Persnickety Puzzling Puzzles….

For your weekend delectation….

A recent Jason Rosenhouse post offers some thought puzzles of a certain style here:

http://scienceblogs.com/evolutionblog/2013/03/20/some-brainteasers/

(here's the style: "If you ask me on Friday what day classes start, and I truthfully reply that they start two days after the day before the day after tomorrow, then on what day do classes start?")
;-) have at it!

And in the oldie-but-goodie brain-stretching department, it probably doesn't get any better than the classic "two envelope problem/paradox" (picking/switching between 2 envelopes of differing dollar amounts; one has n dollars, the other contains 2n dollars -- after you pick one and look inside, is it more advantageous to keep the chosen one or switch to the unchosen?) -- as is often the case, the precise, explicit statement or set-up of the problem is crucial in determining a solution (but even when the set-up is agreed-upon, debate ensues!). Most of you are probably somewhat familiar with the problem, but you can review it at Wikipedia here:

http://en.wikipedia.org/wiki/Two_envelopes_problem

Keith Devlin gives his take here:

http://www.maa.org/devlin/devlin_0708_04.html

I especially like William Briggs' methodical analysis in 2 posts here:

http://wmbriggs.com/blog/?p=2645
http://wmbriggs.com/blog/?p=2649

And another interesting piece:

http://phys.org/news169811689.html

But it goes on and on and on…. a Google search produces copious material on this simple-to-state, insidiously-mind-bending problem:

http://tinyurl.com/c3ojp5v

(...don't blame me if your head explodes)

Friday Fun

Seeing is.... be-fuddling. . . .

Recently highlighted at Grey Matters Blog this version (known as the Winston Freer tile puzzle) of a classic geometry-type puzzle:




On a side-note, KW Regan posted this entry yesterday in honor of the recent Paul Erdös anniversary (with many wonderful links):

https://rjlipton.wordpress.com/2013/03/28/happy-100th-birthday-paul-erdos/

Two-ooo Wonderful!

H/T to Patrick Honner who tweeted this delightful photo of a 72-yr.-old Paul Erdös with a 10-yr.-old Terence Tao (from Terry Tao's Google+ account):

https://plus.google.com/u/0/114134834346472219368/posts/fiZbgKv4Yew

(so great, made my day!... love the two right arms, in synchrony and thought, held to chins)

Devlin on Bonobos etc... ;-)

Yet another great piece from Keith Devlin, this time at Huffington Post, once again taking the educational system to task, while contemplating the value and future of MOOCs:

http://www.huffingtonpost.com/dr-keith-devlin/massive-open-online-courses_b_2946591.html

He compares the stagnant K-12 educational system we have with  "the educational method animal psychologists use to train Bonobo apes."  And continues, "More to the point, it is also the method that was developed (for children) in the early 19th Century, when countries around the world were introducing universal education. Its purpose was to prepare a workforce to fuel the post industrial revolution society. A key requirement was to train millions of people to think inside particular boxes. And that is what it did, very effectively." But, he goes on to say, it is "woefully inappropriate in today's world."

So many trenchant thoughts in the piece, be sure to read the whole article, but here are a few more excerpted lines to whet your appetite:

"Given the stranglehold on U.S. public K-12 education held by various powerful groups with a vested interest in preserving the status quo, buttressed further by others who want to enter the same lucrative market, MOOCs offer a wonderful opportunity to overcome the damage schools do."

"...those of us developing these new courses need to resist the pressures -- from many sides, including many of the students themselves -- to conform to existing educational models."

"I think that with some effort, we can scale enough of it so that MOOCs can make up for much of the damage resulting from putting 21st Century students through a 19th Century school system. And we can do it on a global scale."

Turning Coffee Into Theorems...

Today would be Paul Erdos' 100th birthday!
In his honor, check out this endearing, heartwarming video of one of the most odd, curious, and prolific mathematicians of all time (...it's an hour long, but if you've never seen it, I truly recommend you find time to watch!):

http://www.youtube.com/watch?v=pnCrUroeVlI 
--> Addendum: sorry, taken down apparently due to copyright violation :-(

And here's a wonderful piece just done for the NY Times on Erdös:

http://tinyurl.com/cdv6zzl

 (ohhh, and get your In-honor-of-Erdos coffee mugs here or here.)

