Question

Two more recent views of the rise of statistics:

1) the positive from Marie Davidian here, ending with, "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write" (supposedly a quote from HG Wells):


and 2) a somewhat cautionary take (regarding "big data" and how it leads to increased spurious claims) from the usually irascible, often provocative, always wary, writer Nassim Taleb here:


And on a complete side note a simple question I'm curious about:

The prior post (Monday) referred to the "pigeon-hole principle" and caused me to wonder just how mathematical "principles" and mathematical "axioms" or "postulates" are distinguished by formal definition in math? … both represent (it seems) strong underlying intuitive assumptions. From googling around a bit, it appears to me that axioms/postulates may be a sub-category of "principles," which are specifically implemented to derive theorems through logical steps (while "principles" are a broader category of assumptions that also lead to other results, but not restricted to theorems). Am I on the right track, or if I'm not can someone set me straight with the correct, precise distinction?

Monday, February 11, 2013

For The Genealogy Crowd…

One of my sisters works on our family genealogy, and since there are sometimes interesting results, I've always noted that if you just go back far enough we are ALL related to one another.

John Brockman's latest Edge book is out, "This Explains Everything," with responses from 150 science-y sorts. I've been perusing the more mathematically-related replies and thus far my favorite is from Jon Kleinberg who uses the "pigeon-hole principle" to demonstrate that somewhere well within the previous 4000 years anybody's mother AND father will have a common ancestor:


When Sol Lederman interviewed me for his podcast series, at one point I remarked, tongue-in-cheek, that perhaps somewhere in my distant past I shared some genes with Bernhard Riemann, the great German mathematician who's namesake I use here, and who lived in the same area of the world as my father's ancestors. Nice to apply some mathematics to that thought.

You can view all the responses to the 2012 Edge annual question here:


Sunday, February 10, 2013

Mathematics Via Collaboration

I love that there are so many ways educators are employing the Web to try and improve young peoples' engagement with math! The "Collaborative Mathematics Project" from Jason Ermer is an interesting attempt to involve young math enthusiasts (or enthusiasts-to-be) through the internet's social connectivity. Read about it here:


The object is, as he writes, "to use video as a means of connecting a worldwide community of mathematical problem solvers." And, as he states elsewhere on his site:
"...one of the goals of Collaborative Mathematics is to help cultivate a productive attitude toward challenging problems: one of creativity, resourcefulness, self-confidence, and perseverance.
I believe helping students to develop this productive attitude may be the most important educational objective in the classroom."

I think there may be some glitches with the effort (but I've never been good at predicting the success of such endeavors), but they can likely be overcome along the way, and it will be interesting to see how the project evolves.

Here's Jason giving out the introductory challenge:

Saturday, February 9, 2013

Get To Know Dave Richeson

Sol Lederman has redeemed himself from his previous WildAboutMath podcast ;-)) and now has an hour up with math professor Dave Richeson, proprietor of "Division By Zero" blog and author of the popular volume "Euler's Gem." Sol and Dave talk about many interests (topology, Euler, inquiry-based learning, geometry, Gauss, Dave's new book, and more):


And for an added Saturday bonus, here's a nice little recent geometry puzzle from Mind Your Decisions blog, with a surprisingly simple solution:


Friday, February 8, 2013

Run, Sweat, Bike, Pump Iron…

With a weekend upon us, probably a good time to visit this week-old post from Colm Mulcahy on the link between outdoor exercise and mathematical creativity, specifically a research finding that "endurance sports help with creative/mathematical thinking":


I suspect most of us can attest anecdotally to experiencing increased positive psychological well-being and sharpness from exercise. Mulcahy uses mathematician Ken Ono, who has done significant work with Ramanujan conjectures, and once again, Keith Devlin (he's everywhere, he's everywhere) as examples of  active mathematicians whose craft has benefited from physical exertion.

This weekend... Get movin'!

Thursday, February 7, 2013

Here a Prime, There a Prime

Yesterday I noted the somewhat amazing fact that the latest-discovered (Mersenne) prime number had almost 4.5 million MORE digits than the previously largest known prime! Part of this simply has to do with the fact that Mersenne primes are only one sub-category of ALL primes, and probably also relates back to how the GIMPS project (searching only for Mersennes) operates. But still quite stunning.

I was thinking of writing a short post about Mersenne versus other primes, but heck you can get that info from the Web... more interestingly, Patrick Honner took up the cause to actually get a handle on how many primes may have gone as yet undetected between the newly, and last, found ones.

Mr. Honner did the legwork, by starting with Bertrand's Postulate, and in the end calculating that there must be "at least 14,772,551 primes between the largest and second-largest known primes!" ( 2^{57885161} - 1 and 2^{43112609} -1)
His first commenter takes the argument/logic even further and concludes that there should be around 1.45 * 10^17,425,162 prime numbers between the two known primes, which as he notes "is staggeringly larger" than even Patrick's number [ADDED: actually, this no., as I read it, doesn't make sense to me, but maybe I'm misunderstanding or further clarification will come.][Okay, I think I've worked the numbers roughly enough to better see how the final figure comes about -- what is hard to take in is how "vanishingly small in comparison," as the commenter says, the number of primes below  2^43112609 is…  "staggering" barely does justice to this final number! -- or as Cantor might say, 'I see it, but I can hardly believe it!!']
 Anyway… read all about it:


As an aside, I noticed that on the Wikipedia page for Mersenne primes it mentions that in 2003 the GIMPS project produced a "false positive" for the 40th Mersenne prime, which was shown to be invalid upon failing verification. If someone can explain how such "false positives" can come about (is it due to computer chip glitches or some chance algorithmic flaw or what?) I'd be curious to hear in the comments (...just verifying these GIMPS numbers must be a bit of a story unto itself as well!)...

Wednesday, February 6, 2013

Prime Time With Curtis Cooper

The big news in pure mathematics yesterday was the discovery (actually on January 25, by Dr. Curtis Cooper of Missouri) of the largest prime number yet, with well over 17 million digits... probably too long to use for your Facebook password (amazingly, this number has almost 4.5 million more digits than the last previous largest prime and is the third time Dr. Cooper has been a finder!):


It's always newsworthy when these largest primes are found, even if it doesn't affect our day-to-day lives anytime soon (…if the price of that next cup of Starbucks coffee goes up, well, it's strictly a coincidence… I think). Primes are the "stuff" (atoms as it were) of mathematics, and go on forever.

This latest prime was found by way of the GIMPS (Great Internet Mersenne Prime Search) project, a collaboration of volunteers all across the internet (yes, YOU too can take part) -- a GREAT example of the kind of group-efforts the internet has made possible, and mathematicians have been at the forefront of. If you're not already familiar with it, read about the project at Wikipedia here:


...or at the home site for the project here:


As long as I'm talking prime numbers, might as well go ahead and re-plug a book I love, "The Mystery of the Prime Numbers" by Matthew Watkins. Because it's from Britain and self-published it is not well distributed, but is available through Amazon here:


I believe anyone from bright teenagers to dull mathematicians... er, no, I mean adult mathematicians ;-)... can enjoy and learn some things from this intriguing volume on a subject of eternal interest (and it's actually the first volume in a "trilogy" by the author).

Tuesday, February 5, 2013

Beautiful Math!

Interestingly, this "business" site is listing "the 11 most beautiful math equations" (you have to click through them one-by-one):


EXXXXXCEPT that it DOESN'T include Euler's most famous identity (even though it includes another Euler formula). How is that possible??? Is the below not THE MOST beautiful equation in mathematics by near-unanimous decree!?:

Anyway, if you've practiced enough with Vi Hart, you may be able to follow along with this fellow!:


==> AFTER composing the above post I discovered Keith Devlin, writing on Twitter, saying he had indeed submitted Euler's Identity for the above article but somehow it didn't make the cut... UNbelievable!!

Monday, February 4, 2013

Inspired By Math... Yes, I Is

Ohhhh my, over at WildAboutMath, Sol Lederman has finally scraped the bottom of the (math) barrel:



Meanwhile, at my MathTango site I'm highly recommending Charles Wheelan's new book, "Naked Statistics."

Saturday, February 2, 2013

Calendar Puzzle

Haven't done a puzzle here for awhile, so here's a nifty one -- I took this from THIS SITE (and re-worded it very slightly):
If Friday-the-13th occurs in a given month, then the sum of the calendar dates for the Fridays in that month is 6 + 13 + 20 + 27 = 66. What is the largest possible sum of calendar dates for SEVEN consecutive Fridays in any given year?
.answer below

  142  ....if you didn't get the right answer you'll want to click to the following page that shows the answer:


Friday, February 1, 2013

In Memory....

Stepping away from math for the moment today....

Today marks the 10-year anniversary of the tragic loss of the Space Shuttle Columbia. In commemoration of those we lost that fateful day, and all the marvels that NASA has given us over decades: