Ever think about just how varied individual handwriting is... how many different ways there are to write an "a," as well as every other letter... or number for that matter???

To keep it simple think about just the numbers 0 - 9, and the different ways they may be written, and yet immediately recognized. In fact, we of course have automatic equipment in the Post Office dedicated to do just that: automatically read Zip Codes, in all sorts of handwriting!

Tim Chartier addressed mechanical digit-reading in a recent post here:

http://forum.davidson.edu/mathmovement/2010/08/24/digit-recognition-with-pythagoras/

## Tuesday, August 31, 2010

## Monday, August 30, 2010

### (Don't) Walk This Way

The simple (bust-a-kneecap) geometry of high heels is elucidated over at "mathspig" blog (...the operative term here is "SPLAT"):

Sent out especially for the blogger Isis the Scientist and her readers ;-)

here's some more doozies:

...and here of course, some sensible footwear:

**http://mathspig.wordpress.com/2010/08/25/killer-heels-that-kill/**Sent out especially for the blogger Isis the Scientist and her readers ;-)

here's some more doozies:

**http://tinyurl.com/2eps8nz**...and here of course, some sensible footwear:

**http://tinyurl.com/23gxp62**### Proof? ;-)

A golden oldie...:

Suppose:

a + b = c.

This can then be equivalently written as:

(4a – 3a) + (4b – 3b) = 4c – 3c

After reorganizing (adding/subtracting from both sides):

4a + 4b – 4c = 3a + 3b – 3c

Take the constants out of the brackets:

4(a + b – c) = 3(a + b – c)

Remove the same term left and right:

4 = 3

Ta-daaaaaaa!!

Suppose:

a + b = c.

This can then be equivalently written as:

(4a – 3a) + (4b – 3b) = 4c – 3c

After reorganizing (adding/subtracting from both sides):

4a + 4b – 4c = 3a + 3b – 3c

Take the constants out of the brackets:

4(a + b – c) = 3(a + b – c)

Remove the same term left and right:

4 = 3

Ta-daaaaaaa!!

## Sunday, August 29, 2010

### Fields Medalists... A Breed Apart

Interesting article on the recently-awarded Fields Medals (given "for outstanding mathematical work completed before the age of 40"), and great mathematicians more generally:

http://www.openthemagazine.com/article/nation/masters-of-the-universe

There is, the article notes, an "identification of mathematics with prodigies and troubled minds" which "is reinforced by the few books and movies that have struck a chord with people at large."

While it's true that those at the highest rarified levels of mathematics are often exceptional or quirky individuals, nonetheless, the stereotype painted by popular media is likely overdrawn.

http://www.openthemagazine.com/article/nation/masters-of-the-universe

There is, the article notes, an "identification of mathematics with prodigies and troubled minds" which "is reinforced by the few books and movies that have struck a chord with people at large."

While it's true that those at the highest rarified levels of mathematics are often exceptional or quirky individuals, nonetheless, the stereotype painted by popular media is likely overdrawn.

## Saturday, August 28, 2010

### A Plug For Raymond Smullyan

I was recently re-reading an old work by mathematician/logician Raymond Smullyan, and it occurred to me how relevant many of his writings are today amidst the sudden interest in the P vs. NP problem.

Smullyan is retired, in his 90's now, and it also occurred to me (and I hope this doesn't sound morbid) that I don't know how much longer he'll be around enlightening us. His friend and even-more-famous colleague Martin Gardner of course passed away earlier this year in his mid-90's, after an incredibly productive life. Gardner's death was one of the inspirations for me starting this blog, and I feel I ought acknowledge Smullyan's contributions while he is still among us...

Although (like Gardner) he has written a fair amount of recreational mathematics, Smullyan is probably even better-known as a logician dealing with more abstract issues (recursion, self-reference, paradox) that underlie mathematics, and that when resolved, have significant application. His writings have always been creative, entertaining, original, and generally accessible to lay readers (though probably lacking the "zing" and universality of much of Martin Gardner's output).

His prolific book listings on Amazon here:

**http://tinyurl.com/33325zk**

Smullyan's pursuits also range across astronomy, music, magic, mysticism, and Taoist philosophy, and interestingly he seems to find a measure of unification in Taoism for the abstract mathematical paradoxes/puzzles he ponders (I always find interesting the division between those serious mathematicians who are deeply drawn toward mysticism and those who are not!).

If you're not familiar with Smullyan's work, I'd recommend getting to know him; especially if the whole P vs. NP hoopla intrigued you.

Wikipedia entry for Smullyan here:

**http://en.wikipedia.org/wiki/Raymond_Smullyan**

...and one of his fans has put up a MySpace page dedicated to him as well:

**http://www.myspace.com/raymondsmullyan**

Like Gardner, he is another American gem!

## Friday, August 27, 2010

### ICM Review

A nice commentary on the recently-concluded "International Congress of Mathematicians" Conference (where Fields Medals were awarded) in India from Alex Bellos here:

The Congress meets once every 4 years, and is " the largest, most prestigious and most traditional event in maths;" Bellos adds "no other scientific discipline has an event as big and with as much historic resonance."

Attendees have been blogging and tweeting about the Conference since it opened; Bellos here summarizes a few highpoints (and promises more in follow-up post).

**http://alexbellos.com/?p=1337**The Congress meets once every 4 years, and is " the largest, most prestigious and most traditional event in maths;" Bellos adds "no other scientific discipline has an event as big and with as much historic resonance."

Attendees have been blogging and tweeting about the Conference since it opened; Bellos here summarizes a few highpoints (and promises more in follow-up post).

### Book Review (of Chaitin & Rucker)

Fine older review by Jaron Lanier of two books that have been out for awhile, and worth looking into if you've never read them: "

**Meta Math! The Quest For Omega**" by Gregory Chaitin, and "

**The Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me about Ultimate Reality, the Meaning of Life, and How to Be Happy**" by Rudy Rucker:

http://www.americanscientist.org/bookshelf/pub/two-philosophies-of-mathematical-weirdness

Toward the end of the lengthy review, Lanier summarizes thusly:

"These two books are near opposites even though they appear to explore similar topics. Chaitin loves negative results and is thrilled by the prospect of future generations of mathematicians finding ever weirder math. The Chaitinesque intellectual future will be eternally youthful and anarchic. Neither mathematicians nor computer scientists will settle down into a single preferred pattern of thought. Rucker, in contrast, is reaching as high as he can to try to use available computer science and math metaphors to create a new, comprehensive, multidisciplinary sensibility. The Ruckerian future is one in which new guiding explanatory ideas will connect all areas of intellectual curiosity."(Of these two works, I enjoyed Chaitin's volume more than Rucker's, but it's largely a personal preference, and I've enjoyed some of Rucker's other books more than this particular one.)

And a bit more current, Murray Bourne's latest "IntMath Newsletter" is up here (with a little something for everybody who enjoys math):

**http://tinyurl.com/2euddty**

## Thursday, August 26, 2010

### Neuro-mathematics...

The human brain is probably the 'final frontier'...

Blogger Jason Goldman reviews some of what we know/believe about mathematical dysfunction of the brain here, based in part on case-studies of patients with different brain lesions and on fMRI studies:

This was actually the

Blogger Jason Goldman reviews some of what we know/believe about mathematical dysfunction of the brain here, based in part on case-studies of patients with different brain lesions and on fMRI studies:

**http://tinyurl.com/2dz2m3a**This was actually the

**4th**in a*series*of related posts Jason did that can be looked up here:**http://tinyurl.com/266o57k**## Wednesday, August 25, 2010

### An "Incredible Time For Mathematicians"

Hmmmm... looks like there may be a lucrative future in this thing we call mathematics:

http://www.livemint.com/2010/08/24001310/Math-becomes-fashionable-focu.html?h=A1

....if the movie "The Graduate" were produced today, I guess the one-word that might get passed along to Ben wouldn't be "plastics," but "

And in a slightly related vein, "

http://www.theatlantic.com/science/archive/2010/08/when-computers-predict-crime/61870/

Finally, on a side-note, the 29th "

http://numberwarrior.wordpress.com/2010/08/23/math-teachers-at-play-29/

http://www.livemint.com/2010/08/24001310/Math-becomes-fashionable-focu.html?h=A1

....if the movie "The Graduate" were produced today, I guess the one-word that might get passed along to Ben wouldn't be "plastics," but "

*algorithms*."And in a slightly related vein, "

**The Atlantic"**has a current article relating to applied algorithms:http://www.theatlantic.com/science/archive/2010/08/when-computers-predict-crime/61870/

Finally, on a side-note, the 29th "

**Math Teachers At Play**" Blog Carnival is up for your perusal, with a variety of topics, here:http://numberwarrior.wordpress.com/2010/08/23/math-teachers-at-play-29/

## Tuesday, August 24, 2010

### Math In the Arts...

Ivars Peterson interestingly reviews a play he recently took in in India, "

The play originated in 2007 (and won several "Best New Play" awards that year), and has been staged in many U.S. and worldwide venues, virtually always to rave reviews. The

"It’s not fundamentally about numbers, either, but about the search for meaning and the consoling satisfaction of finding the patterns that define and describe both the physical universe and individual human lives."

Another reviewer simply calls it "a thrilling, thrilling, thrilling play." And there are many additional reviews on the Web, including this 7-minute NPR podcast review:

http://www.npr.org/player/v2/mediaPlayer.html?action=1&t=1&islist=false&id=128513493&m=128513513

It's nice to know that the wonder/beauty of mathematics has been translated to the acting stage so effectively!

And this October the play will be broadcast live across the U.S. (and the world) in select theaters. If you are near one... and you are a reader of this blog... you ought probably look for it!

**A Disappearing Number**," centered around mathematics and especially the life of Indian prodigy Ramanujan:**http://mathtourist.blogspot.com/2010/08/english-play.html**The play originated in 2007 (and won several "Best New Play" awards that year), and has been staged in many U.S. and worldwide venues, virtually always to rave reviews. The

**NY Times**called it "lucid, dynamic and continuously engaging," and then continued:"It’s not fundamentally about numbers, either, but about the search for meaning and the consoling satisfaction of finding the patterns that define and describe both the physical universe and individual human lives."

Another reviewer simply calls it "a thrilling, thrilling, thrilling play." And there are many additional reviews on the Web, including this 7-minute NPR podcast review:

http://www.npr.org/player/v2/mediaPlayer.html?action=1&t=1&islist=false&id=128513493&m=128513513

It's nice to know that the wonder/beauty of mathematics has been translated to the acting stage so effectively!

And this October the play will be broadcast live across the U.S. (and the world) in select theaters. If you are near one... and you are a reader of this blog... you ought probably look for it!

### The Math of DNA Matches

CSI, DNA, and what else... MATH!

...or

Why a 'DNA match' in forensics may NOT be all that it seems... what it does and doesn't mean mathematically speaking:

...or

Why a 'DNA match' in forensics may NOT be all that it seems... what it does and doesn't mean mathematically speaking:

**http://plus.maths.org/content/os/issue55/features/dnacourt/index**## Monday, August 23, 2010

### Men... Take Heart...

Hmmmm... another study at the intersection of set theory and evolution ;-))) :

(There are apparently more beautiful females around than ever before for we unsightly male counterparts!)

http://www.telegraph.co.uk/news/newstopics/howaboutthat/5912250/Women-getting-more-beautiful-say-scientists.html

The report begins thusly:

(There are apparently more beautiful females around than ever before for we unsightly male counterparts!)

http://www.telegraph.co.uk/news/newstopics/howaboutthat/5912250/Women-getting-more-beautiful-say-scientists.html

The report begins thusly:

"Researchers found that attractive women have more children than their less attractive counterparts and that a higher proportion of those children are female.

Once those daughters become adult they tend to be good looking themselves and so the pattern is repeated as women over the generations become steadily more aesthetically pleasing.

As attractive couples are less likely to have a boy than a girl,men, in contrast, remain as aesthetically unappealing as their caveman ancestors, the scientists claim." [emphasis added]

### 100-Meter Race Problem

Just a simple word problem to start the week off; adapted from "Futility Closet" (http://www.futilitycloset.com/2010/08/02/the-handicap/):

Michael challenges his brother Ryan to a 100-meter foot race. When Ryan crosses the finish line Michael has only covered 97 meters.

Michael challenges Ryan to a re-match, but this time Ryan has to start 3 meters

Assuming the brothers run at the identical speed they did in the first race, who will win?

Or, if you want a bigger challenge, Matt Parker asks on Twitter recently:

.

.

.

.

.

.

.

.

.

Ryan wins again. Ryan is able to cover 100 meters in the time it takes Zachary to run 97 meters. By starting 3 meters behind the starting line, Ryan will be dead-even with his brother at the 97-meter mark, at which point he (Ryan) will then pull ahead for the victory!

Michael challenges his brother Ryan to a 100-meter foot race. When Ryan crosses the finish line Michael has only covered 97 meters.

Michael challenges Ryan to a re-match, but this time Ryan has to start 3 meters

*behind*the starting line.Assuming the brothers run at the identical speed they did in the first race, who will win?

*answer down below...*

"Is any number palindromic in bases 2, 3 and 5? I know @stecks checked up to a million. But still, it feels like there should be one..."...have at it??? ;-)

.

.

.

.

.

.

.

.

.

*Solution (1st problem):*

## Saturday, August 21, 2010

### "Patterns & Beauty: The Mathematics of Music"

Nice 6-minute video on the mathematics of music:

http://videos.howstuffworks.com/hsw/11978-patterns-and-beauty-the-mathematics-of-music-video.htm

(...if you search "mathematics" on this site, "HowStuffWorks," there are 270

http://videos.howstuffworks.com/hsw/11978-patterns-and-beauty-the-mathematics-of-music-video.htm

(...if you search "mathematics" on this site, "HowStuffWorks," there are 270

*more*offerings!)## Friday, August 20, 2010

### Evolution of the 'Recycle' Logo

Ivars Peterson of "The Mathematical Tourist" reviews various designs for the ubiquitous "recycle" symbol that most of us are used to seeing, pointing out that the original logo was based on a Mobius strip design, but was followed by several 'mutant' designs that lacked the Mobius feature:

**http://mathtourist.blogspot.com/2010/08/recycling-arrows.html**

### Fox News... Math Illiterate

How

http://scienceblogs.com/corpuscallosum/2010/06/tricks_with_graphs_deliberate.php#more

You may also want to check out this

http://www.ted.com/talks/lies_damned_lies_and_statistics_about_tedtalks.html

**NOT**to graph! (...or just an example of "lies, damned lies, and statistics)":http://scienceblogs.com/corpuscallosum/2010/06/tricks_with_graphs_deliberate.php#more

You may also want to check out this

**TED Talk**for a chuckle, as well:http://www.ted.com/talks/lies_damned_lies_and_statistics_about_tedtalks.html

## Thursday, August 19, 2010

### I'll Drink To That!

Tiny bubbles....

Back in 2002, some physicists won an "

I can hardly wait to learn this year's illustrious winners come Sept. 30 at Harvard.

Back in 2002, some physicists won an "

**Ig Nobel Prize**" ;-) for their study of the mathematics of beer bubbles:**http://tinyurl.com/2elosmd**I can hardly wait to learn this year's illustrious winners come Sept. 30 at Harvard.

### Heron's Formula

Pythagoras gets all the publicity but Heron was no slouch either... Heron was another ancient who is credited with devising a formula for computing the area of a triangle from knowing only the lengths of the 3 sides involved. The formula appears in a few different forms, 2 common ones below (a, b, and c represent the 3 sides of a triangle, and "s" is the perimeter value for same).

more on Heron's formula here:

more on Heron's formula here:

**http://en.wikipedia.org/wiki/Heron%27s_formula**An offshoot of Heron's formula is Brahmagupta's formula for the area of any 'cyclic'

quadrilateral (one that fits inside a circle):

- or

_________________________________

`1/4 √(a+b+c-d) (a+b-c+d) (a-b+c+d) (-a+b+c+d)`

## Wednesday, August 18, 2010

### Surprise Me!

I suspect this post on 'mathematical surprises' from Dave Richeson may get passed around a lot:

He lists a number of varied mathematical discoveries over the years that surprised the math community and offered new perspectives; his commenters chime in with still more examples.

I think this gets at what mathematicians love about their subject: it is full of never-ending surprises out of the blue; unlike the stereotyped image (all too often held by lay people) of math as boring, tedious, routine, cut-and-dry....

**http://divisbyzero.com/2010/08/18/mathematical-surprises/**He lists a number of varied mathematical discoveries over the years that surprised the math community and offered new perspectives; his commenters chime in with still more examples.

I think this gets at what mathematicians love about their subject: it is full of never-ending surprises out of the blue; unlike the stereotyped image (all too often held by lay people) of math as boring, tedious, routine, cut-and-dry....

### Book Re-View...

...For starters, let me assure readers (as many will be happy to hear) that I promise today's first post will contain absolutely no mention at all... of P vs. NP! ;-)

Instead, with so many popular math books appearing in bookstores, I'll take a moment, every month-or-two, to re-mention some of the volumes I've given a positive reference to on the blog recently. Here are some from the last several weeks, in no particular order:

"

"

"

"

"

"

"

(p.s. --- I make no promises whatsoever regarding the content of any second posts I might do later today!)

Instead, with so many popular math books appearing in bookstores, I'll take a moment, every month-or-two, to re-mention some of the volumes I've given a positive reference to on the blog recently. Here are some from the last several weeks, in no particular order:

**"The Pythagorean Theorem**" by Alfred Posamentier"

**The Number Mysteries**" by Marcus du Sautoy"

**Here's Looking At Euclid**" by Alex Bellos"

**The Calculus Diaries**" (due out soon) by Jennifer Ouelette"

**Everything and More**" (2003) by David Foster Wallace (I've heard there will be a new revised edition this year in memory of the author who died in 2008)"

**101 Things Everyone Should Know About Math**" by Marc Lev*et. al.***"PopCo**" (novel, 2004) by Scarlett Thomas**"Hot X: Algebra Exposed**" by Danica McKellar"

**The Number Sense**" (1997) by Stanislas Dehaene"

**Proofiness: The Dark Art of Mathematical Deception**" (due out soon) by Charles Seife(p.s. --- I make no promises whatsoever regarding the content of any second posts I might do later today!)

## Tuesday, August 17, 2010

### MIT Math Courses (Free!) & Still MORE P vs. NP

Interested in a math course? Why not go with one of the best!... Massachusetts Institute of Technology shares many mathematics courses in some form online for free (...one of my favorite words!) under their "open courseware" program:

And hope readers don't find these links I keep providing too redundant, but RJ Lipton has a new post nicely recapitulating the interesting events of the first week of the P ≠ NP proof announcement:

Meanwhile, the

For any who still don't 'get' why so much attention is being lavished on this abstruse problem, I'll just note that Scott Aaronson sums it up as the

And for those who find audio-visual material easier to follow than written words, there are several P vs. NP video presentations on the Web. Of those I've briefly looked at so far, I think the one below is particularly good/helpful (hour-long talk by Michael Sipser at the Clay Institute For Mathematics). I recommend it to anyone interested in, but still having difficulty grasping, the essence of the discussion:

**http://ocw.mit.edu/courses/mathematics/**And hope readers don't find these links I keep providing too redundant, but RJ Lipton has a new post nicely recapitulating the interesting events of the first week of the P ≠ NP proof announcement:

http://tinyurl.com/25rbpmuhttp://tinyurl.com/25rbpmu

Meanwhile, the

**NY Times**covers the story**HERE.**For any who still don't 'get' why so much attention is being lavished on this abstruse problem, I'll just note that Scott Aaronson sums it up as the

**"the biggest unsolved problem of theoretical computer science, and one of the deepest questions ever asked by human beings!"**And for those who find audio-visual material easier to follow than written words, there are several P vs. NP video presentations on the Web. Of those I've briefly looked at so far, I think the one below is particularly good/helpful (hour-long talk by Michael Sipser at the Clay Institute For Mathematics). I recommend it to anyone interested in, but still having difficulty grasping, the essence of the discussion:

**http://claymath.msri.org/sipser2006.mov**## Monday, August 16, 2010

### Better Than Making License Plates...

Just another Magic Square (compiled by a prisoner)... that's a tad bigger than most:

(I don't have time to myself, but anyone else please feel free to verify that everything the post claims is true for it is!? ;-))

**http://www.futilitycloset.com/2010/08/14/time-well-spent/**(I don't have time to myself, but anyone else please feel free to verify that everything the post claims is true for it is!? ;-))

### A Sequence Playground... +more P vs. NP

An encyclopedia of mathematical sequences... why not! "Math Trek" blog reports on an AT&T mathematician, Neil Sloane, who has been collecting/cataloguing math sequences for much of his life... to the joy and wonderment of various number-obsessives everywhere. As the post says, "he now has nearly 200,000 number sequences in a searchable online database [the OEIS --- Online Encyclopedia of Integer Sequences], and his personal obsession has become a treasure for the entire mathematical community":

Meanwhile, RJ Lipton's latest post covering problems with the recent

...It's all Greek to me (and I suspect to most of my readers)... but again, what is fascinating is just how much thoughtful/insightful discussion of the matter has already been generated at just the one-week-old stage!

...and here is Scott Aaronson's attempt (with lots of good additional links) to translate the discussion from Greek to say, Latin, for those of us who have wandered 'into the tent' a bit unprepared:

**http://www.sciencenews.org/view/generic/id/61870/title/Math_Trek__The_pattern_collector**Meanwhile, RJ Lipton's latest post covering problems with the recent

**P ≠ NP**proof at the one-week point is up:**http://rjlipton.wordpress.com/2010/08/15/the-p%E2%89%A0np-proof-is-one-week-old/**...It's all Greek to me (and I suspect to most of my readers)... but again, what is fascinating is just how much thoughtful/insightful discussion of the matter has already been generated at just the one-week-old stage!

**Congratulations to all involved! This is something the Web is ideally suited for.**...and here is Scott Aaronson's attempt (with lots of good additional links) to translate the discussion from Greek to say, Latin, for those of us who have wandered 'into the tent' a bit unprepared:

**http://scottaaronson.com/blog/?p=459**## Sunday, August 15, 2010

### New High-Scoring Scrabble Word?

I sincerely hope that none of my readers suffer from "hexakosioihexekontahexaphobia"... or for that matter any of the other maladies mentioned herein:

**http://tinyurl.com/38n98zl**### "Math Brats" and Texas Hold 'em

Even if math usually bores you, this article from

**Time Magazine**about math whizzes using applied math and Internet experience to win big-time card tournaments and change the face of poker, ought to prove interesting; turning a game of chance into a game of stricter probabilities:**http://www.time.com/time/magazine/article/0,9171,1997467-1,00.html**## Saturday, August 14, 2010

## Friday, August 13, 2010

### The Web Engaged... (P vs. NP)

Almost feel like I should apologize for expending so much attention on a matter the majority of us can't even comprehend, but recognizing the potential significance of any Millennium Problem being solved, I can't help myself!... Even not grasping the mathematics involved, just watching this polymath, collaborative, open-source endeavor unfold on the Web (to validate or refute Deolalikar's 'P vs. NP' paper) is fascinating stuff to behold, and likely treads new ground for similar efforts in the future.

If you haven't already seen them, Scott Aaronson has 2 great posts at his blog on the matter (with boatloads of comments as well):

Moreover, an ongoing wiki is set up specifically for further (varied and serious) discussion here:

I presume RJ Lipton will continue to pass along abbreviated, fairly reader-friendly updates of news on the subject at his blog as well:

....and now, if one of you blokes (in your spare time) could just puhh-leeeze turn in a proof of the Riemann Hypothesis ;-)

If you haven't already seen them, Scott Aaronson has 2 great posts at his blog on the matter (with boatloads of comments as well):

**http://scottaaronson.com/blog/?p=456****http://scottaaronson.com/blog/?p=457**Moreover, an ongoing wiki is set up specifically for further (varied and serious) discussion here:

**http://michaelnielsen.org/polymath1/index.php?title=Deolalikar%27s_P!%3DNP_paper**I presume RJ Lipton will continue to pass along abbreviated, fairly reader-friendly updates of news on the subject at his blog as well:

**http://rjlipton.wordpress.com/**....and now, if one of you blokes (in your spare time) could just puhh-leeeze turn in a proof of the Riemann Hypothesis ;-)

### Quote... Unquote

First, once again, thanks to RJ Lipton for another update on Deolalikar's

This will be well beyond the comprehension of most of us... which simply makes it all the more fascinating that there are those who do grasp the reasoning communicated!

On to other things... Just a few miscellaneous quotes to ponder for today, as food for thought!:

"[Alain] Connes thinks that expert mathematicians are endowed with a clairvoyance, a flair, a special instinct comparable to the musician's fine-tuned ear or to the wine taster's experienced palate that enables them to directly perceive mathematical objects: 'The evolution of our perception of mathematical reality causes a new sense to develop, which gives us access to a reality that is neither visual, nor auditory, but something else altogether'."

"The sciences have developed in an order the reverse of what might have been expected. What was most remote from ourselves was first brought under the domain of law, and then, gradually, what was nearer: first the heavens, next the earth, then animal and vegetable life, then the human body, and last of all (as yet very imperfectly) the human mind."

"Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings."

"It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class."

"All generalizations are false, including this one."

**P ≠ NP**proof, in which Neil Immerman expresses*major*concerns (2 major, fatal?, flaws):**http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/**This will be well beyond the comprehension of most of us... which simply makes it all the more fascinating that there are those who do grasp the reasoning communicated!

On to other things... Just a few miscellaneous quotes to ponder for today, as food for thought!:

"[Alain] Connes thinks that expert mathematicians are endowed with a clairvoyance, a flair, a special instinct comparable to the musician's fine-tuned ear or to the wine taster's experienced palate that enables them to directly perceive mathematical objects: 'The evolution of our perception of mathematical reality causes a new sense to develop, which gives us access to a reality that is neither visual, nor auditory, but something else altogether'."

*--- from "The Number Sense" by Stanislas Dehaene*"The sciences have developed in an order the reverse of what might have been expected. What was most remote from ourselves was first brought under the domain of law, and then, gradually, what was nearer: first the heavens, next the earth, then animal and vegetable life, then the human body, and last of all (as yet very imperfectly) the human mind."

*--- Bertrand Russell*"Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings."

*--- Alfred North Whitehead*"It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class."

*--- Author Unknown*"All generalizations are false, including this one."

*--- Mark Twain*## Thursday, August 12, 2010

### 'Number Sense'

**SquareCircleZ**blog has a nice post up on "Number Sense," the subject and title of a 1999 book by psychologist Stanislas Dehaene that I too would recommend about the innate "sense" of numbers/mathematics we (and even some animals to a limited extent) seem to be born with (also, Keith Devlin covers some of the same subject matter in his 2000 volume "The Math Gene"):

**http://www.squarecirclez.com/blog/number-sense/4957**

...interesting reading for the scientist/mathematician and non-scientist alike.

## Wednesday, August 11, 2010

### P vs. NP Issues...

For those with the cerebral math stamina to follow it, RJ Lipton has been monitoring some of the initial critical issues emerging with the recent

(...not easy to comprehend, but still in its own way, fascinating stuff.)

...also worth reading, the many (and varied) comments to Mark Chu-Carroll's initial post on the subject:

The proof's author,

**P ≠ NP**proof (several key mathematicians having doubts that the proof, as given, will stand):**http://rjlipton.wordpress.com/2010/08/09/issues-in-the-proof-that-p%E2%89%A0np/****http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%E2%89%A0np/**(...not easy to comprehend, but still in its own way, fascinating stuff.)

...also worth reading, the many (and varied) comments to Mark Chu-Carroll's initial post on the subject:

**http://scientopia.org/blogs/goodmath/2010/08/09/holy-freaking-cow-p-np/#comments**The proof's author,

**Deolalikar, is busily working on answers/clarifications to questions raised, and likely there will be much more commentary in the next week. Unless an absolutely fatal flaw is detected early-on, the entire process of examination will no doubt be lengthy. (Since this is well-beyond my pay-grade or IQ :-(, not sure just how closely I'll continue to report on it here, though anything regarding Millennium Problem solutions can be interesting stuff, especially when a lesser-known mathematician is involved).***another new update from Lipton***ADDENDUM:****HERE**.### This Sentence Is False...

List:

1. This sentence contains eight words.

2. This sentence contains five words.

3. Exactly one sentence in this list is true.

2. This sentence contains five words.

3. Exactly one sentence in this list is true.

### And ANOTHER Proof

Image via Wikipedia

Alex Bellos has a post up about a new proof that Rubik's Cube can be solved from

*any*starting position in 20 moves or less (known in some circles as "God's number" because it was considered so difficult to calculate that only God could know it --- there are around 43,000,000,000,000,000,000 different possible positions for Rubik's Cube, so you get some sense of the task). In the post, Bellos interviews one of the collaborators who solved the puzzle:

**http://alexbellos.com/?p=1327**

## Tuesday, August 10, 2010

### Factoid...

From the 'Who-the-heck-has-the-time-to-think-this-stuff-up' Dept.

(...no, actually it's from a recent 'MAA NumberADay' blog post):

(...no, actually it's from a recent 'MAA NumberADay' blog post):

**"819**is the largest three-digit number whose square (670761) and cube (549353259) use different digits."

Sure, I knew that....

### 'nuther Book Upcoming

Boy, sometimes I wanna start a separate blog (...but I won't) just to more fully cover all the popular math trade books that keep appearing (...given the bad rap math gets from so many students as their most hated school subject, it's a wonder there's such an audience for popularized math writing!).

One of my favorite science writers, Charles Seife, has an upcoming new volume focused on numbers and statistics (especially in politics), "

**Proofiness: The Dark Art of Mathematical Deception**" (gotta love it, for the title alone!). A couple of quickie takes on it here:

**http://www.worldsciencefestival.com/blog/85_stats**

**http://tinyurl.com/37lktga**

(And perhaps I should quickly add that the "proofiness" of which Seife writes has

*nothing*whatsoever to do with the 'proof' which has been under much discussion in math circles over the last 48 hours.)

## Monday, August 9, 2010

### P vs. NP cont'd...

Mark Chu-Carroll over at "

The just-released 'proof' is already creating a tremendous buzz in the mathworld, as it gets analyzed by those who understand the abstract depths and intricacies of the problem. It's commonplace for glitches or shortcomings to be found in such complex proofs that require fixes or work-arounds, but will there be a fatal flaw??? If not, the Millennium Prize Problems will be down, in due time, to just five!

**Good Math, Bad Math**" blog takes a stab at explaining the basics of the 'P vs. NP' problem and its significance to us math also-rans:**http://scientopia.org/blogs/goodmath/2010/08/09/holy-freaking-cow-p-np/#more-952**The just-released 'proof' is already creating a tremendous buzz in the mathworld, as it gets analyzed by those who understand the abstract depths and intricacies of the problem. It's commonplace for glitches or shortcomings to be found in such complex proofs that require fixes or work-arounds, but will there be a fatal flaw??? If not, the Millennium Prize Problems will be down, in due time, to just five!

*and here's another decent attempt, from M.I.T. this time, to explain P vs. NP to us layfolk:*

**ADDENDUM:****http://web.mit.edu/newsoffice/2009/explainer-pnp.html***just an interesting quotation from Keith Devlin's opening to his chapter on the P vs. NP problem in his 2002 book, "***and a 2nd ADDENDUM!:****The Millennium Problems**":"Of all the Millennium Problems, the P versus NP puzzle is the one most likely to be solved by an 'unknown amateur' --- someone largely untrained in mathematics, possibly someone very young, who is unknown to the mathematical community. All the other Millennium Problems are buried deep within a mass of heavy-duty mathematics, which has to be mastered before you can begin working on the problem itself. This is not the case for the P versus NP problem, which deals with how efficiently computers can perform certain kinds of tasks. Not only is it relatively easy to understand what the problem says, it is possible that all it will take to solve it is one good new idea. And you don't need lots of knowledge to have a good idea, just imagination."

### Math Instruction From Winnie Cooper

Actress (and child star of "The Wonder Years," where she played 'Winnie Cooper') Danica McKellar is out with her third volume aimed at encouraging middle-school-aged girls to pursue math, "

**Hot X: Algebra Exposed**." Her first two successful books were "

**Math Doesn't Suck**" and "

**Kiss My Math,**" speaking to middle-schoolers in their own language. Danica went to UCLA where she majored in mathematics, all the while maintaining her actress credentials.

Hoping in due time she'll be bringing us volumes on Trig and Calculus!!

## Sunday, August 8, 2010

### P vs. NP

WOW!! the flash on Twitter (and Slashdot) tonight that one of the complex Clay Institute Millennium problems, "P vs. NP" (worth a million dollars to solve, and especially of interest to computer scientists), is being claimed proved by an HP Labs researcher as

Story originally sourced here:

Another early report on the news

100-page proof likely to take months or more to be verified.

**P ≠ NP.**He notes his solo effort involved "uncovering a chain of conceptual links between various fields and viewing them through a common lens," as might be expected.Story originally sourced here:

**http://gregbaker.ca/blog/2010/08/07/p-n-np/**Another early report on the news

**HERE.**100-page proof likely to take months or more to be verified.

### Daniel Tammet Has His Work Cut Out For Him Now ;-)

Japanese and US computer scientists claim to have calculated Pi to

*five trillion*decimal places... just a few trillion more than the previous record! (...but I'll save some space here by

*not*typing it out for you):

**http://pda.physorg.com/trilliondecimal-decimalplaces-computer_news200209829.html**

[Daniel Tammet, BTW, is the autistic savant famous for having recited Pi accurately from memory to over 22,500 digits back in 2004.]

### Students Will Be Thrilled to Learn...

...about Scott McNealy's (formerly of Sun Microsystems' fame) latest enterprise as reported on "Mathematics Under the Microscope" blog:

**http://micromath.wordpress.com/2010/08/07/200-textbook-vs-free-you-do-the-math/**

(the post is actually a full quote of the original

**NY Times**' article covering same subject)

His goal isn't simply to bring down the forever skyrocketing cost of textbooks... but to make them available on the Web for

*free*, currently through an internet-based non-profit called "Curriki"... Music to any student's ears!!! (So far as I can tell though only high-school-and-below textbooks are involved at this point, not college texts.)

## Saturday, August 7, 2010

### Free Math Film

From the Equalis Math Community site I just learned of the math film entitled "

...enjoy!

**Dimensions**" (originally released in 2008) they recommend as a sort of mathematical tour. It is available in various forms, including online for free here (in 9 parts):**http://dimensions-math.org/Dim_regarder_E.htm**...enjoy!

### 68th Carnival of Mathematics and...

The latest edition of the blogging "Carnival of Mathematics" is up for your reading perusal and pleasure here:

...and, again over at "Wild About Math" blog an announcement of the new "Equalis Math (Web) Community" which looks to be offering a lot of rich content and resources, as well as several good blogs for those interested in, as-they-say, "math-centric endeavors

**http://plus.maths.org/content/carnival**...and, again over at "Wild About Math" blog an announcement of the new "Equalis Math (Web) Community" which looks to be offering a lot of rich content and resources, as well as several good blogs for those interested in, as-they-say, "math-centric endeavors

**"**(looks very promising!):

**http://tinyurl.com/3yvwkd6**## Friday, August 6, 2010

### Another Puzzle That Keeps On Giving

The 'boy born on Tues." problem has been gaining the same sort of traction and attention formerly accorded to the "Monty Hall" problem; both involve issues of probability which is not only one of the trickiest realms of math for lay people to understand, but indeed one of the trickiest areas for mathematicians themselves; and the 'boy born on Tues.' riddle is trickier than its 'Monty Hall' predecessor. "Division By Zero" blog has done another post on the problem, interestingly playing around a bit more with the variables:

If you haven't yet had your fill of this puzzle check it out. It was also interesting to learn from the post that this problem (in its usual form) originated with a 3-sentence presentation at the last "Gathering For Gardner," just a few months back.

**http://divisbyzero.com/2010/08/05/the-left-handed-boy-problem/**If you haven't yet had your fill of this puzzle check it out. It was also interesting to learn from the post that this problem (in its usual form) originated with a 3-sentence presentation at the last "Gathering For Gardner," just a few months back.

### Practical Math and Novel Math

Books... Math Books, That Is...

Sol over at 'Wild About Math' blog just reviewed a very practical-sounding book I haven't come across previously: "

Looks like a good book for anyone interested in math... or anyone who

And from the practical to the fictitious... Not into novels much myself, but for those who are, one volume of mathematically-inclined fiction that Martin Gardner (and others) have reviewed very favorably was a novel of intrigue entitled "

a Web review here:

At the end of his own review of the work, Martin Gardner originally wrote, "Scarlett Thomas writes beautifully. Her novel is not easy to put down." ...Could there be a better endorsement?!

Sol over at 'Wild About Math' blog just reviewed a very practical-sounding book I haven't come across previously: "

**101 Things Everyone Should Know About Math**." See his positive review here:**http://tinyurl.com/26v6yp3**Looks like a good book for anyone interested in math... or anyone who

*should*be interested in math! (actually, the publisher's target audience looks to be a young, middle-school-ish crowd... but hey, we were all young once ;-)).And from the practical to the fictitious... Not into novels much myself, but for those who are, one volume of mathematically-inclined fiction that Martin Gardner (and others) have reviewed very favorably was a novel of intrigue entitled "

**PopCo**" by Scarlett Thomas from 2004, containing many mathematical allusions.a Web review here:

**http://tinyurl.com/239kq5y**At the end of his own review of the work, Martin Gardner originally wrote, "Scarlett Thomas writes beautifully. Her novel is not easy to put down." ...Could there be a better endorsement?!

## Thursday, August 5, 2010

### Triangle Riddle

Triangle ABC has AB side-length = 5, BC side-length = 6, and AC side-length = 7. Two snails start at point A and crawl in opposite directions at equal speeds along the triangle 'til they meet at point D. What is the length of BD?

. answer below

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answer: 4

. answer below

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answer: 4

### More Polymath...

I've previously referenced Tim Gowers' collaborative Polymath Project (using the power of many to solve difficult math problems), and here's another nice article on it from

The polymath blog is here:

And here's a more general wiki-page of various polymath projects:

Lastly, in a related note, this recent interesting

**Scientific American**:**http://www.scientificamerican.com/article.cfm?id=problem-solved-tic-tac-toe-blog**The polymath blog is here:

**http://polymathprojects.org/**And here's a more general wiki-page of various polymath projects:

**http://michaelnielsen.org/polymath1/index.php?title=Main_Page**Lastly, in a related note, this recent interesting

**NY Times**story reports how (non-scientist) video gamers have contributed to the understanding of complex biological protein-folding through the competitive (and collaborative) use of a video game created for such purposes:**http://tinyurl.com/2gx3hw4**## Wednesday, August 4, 2010

### 'Synchronization'

This demonstration in (metronome) "synchronization" is supposed to be a physics demonstration, but of course what underlies physics, but mathematics (and in this case, signal transference):

### Circumference Classic

I've adapted this classic anti-intuitive puzzle from one of Clifford Pickover's renditions of it:

Imagine a rope tightly encircling the 'equator' of a basketball. How much longer would you have to make the rope for it to now be (when stretched out as a circle) one foot from the surface at all points?

Now, imagine a rope similarly around the equator of an

The AMAZING answer is 2π (or approximately 6.28) additional feet for

Imagine a rope tightly encircling the 'equator' of a basketball. How much longer would you have to make the rope for it to now be (when stretched out as a circle) one foot from the surface at all points?

Now, imagine a rope similarly around the equator of an

*Earth-sized*sphere (around 25,000 miles long). How much longer would you now have to make that rope for it also to be one foot off the ground all the way around the equator?The AMAZING answer is 2π (or approximately 6.28) additional feet for

**the basketball and the Earth! If r is the radius of the Earth (in ft.), and 1 + r is therefore the radius in feet of the enlarged circle, we need only compare the rope circumference (length) before (2π r) and after [2π (1 + r)] (and ditto for the basketball!)***BOTH*## Tuesday, August 3, 2010

### 4 is....

Factoid: In English, "4" is the

**number with the same number of letters in it as the number it represents.***only*### "What Is Mathematics?"

An ongoing debate:

A discussion (actually review of a John Barrow book) centering around the long-time debate over whether mathematics objectively exists in reality or is merely a creation of the human mind.

...And plenty more food-for-thought here:

Nice article at Wikipedia giving overview of "philosophy of mathematics;" includes many links:

...and similar "Foundations of Mathematics" article also at Wikipedia:

Relatedly, possibly worth mentioning a 2009 book from Jeremy Gray "

an online review here:

**http://members.cox.net/mathmistakes/what_is_mathematics1.htm**A discussion (actually review of a John Barrow book) centering around the long-time debate over whether mathematics objectively exists in reality or is merely a creation of the human mind.

...And plenty more food-for-thought here:

Nice article at Wikipedia giving overview of "philosophy of mathematics;" includes many links:

**http://en.wikipedia.org/wiki/Philosophy_of_mathematics**...and similar "Foundations of Mathematics" article also at Wikipedia:

**http://en.wikipedia.org/wiki/Foundations_of_mathematics**Relatedly, possibly worth mentioning a 2009 book from Jeremy Gray "

**Plato's Ghost: The Modernist Transformation of Mathematics**," which narrates the evolution of mathematical thought from 1880 to the 1920s:an online review here:

**http://www.americanscientist.org/bookshelf/pub/modernism-in-mathematics**## Monday, August 2, 2010

### A Tisket, A Tasket, Another Fractal Gasket

Math

*IZ*a beautiful thang, and perhaps geometry is the most beautiful part of math, with circles just possibly the most beautiful part of geometry, in which case the fractal "Apollonian gasket' 'tis a very beautiful thang indeed!:

**http://en.wikipedia.org/wiki/Apollonian_gasket**

**(P.S... I now have a CORRECTION up on the "Coffee/Tea Mixup" post from yesterday, you may wish to check out!)**### Musings of a Polymath

"The Nexus of Wonder".... an interview with prolific writer/thinker/math-elucidator Clifford Pickover, from awhile back:

**http://sherryaustin.livejournal.com/3348.html**## Sunday, August 1, 2010

### Coffee/Tea Mixup

Richard Wiseman offered up another nice little 'Friday Puzzle' for the weekend, over at his blog:

For those of us on this side of the pond, I'd substitute 50 cents for his 50 pence, in which case I've given the answer below (I'll let him explain the reasoning on Monday ---:

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answer:~~$1.00~~ 50 cents!

**http://tinyurl.com/39f59ej**For those of us on this side of the pond, I'd substitute 50 cents for his 50 pence, in which case I've given the answer below (I'll let him explain the reasoning on Monday ---

*there's actually some slight ambiguity in the problem as stated, and depending on the potential interpretation my given answer below would be different*)

*ADDENDUM:*I originally stated a WRONG answer below (now corrected) --- Wiseman's explanation here:**http://tinyurl.com/22k4mpv**.

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answer:

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