Mathematical Mystery...

just a little bit of bizarreness to imbibe in…

Leila Schneps is in the news lately (including the prior post!) as she and her daughter actively promote their new book, "Math On Trial." It turns out that she is also active with 'The Grothendieck Circle,' a small group which focuses on the work of French mathematician Alexandre Grothendieck, as famous now for his reclusiveness and departure from mathematics (another genius mathematician who couldn't cope well with society), as for his advanced contributions to mathematics. And, it turns out that Schneps may have been one of the last mathematicians to have contact with Grothendieck before he went into seclusion.
Grothendieck's whereabouts have been a source of speculation over the years, and the following fascinating, almost cloak-and-dagger post tells a tale while hinting at some possibilities regarding the mystery:

http://matrix.cmi.ua.ac.be/g-spots-saint-girons

And the above post is just the latest in an interesting series by the blog writer on what he christens "G-Spots" (Grothendieck locations)... you can backtrack through them:

http://matrix.cmi.ua.ac.be/tag/grothendieck

(Fun reads... not exactly Agatha Christie, but about as good as it gets in mathematical mystery.)

Grothendieck's 85th birthday is coming this Thursday.

Start-of-the-Week

Sol Lederman's latest podcast is with the interesting mother-daughter team who co-wrote "Math On Trial," which takes a critical look at the (ab)use of math/statistics in courtroom proceedings:

http://wildaboutmath.com/2013/03/24/leila-schneps-coralie-colmez-inspired-by-math-26/

My own latest interview is with writer-extraordinaire Clifford Pickover here:

http://mathtango.blogspot.com/2013/03/clifford-pickover-one-man-carnival.html

(I still have a list of folks to possibly contact for interviews, but again, if any readers have suggestions they'd like to send along via comments or email, let me know.)

And to get your thinking caps revved up on a Monday morning here's a simple puzzle I've taken directly from Futility Closet a bit ago:

The digits 123456789 can be arranged to form 362,880 distinct 9-digit numbers. How many of these are prime?
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.answer below
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ANSWER:  
since the sum of those 9 digits is 45, no matter how they are rearranged, any resulting number will be divisible by 9. Thus, NONE are primes.

Saturday Snippets

A little weekend fare...:

1) Just discovered that David Wells' great "Prime Numbers: The Most Mysterious Figures In Math" book is available as a free download here:

http://www.scribd.com/doc/131913331/Prime-Numbers-the-Most-Mysterious-Figures-in-Math-D-Wells

This is a great resource on primes.

2) Listen to math writer Marcus du Sautoy briefly expound (less than 2 minutes) on why Andrew Wiles' proof of Fermat's Last Theorem is the "most important British innovation of the 20th century" here:

http://www.topbritishinnovations.org/PastInnovations/ProofofFermatsLastTheorem.aspx

3) From the brief to the extended... tangential to math but too good not to pass along… the 2-hour "2013 Isaac Asimov Memorial Debate: The Existence of Nothing;" cosmology/philosophy discussion with 5 participants, Richard Gott, Jim Holt, Lawrence Krauss, Charles Seife and Eva Silverstein, moderated by Neil deGrasse Tyson (a real mix of ideas to chew on):




http://www.amnh.org/explore/amnh.tv/%28category%29/52915

ADDENDUM: I now have an interview up with prolific writer Clifford Pickover over at the MathTango blog:

http://mathtango.blogspot.com/2013/03/clifford-pickover-one-man-carnival.html

Good Wednesday Schtuff...

1) In the news... the winner of this year's Abel Prize (here & here) in mathematics (considered by many, as the math Nobel) is Pierre Deligne,  algebraic geometrician. The news announced at Tim Gowers' blog among many places:

http://gowers.wordpress.com/2013/03/20/pierre-deligne-wins-the-2013-abel-prize/

2) The Simons Foundation has an ongoing list of brief video interviews with various accomplished international mathematicians below (looks quite interesting):

https://simonsfoundation.org/category/features/science-lives/alphabetical-listing/

3) I'm about halfway through Lance Fortnow's new volume "The Golden Ticket," all about the P vs. NP Millennium Problem. It's a fantastic read on a topic that is not easy to write about for a mass audience. Even at just the halfway point I feel very comfortable recommending it, especially if you're interested in the Clay Millennium Problems, specifically P vs. NP, or computer science theory more generally.
"CTK Insights" already has a great review of it up, saying several of the things I had jotted down to mention (I may still write some sort of review later… or may just skip it, since CTK has covered it so well). His review starts off saying, "...when the book by Lance Fortnow arrived the day before yesterday, I opened it for cursory inspection, just to get a first impression. Yesterday I finished reading it"… that gives you some idea how good it is!!:

http://www.mathteacherctk.com/blog/2013/03/the-golden-ticket-p-np-and-the-search-for-the-impossible/

4) Finally, Keith Devlin's latest update on his current Coursera MOOC course ("Introduction to Mathematical Thinking") is here:

http://mooctalk.org/2013/03/20/overcoming-the-legacy-of-prior-education/

Once again, a fabulous, if long, read. The early part of the post is largely a bit re-hash of things Keith has expressed before, but then he gets into a ranty section about the education system that really offers a lot of food-for-thought (and probably debate). ...Recommended reading for all educators especially, but for students, parents, and observers as well.

Four For Tuesday

A few of the things that crossed my screen yesterday…:

1) Another interesting article from the Simons Foundation, this time on the nature and fragility of networks, or as they say, "extreme fragility of interdependency" in the modern world. Be it urban infrastructure, the stock market, even the human body, or other networked systems the risks of small failures leading to chaotic cascading effects seems ever-present, as scientists try to figure out ways to evade catastrophes. From the article:

"While scientists remain cautious about using the results of simplified mathematical models to reengineer real-world systems, some recommendations are beginning to emerge. Based on data-driven refinements, new models suggest interconnected networks should have backups, mechanisms for severing their connections in times of crisis, and stricter regulations to forestall widespread failure."

Read further here: http://tinyurl.com/bom9eym

2) Hat tip to The Aperiodical for bringing this to my attention… a 40-page Shinichi Mochizuki paper intended to assist those trying to understand his "proof" of the ABC conjecture. I certainly don't grasp the paper (although there are a few words I can understand: "the," "of," "with," "for"… ;-) and 99% of folks won't comprehend it, but what's absolutely amazing is that there exists human brain wiring that IS capable of such production/comprehension!:

http://tinyurl.com/d6pje5l

3) And a hat tip to Patrick Honner for pointing me to this NY Times piece about "online proctoring" for MOOC course testing, a topic that is sure to draw more discussion as time goes on:

http://tinyurl.com/cstz6xa

4) Finally, even if you're tired of the question, "Is mathematics discovered or invented?" you may find the below New Scientist article interesting… it moves that question into the realm of business, software, and the debate over patentability, before concluding: "The rallying cry of a good many critics remains 'software is mathematics', meaning that software shouldn't be patentable. The odds are stacked against them, though – there's too much money at stake":

http://www.newscientist.com/article/mg21729086.300-should-business-be-allowed-to-patent-mathematics.html

Conway, Collatz, Chaos

19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Ivars Peterson discusses a recent "provocative" article from the always-interesting John Conway on the provability of various mathematical claims, with a focus on the Collatz conjecture (one of those famous easy-to-state, difficult-perhaps-impossible-to-prove conjectures):

http://mathtourist.blogspot.com/2013/03/wild-beasts-around-corner.html

It starts off thusly:

"Some mathematical problems are easy to describe but turn out to be notoriously difficult to solve. In some instances, these difficulties may stem from fundamental issues of provability, especially for mathematical problems apparently poised between order and chaos."

Even xkcd has taken notice of the Collatz conjecture.

Simpsons' Math

               
 
Yesterday, physicist Paul Halpern tweeted a link to this talk on the inclusion of STEM-related content in the creative and popular animated "Simpsons" TV series. Fermat's Last Theorem, in the form of the above equation ;-) shows up early among a variety of other examples:

...and I mean it

I have absolutely, positively, unequivocally no intention whatsoever (and I mean that emphatically) of acknowledging in any way on this blog... that some folks out there designate today as Pi Day. (...So don't ask!)

The Practical... and the Philosophical

1) Guillermo Bautista recently posted a list of 13 online sites giving calculus tutorials... might be useful for some (I'm not necessarily endorsing them, but just passing along, and also adding the link to the right-hand column "math instruction" list):

http://tinyurl.com/ajjy54d

2) A different blogger has interestingly spotlighted an older post by Terry Tao on 'rigor in mathematics' and intuitions:

http://m-phi.blogspot.com/2013/03/terry-tao-on-rigor-in-mathematics.html

May be a bit too philosophical for some, but anything from Dr. Tao of course is worth consideration.

Tao divides mathematical education into three stages: the "pre-rigorous," "rigorous," and "post-rigorous" stages. I like the approach he takes:

"The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field."

Read the entire post here:

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

The Mystery of... Matthew Watkins

Sol Lederman's podcast with British mathematician Dr. Matthew Watkins (author of "The Mystery of the Prime Numbers") is now up:

http://wildaboutmath.com/2013/03/10/matthew-watkins-inspired-by-math-25/

It's 74 mins. long, so again you'll need to set aside some time to hear the whole thing, as Matthew expounds on his intuitive feel that there is more to the number system than meets the eye... or brain.

In trying to sum up his "Secrets of Creation" book trilogy Matthew notes that the third volume really contains the key ideas he's wishing to transmit, but volumes 1 and 2 are the necessary prelude to comprehending that final volume (which comes out shortly). His books are suited to a surprisingly wide range of ages and knowledge levels (despite the playful illustrations incorporated). In Matthew's words the books attempt "to undermine the conventional certainties" and get at the "ultimate nature of the number system" which he believes is different from how commonly perceived. He notes the world has become intensely and overly "mathematicized" with a near-worship of "quantification" which represents a detrimental, even "alarming trend" in human society.

Dr. Watkins' almost 'mystical' outlook on mathematics may put some listeners off, but I (and Sol) find it fascinating and rewarding... nor is it really much different than the deeply reverential view or tone of math that many of the greatest mathematicians of the past have likewise expressed.

When I interviewed Dr. Watkins here at Math-Frolic I called him someone "off the beaten path" of mathematics (and meant that in a positive way). That will come through all-the-more by the conclusion of Sol's extensive interview.

Tidbits...

Saturday Mathy Morsels:

1) A heads-up: Sol Lederman is expecting to interview Matthew Watkins (author of "The Mystery of the Prime Numbers") on his Sunday (or Monday) podcast… should be interesting!

2) One of the chapters in Ian Stewart's latest book (which I recently reviewed) is on the 300-year-old "three-body problem," trying to predict the complicated movement of 3 bodies that are orbiting one another in space. Well now Ian has to get busy on a rewrite... incredibly, physicists have added 13 new families of solutions to the 3 families of solutions previously known! Read all about it:

http://news.sciencemag.org/sciencenow/2013/03/physicists-discover-a-whopping.html

3) Again, if any readers here happen to be taking Keith Devlin's current MOOC course in 'mathematical thinking' I'd be curious to hear how that's going or overall impressions (...I realize it's early on).

4) Finally, I've seen some YouTube clips before of game theory in action, so when Steven Strogatz tweeted yesterday, "For a stunning display of game theory, watch this clip…" I wasn't expecting to be gobsmacked… BUT... I was! (…and 'stunning' is the right word):

Of Primes and Savants Again

A couple weeks back I referenced George Johnson's recounting of an Oliver Sacks' story about autistic twin savants who seemed to have an intuitive or mental sense of prime numbers. The whole subject area is fascinating, even though largely unresolved. I just came across another 1994 study of the subject by psychologist, Hans Welling, HERE. Still no firm conclusions, though he does note, in the end, that some savants may be capable of more 'abstract' thinking than is usually recognized. He also makes note of memory capacities and perceptual (visual or synaesthetic) skills possibly playing a role, but a truly satisfactory explanation of the savant skills seems elusive, and the basic mystery remains: Is there an underlying simplicity to prime numbers that eludes most human rationality, but is accessible to some savants (or does their talent have a simpler explanation)?

More Blogs Than You Can Shake a Protractor At

Probability enthusiasts may enjoy this blog, "Probability Puzzles," that I've only recently discovered (thanks to a @pickover tweet).

http://bayesianthink.blogspot.com/

Or, if your interest is specifically math education, David Wees has put up a list of well over 300 blogs dealing with the subject:

http://davidwees.com/content/mathematics-education-blogs

Great list... many wonderful blogs... finding the time to sort through them all; aye, there's the rub!

...Meanwhile, my review of Ian Stewart's latest book, "Visions of Infinity," is over at MathTango here. (Bottom line: Get it!)

Rooms and Lines and Angles, Oh My

(via Wikipedia)

If you're still catching up on yesterday's material, a lighter read today, but none-the-less interesting, especially for geometers... called "the illumination problem" and goes back originally to the 1950s, but MathFail blog just posted about it:

http://math-fail.com/2013/03/the-illumination-problem.html

or, you can just look it up on Wikipedia:

http://en.wikipedia.org/wiki/Illumination_problem

Question is, can you create a room of some shape completely lined with mirrors (but with no interior rooms or closed-off areas) such that a light source could be positioned at some one point, and the light beams therefrom not be observable from some other point in the room space; in short, normally any light source would fill the entire room with light beams -- is there a geometric way to prevent that from happening? (alternatively, the problem is sometimes stated in terms of creating a billiard table such that a ball taking off from one point would never drop into a hole at another point on the surface, even given infinite wall-bounces). Can it be done?...

Yes! (theoretically).

Monday Math Buffet...

Another potpourri of offerings, if you've missed any of these…:

1) I love this relatively brief recent post from a secondary educator:

http://practicaltheory.org/blog/2013/02/28/we-dont-know-how-to-teach-math/

Also have to smile a bit at this particular line from the piece: "Seymour Papert said that math represents the failure of progressive education because the way we teach math always reintroduces coercion back into education."

But every line is good, and it largely reminds me of a fantastic, (looong) older post by Fields Medalist Timothy Gowers (which drew over 160 comments) that deserves frequent re-visiting:

http://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/

2) Latest (#96) Carnival of Mathematics is now up at "Math Mama Writes" blog:

http://mathmamawrites.blogspot.com/2013/03/carnival-of-mathematics-96.html

...Plenty of variety!

3) "Futility Closet" recently highlighted the delightful (to me) Yablo's Paradox:

http://www.futilitycloset.com/2013/02/22/yablos-paradox/

4) On the same day that I interviewed Evelyn Lamb over at MathTango, she put up a new post at her blog on the four-color theorem:

http://blogs.scientificamerican.com/roots-of-unity/2013/03/01/4-color-map-theorem/

5) Once again Keith Devlin covers some aspects/difficulties of running a MOOC at his "Devlin's Angle" blog (Keith is running at least 3 separate blogs, plus a Huff. Post column!):

http://devlinsangle.blogspot.com/2013/03/can-we-make-constructive-use-of-machine.html

...and over at Huffington Post Keith writes about the dropout rates for MOOCs, and why an 80+% dropout rate isn't necessarily a problem:

http://www.huffingtonpost.com/dr-keith-devlin/moocs-and-the-myths-of-dr_b_2785808.html

6) Math educator Maria Droujkova, with a focus on toddlers and youngsters, is Sol Lederman's 24th podcast interviewee here:

http://wildaboutmath.com/2013/03/01/maria-droujkova-inspired-by-math-24/

7) And finally, if you have any mental energy/time left, a long, thought-provoking read from the Simons Foundation on the future of computers in mathematical proofs (can we trust computers, as we turn to them more and more in the future for proofs of highly-complex theorems?):

https://simonsfoundation.org/features/science-news/in-computers-we-trust/

Evelyn Lamb and more...

 Into the weekend....:

1) Hope this isn't too confusing, but for now will still be calling the interviews I do the "Math-Frolic Interviews" but posting them over at my companion blog (for longer pieces), MathTango. The newest one, with Evelyn Lamb of Scientific American is now up:

http://mathtango.blogspot.com/2013/03/evelyn-lamb-of-scientific-american.html

So grateful to Evelyn for taking part and sharing her story. Check it out!

2) Below, a delightful post from the Wolfram folks explaining why the newly-opened Museum of Mathematics has no logo (...because it has an infinity of them):

http://blog.wolfram.com/2013/02/28/behind-the-scenes-at-the-national-museum-of-mathematics-meta-logo/

3) "The Aperiodical" blog has begun a new podcast series, they're calling "All Squared," with first edition (a diversity of topics) here:

http://aperiodical.com/2013/02/all-squared-number-1-maths-out-loud/

4) Finally, here's more of Keith Devlin talking about the evolution of his math MOOC: