The Power of Google....

Kind of funny that my most highly-trafficked post in the last week, by far, was this brief one that goes all the way back to Nov. 20th, related to the P vs. NP problem!:

http://tinyurl.com/42456wl

The post simply links to a wonderful Matt Parker piece in the Guardian that briefly references certain nuptial invitations while discussing the P vs. NP conundrum... ohhh, the title of the post was, "A Royal Wedding... and Mathematics."

Penny-pitching

For a Friday workout, here's a post about an old Marilyn vos Savant problem (and NO, it has nothing to do with Monty Hall, but does pertain to game theory):

http://tinyurl.com/3np5oh9

A fuller treatment of the problem here, from Scientific American (pdf):

http://siam.org/pdf/news/331.pdf

Thursday Math Miscellany

1) Sol Lederman's new site for "playing" with Wolfram Mathematica is now up-and-running, and looks gooood! ...but then that would be expected from Sol!!:

http://playingwithmathematica.com/

2) SquareCircleZ blog offers a list of 10 free-for-the-asking help sites for math questions:

http://www.squarecirclez.com/blog/top-10-math-help-sites/5966

3) Finally, Jason Rosenhouse astutely explained Mersenne primes in this recent post:

http://scienceblogs.com/evolutionblog/2011/04/monday_math_mersenne_primes.php#more

A Video Rerun

I've shown this before, but have several new readers now, and consider it worth repeating... An interesting 9-minute clip from an old "BloggingheadsTV" edition with John Horgan and Jim Holt discussing the nature of mathematics ('are mathematical truths discovered or invented?'), the Riemann Hypothesis, and similar topics:

"Making Math Fun"

Alfred Posamentier is one of my favorite math writers, but I'd never actually seen him... until now, that is (courtesy of YouTube):



...and here's an Amazon page for his books:

http://tinyurl.com/3d8ef9c

Axiom of Choice... and Pigeonholes

The "Axiom of Choice" is one of those simple-seeming notions that is difficult to fully-grasp... or at least grasp its profound implications (I don't claim to). It probably requires a fairly deep grounding in set theory in order to fully wrap your brain around its significance.
Anyway, here is an older post I recently ran across that just sort of struck my fancy on the topic, so maybe it will yours as well (even though the author, Adam Bohn, admits at one point that the axiom of choice falls into the category of “things it is best to not think about”!):

http://yabm.wordpress.com/2010/04/22/the-axiom-of-choice/#more-456

Meanwhile, the best overall treatment of the Axiom of Choice (for layfolks) that I've seen on the Web is here (with plenty of further links):

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

Statistics, Causation, Association... uhhh, Cell Phones

Do cell phones cause cancer?... the question that won't go away... and probably won't be settled conclusively anytime soon (those who think it's been settled, and that the answer is 'no,' DON'T understand the full nature of science or synergistic effects in biology).

Good (long) article from oncologist Siddhartha Mukherjee in the NY Times a bit ago, indicating why there is yet little good/consistent scientific evidence to support a cancer/cell-phone-use link... but why it also can't yet be ruled out with our current limited understanding and assumptions:

http://tinyurl.com/42tbnuf

Couple of Videos

For your Friday entertainment, a fun video clip from YouTube based on one of the old "liar" paradoxes (h/t to Ben Vitale):




And here a simple mathematical card trick:

http://www.wimp.com/mathcard/

More On Math Education

Interesting NY Times piece on math education here:

http://tinyurl.com/3vvu7mm

It's centered around a Canadian-promoted program called "JumpMath" (produced by "a charitable organization working to create a numerate society") which seeks to achieve more equitable math success for all in grade levels 1 through 8:

http://jumpmath.org/

(...I'm not endorsing JumpMath here, by the way, since I don't know enough about it, but simply passing along its existence for any who may wish to pursue it further.)

...Note also that there is to be a follow-up to the above article in tomorrow's Friday edition of the Times as well.

ADDENDUM: the Friday article, further explaining the "Socratic" approach of JumpMath, is now available here: 

  http://tinyurl.com/3j4vang

Wild About... Sol Lederman

"To me, math is about play... Math is art. I can look at a formula or solution to a problem and be in absolute awe, as if I were looking at a beautiful painting. I get a thrill from making connections, and from guiding people to make connections." -- Sol Lederman quoted in a Sante Fe, NM. news article

One of the things I've learned in less than a year of doing a "math" blog is the wonderful variety of blogs that are out there, from those geared to children to those intended primarily for PhDs., and everything in-between. Too many people don't realize the sheer range and diversity (and even fun) of approaches to mathematics that are possible; there are math blogs to suit almost anyone's taste.

One of the primary inspirations for me even taking a stab at a math blog was witnessing the popularity of Sol Lederman's "Wild About Math" blog --- it clearly demonstrated there was an audience for those who likely couldn't participate in math's higher applications, but who still find the subject fascinating, as an integral part of human life, and not merely as a classroom endeavor.

Anyway, congratulations to Sol for being profiled in a recent New Mexico news piece here:

http://tinyurl.com/3snobxp

And on a related note, Sol just announced that he is initiating a new math blog centered around programming with Wolfram Mathematica, as a further way to engage people in mathematics. He's specifically requesting assistance from any who may be willing to help him write "some simple animations," as he goes about educating himself more on this widely-used creative software program:

http://wildaboutmath.com/2011/04/19/seeking-mathematica-experts/

Strange Genius

In his 2001 volume "Wonders of Numbers," Clifford Pickover ranked the "5 Strangest Mathematicians Who Ever Lived." One could offer a lot of suggestions and debate on this subject, but here is Pickover's slate, with links to their Wikipedia pages:

1. Paul Erdos

2. Srinivasa Ramanujan

3. Pythagorus

4. Theodore Kaczynski

5. John Nash

(I'll throw in a couple more that quickly come to mind: Grigori Perelman and Alexander Grothendieck.)

The oddest member in this whole group, from my viewpoint, is Kaczynski, who in case you've forgotten or didn't know, was a loner, domestic terrorist also known as the "Unabomber" (and also a Harvard grad and Michigan math PhD.) who eluded law enforcement for over 17 years, living a spartan, reclusive life in rural Montana, while sending out his occasional mail bombs and manifestos.

More Books... Old and New

Information... Is That All There Is?

James Gleick is hands-down one of the best science writers in America today and his latest opus/book, "The Information," looks to be another major intellectual compendium, worthy of every science reader's attention. Its subject matter is in some ways tangential to mathematics, and yet in other ways quite integral to the subject of math. I haven't read the volume myself (hope to get around to it at some point), but in the meantime here are various reviews of it from around the Web:

From John Horgan for Wall Street Journal:

http://online.wsj.com/article/SB10001424052748703409304576166880701416672.html

David Ulin in the LA Times:

http://articles.latimes.com/2011/mar/13/entertainment/la-ca-james-gleick-20110313

Freeman Dyson in NY Review of Books:

http://www.nybooks.com/articles/archives/2011/mar/10/how-we-know/?pagination=false

And 2 reviews from the NY Times, from Geoffrey Nunberg and Janet Maslin:

http://www.nytimes.com/2011/03/20/books/review/book-review-the-information-by-james-gleick.html

http://www.nytimes.com/2011/03/07/books/07book.html 



I'm waaay late on this one... "In Code" is a 2001 book by Sarah Flannery and her dad telling the interesting story of this award-winning Irish-teenage mathematician's work in cryptography. As cryptography and modular arithmetic are not major interests of mine, but are a big chunk of the book, I skipped this volume when it came out 10 years ago. Recently though had occasion to read it, and definitely recommend it to others who have bypassed it; all the moreso if you ARE interested in cryptography (it offers, by the way, one of the best, layman-friendly treatments of RSA encryption I've come across). Sarah's story-line is interesting in its own right, and the integral discussion of prime numbers is also worthwhile.

Large parts of it are available (for free) from Google books on the Web here:

http://tinyurl.com/3qtz8ka  

And in Britain, Tony Crilly has a new basic math book out (not available in States yet) -- "The Big Questions: Mathematics." A review here:

http://plus.maths.org/content/big-questions-mathematics

You Say You Wanna Go Ivy League

You can watch a variety of free math course videos from top-notch Ivy League-type schools as listed here (...and forgo the $35,000+ tuition):

http://www.openculture.com/math_free_courses

Two Oldie-But-Goodies...

First, the puzzle that never dies....

The Monty Hall problem covered yet again, this time in Scientific American:

http://www.scientificamerican.com/blog/post.cfm?id=lets-make-a-deal-revisiting-the-mon-2011-04-15

... and (unrelatedly) this great old piece on "Kaprekar's operation" (or Kaprekar's number "6174") from plus.maths.org:

http://plus.maths.org/content/os/issue38/features/nishiyama/index

Friday Puzzles...

I've adapted this one from "Math Charmers" by Alfred Posamentier:

Before him, Newt has 3 boxes, one of which contains ONLY nickels, one containing ONLY dimes, and one containing a mix of nickels and dimes. But the labels for the boxes ("Nickels," "Dimes," and "Mixed") fell off and when they were replaced they were all placed on WRONG boxes. Without looking, Newt can draw ONE coin from one of the now mislabeled boxes and see what it is. If only he were bright enough, which box should he draw one coin from in order to insure that he can determine what is actually in all 3 boxes? (answer down  below)

NICKELS          DIMES          MIXED


And a simple bonus logic chestnut today adapted from a current Peter Cameron post HERE:

On a table are 4 cards marked (on the sides that are showing) "A," "B," "2," and "3.". It is known that each of the 4 cards has a letter on one side and a number on the other side. Which 2 cards should you turn over in order to determine if all cards with a vowel on one side have an even number on the opposite side?

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dimes/nickels answer: draw from the box labeled "mixed"

card answer:  "A" and "3"

DNA Code... Is There a Pattern?

Image via Wikipedia

"...the impulse to find order amid chaos". . . . .

Interesting post from John Baez on mathematical physicists exploring the genetic code for any underlying pattern:


http://johncarlosbaez.wordpress.com/2011/04/14/the-genetic-code/

The Polymath Project... Redux

"The practice of mathematics is changing. While in the past mathematics was predominantly a solitary effort, now it has become increasingly collaborative. How does this growing inclination for teamwork translate on the Internet and what benefits and challenges does it bring?"

That's the opening of an interesting article on Tim Gowers "Polymath Project" -- essentially a collaborative forum for solving difficult math problems over the Web. I've reported on this topic in the distant past, and am a huge fan of (but non-participant in) the approach. The full article here:

http://polymathprojects.files.wordpress.com/2011/03/polymathias.jpg

The Project's blog is here:

http://polymathprojects.org/

"March Mathness Wrap-Up"

From 'Princeton University Press Blog' a wrap-up of how a group of Davidson College mathematicians approached bracket-forecasting (based on "math modeling") for the March Madness NCAA Tournament:

http://press.princeton.edu/blog/2011/04/12/march-mathness-wrap-up/?sms_ss=twitter&at_xt=4da49fc12f0f177a,0

...I'm fascinated to read that out of 5.9 million entries to ESPN two people actually correctly predicted this year's surprise final four (statistically perhaps it may stand to reason, though still seems amazing to me), but they were not from the math modelers reported on above.

Cantor Set

Georg Cantor derived some of the most mind-blowing insights of any mathematician; indeed many of his contemporaries found his ideas so wackalooney (excuse the technical language ;-)) that they simply dismissed his notions out-of-hand. But today, while his ideas are no less mind-boggling, they are widely accepted.
One of the most fascinating discoveries, though probably less famous than many of his conclusions regarding infinity, was the "Cantor Set." It is commonly depicted by taking a unit length and then deleting the middle third, to leave two bookend pieces, which in turn have their middle third removed, and on and on and on, infinitely. Many of the properties of this simple, final Cantor Set are quite remarkable...

From the mathacademy.com site (see: http://www.mathacademy.com/pr/prime/articles/cantset/ ):

"What remains after infinitely many steps is a remarkable subset of the real numbers called the Cantor set, or “Cantor’s Dust.”
At first glance one may reasonably wonder if there is anything left. After all, the lengths of the intervals we removed all add up to 1, exactly the length of the segment we started with:

 

Yet, remarkably, we can show that there are just as many “points” remaining as there were before we began! This startling fact is only one of the many surprising properties exhibited by the Cantor set."

The Cantor Set also exhibits self-similarity or fractal-like properties throughout.

...and another site here to explore the Cantor Set step-by-step:

http://personal.bgsu.edu/~carother/cantor/Cantor1.html

One often wonders, in the case of Cantor (and certain other mathematicians), did some sort of madness drive him to many of his remarkable, penetrating insights... or, did his unique insights drive him to madness???
Occasionally (and luckily, only occasionally) it seems as if there is a fine line between succeeding at the highest levels of math... and insanity. Chapter 4 (entitled "Mathematics as an Addiction") of Reuben Hersh's recent "Loving and Hating Mathematics" interestingly addresses the link (if any) between math and mental illness. (He concludes "No," but does add: "Still, there is something different about mathematicians compared to, say chemists or geologists or even English professors. It is possible to be 'crazy' -- that is conspicuously eccentric, very odd, even antisocial -- and still hold a job as a math professor.")
...It all reminds me a bit of Steven Wright's oft-cited observation that "There's a fine line between fishing... and standing on the shore looking like an idiot." ;-)

Alan Sokal Revisited???

Wow! A somewhat fascinating case of a math paper in a legitimate serious journal being recently retracted, as most likely a hoax...
I link in my right-hand column to physicist Alan Sokal's deliciously-famous parody/hoax article he submitted to an academic journal (and had accepted) to show how some nonsensical postmodernistic verbiage could be passed off to an unwitting editor (and referees) as scholarly work. This new instance almost seems like a 'Sokalism' in reverse... someone passing off nonsense math jargon well enough to have it accepted by a mathematics journal editor (not usually known for being swayed by jargon).

http://retractionwatch.wordpress.com/2011/03/15/faked-data-unsubstantiated-claims-and-spirituality-add-up-to-a-math-journal-retraction/

"CTK Insights" blog originally reported on this another paper from the same author back in October:

http://www.mathteacherctk.com/blog/2010/10/a-computer-application-in-mathematics/

Still not totally clear to me if this is a case of a flat-out hoax, or some form of postmodern mathematical crackpottery, but either way rather entertaining... except for those suffering embarrassment.

(Addendum: as a side-note I might mention that I consider Sokal's book, "Beyond the Hoax: Science Philosophy and Culture,"  though it won't suit everyone's taste, one of the greatest, thought-provoking academic reads out there, especially if you're at all interested in the philosophy of science.)

Do the Work, Don't get the Credit

RJ Lipton discusses "Stigler's Law," today (from professor Stephen Stigler) which states that "no scientific discovery is named after its original discoverer"... and Stigler notes Robert K. Merton as the discoverer of “Stigler’s Law.” :-)

http://rjlipton.wordpress.com/2011/04/08/why-is-everything-named-after-gauss/

Wikipedia offers a list of examples here:

http://en.wikipedia.org/wiki/List_of_examples_of_Stigler%27s_law

and a similar list here (specifically-related to mathematics):

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

(...just don't anyone try to tell me that Georg Friedrich Bernhard Riemann doesn't get full credit for the Riemann Hypothesis, okay!)

Friday Puzzle... and Cicada Math

Another Friday puzzle to rev up your brains before the weekend:

 A book requires 552 digits to be numbered from its first page (page 1) to its last page (???).  How many pages are there in the book? (answer below)

(I've adapted the problem from this site:

http://www.numericana.com/answer/recreational.htm#pages )

...and to additionally stretch your mind, especially if you're a computer scientist or programmer, an interesting piece (that's making its way around the Web) here applying the "cicada principle" to the world of web design:

http://tinyurl.com/3j4r2oj
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book answer: 220

Medical Screening: The Good, the Bad, the Statistical

A number of webpages address the problems/misunderstanding that often surrounds the interpretation of medical screening tests. The below recent blog post from "Understanding Uncertainty" blog, does a reasonably good job of it:

http://tinyurl.com/3snndv9

Murder and Math?

For any cryptography buffs out there...:

The FBI is requesting help trying to decipher some encrypted handwritten notes found on a murder victim from a 12-year-old unsolved case!

Hat tip to the "Fun With Numbers" site for originally leading me to the story:

http://benvitale-funwithnum3ers.blogspot.com/2011/04/cracking-code-in-unsolved-murder-case.html

Another version of it here:

http://tinyurl.com/43p6geu

...and finally the Wikipedia page on same story:

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

Seeing and Believing...

Math can be shocking!...

...that is the point RJ Lipton makes in another wonderful post, this time on certain "shocking" results in the analysis of spherical constructions. There's some higher-level, abstract math involved here, but even without fully knowing/understanding it, one can sense the 'shocking' quality of these ideas/conclusions:

http://rjlipton.wordpress.com/2011/04/04/i-see-it-but-i-dont-believe-it/

(...an interesting short side-note is included too, on brilliant mathematician Steven Smale once having his government funding retracted for having stated that he did his best work "on the beaches of Rio." :-))

Let's Hear It For Algebra II

Most folks know that the S.A.T. has never been a great predictor of success in either college or life. According to the below interesting Washington Post article, though, Algebra II does fill the bill, and is being looked at more and more as a requirement for graduation, as it tends to heighten critical thinking of the sort that will be needed increasingly in workplaces of the future:

http://www.washingtonpost.com/business/economy/requiring-algebra-ii-in-high-school-gains-momentum-nationwide/2011/04/01/AF7FBWXC_story.html

(interestingly too, the article has already garnered 360 comments!)

Bookshelf: William Byers

<--(Wm. Byers with his first book) 

I'm currently re-reading (and re-relishing) mathematician William Byers' "How Mathematicians Think" -- for me, the richest, most interesting mathematical read in my entire library (though I recognize it won't be everyone's 'cup-o-tea' --- if you have little interest in the philosophical underpinnings of math, or if talk of Godel and Cantor tends to send you screaming out of the room, tearing your hair out by the roots.... well, then, this may not be a volume for you!)... but it is a great and accessible, thought-provoking treatment of ideas/issues underlying mathematical thinking (centering not around logic, but around ambiguity, contradiction, paradox, uncertainty, and creativity as being at the core of math).
But I'm also re-reading and refreshing myself on Byers because he has a new book out, "The Blind Spot: Science and the Crisis of Uncertainty," wherein he tackles many of the same concerns once again. 

http://press.princeton.edu/titles/9406.html

I can't imagine that his new book will surpass or even equal "How Mathematicians Think," but still looking forward to it. Additionally, it has a Facebook page devoted to it here:

http://www.facebook.com/pages/The-Blind-Spot-by-William-Byers/119855461412669

Below, a passage from the older work, "How Mathematicians Think":

"Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary...
"Now the mathematician can sense the presence of an idea even when the idea has not yet emerged... It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be. This is a tangible feeling...
"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name, 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medal winner Richard E. Borcherds. Thus the initial observation plus the recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research...
"McKay noticed that there was something going on... Understanding what is going on is an ongoing process -- the very heart of mathematics."

Friday Puzzle(s)

To get your Friday brain cells in motion, a straightforward problem I've adapted from the puzzles4you.com site:

 1/4 of the hats in Hank's Hat Haberdashery are brown. During a weekend sale, the store sells 2/3 of all their hats, including 4/5 of the brown hats. What fraction of the UNSOLD hats remaining are brown? (answer below)

...and if that one warmed you up, you can give your neurons a further work-out with this "proof" over at Equalis Community Blog that the hypotenuse of a right triangle equals the sum of the two sides!:

http://www.equalis.com/members/blog_view.asp?id=565749&post=122969

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--->  hat answer: 15%