Friday Puzzle

For the probability buffs out there, here's a tricky problem from another site:

What is the probability that in a group of 31 people, NONE of them have birthdays in February or August? (assume equal probability of birthday for all days throughout the year)
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the answer (I presume correct), 0.41%, is computed here:
http://tinyurl.com/7h3nh23

Delightful Math

I've seen this particular puzzle presented many times in the form of a typical word or story problem, but don't recall if I'd seen it rendered in poetical form before:

http://tinyurl.com/7j4dlcs

6 and Primes

It is an interesting fact that ALL prime numbers beyond 2 and 3 are either 1 more, or 1 less, than a multiple of 6. If you were unaware of this simple factoid you can check out these links for a discussion/explanation of it:

a short explanation: http://www.jimloy.com/number/prime6.htm

a longer explanation: http://tinyurl.com/77l94cc

Quickie Friday Puzzle

Just a quickie on this day of Festivus! ;-)

How many Santa's helpers must there be total in a group, if all but 2 are named Smith, all but 2 are named Jones, and all but 2 are named Wilson?
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answer: 3 total (1 Smith, 1 Jones, 1 Wilson)

Year-end Review...

For newer readers here, just a year-end list of some personal favorite posts from the last 12 months. Nothing particularly profound or deep about most of these; just touching on matters I happened to find interesting:

1) A Seemingly Impossible Task That Isn't
(a favorite paradox/puzzle coming from Raymond Smullyan)

2) Of Wheels and Circles and Points, Oh My
(another old mind-bending paradox)

3) Couple of Videos 
(couple of fun YouTube videos)

4) Fermat's Last Theorem
(a video on Andrew Wiles solving Fermat's Last Theorem)

5) Marilyn Rolls the Dice
(Marilyn vos Savant introduces another puzzle)

6) The Cantor Set
(the Cantor Set)

7) A Video Re-run
(A Bloggingheads/TV video with John Horgan and Jim Holt)

8) The Blind Spot
(Review of William Byers' book "The Blind Spot")

9) Kickin' Back With Cantor
(BBC documentary on Cantor)

10) Loser…
(just for fun)

Explaining the Riemann Hypothesis

For anyone who woke up this morning and muttered to themselves, "Hmmm, what is that there Riemann Hypothesis all about anyway?...":

http://www.askamathematician.com/2011/12/q-what-is-the-reimann-hypothesis-why-is-it-so-important/

Are These Folks Humans or Cyborgs!?

Some people's brains are definitely wired very differently from mine....

When I first read of autistic savant Daniel Tammet's recitation of pi to over 22,500 decimals back in 2004, it seemed like a stunt beyond human comprehension, likely not to be surpassed in my lifetime… BOY, WAS I WRONG!!
It was a couple years later that I learned that Tammet's effort merely set a British and European record for pi recitation, but wasn't even close to the world record. As far back as 1981, an Indian had already recited pi to almost 32,000 digits, and a couple of Japanese blokes later greatly surpassed that endeavor. Then in 2005, a Chinese fellow, Chao Lu, would more than TRIPLE Tammet's puny ;-) performance, by reciting from memory, a phenomenal 67,890 digits of pi, and is still listed as the official record-holder by some accounts. Here's one listing of the world's top pi record-holders:

http://pi-world-ranking-list.com/lists/memo/

I'm not sure how these "official" designations are established though, because the current proclaimed record-holder is a former Japanese engineer, named Akira Haraguchi, who recited pi to 100,000 digits in 2006 (breaking his own prior record of almost 84,000):

http://www.foxnews.com/story/0,2933,217765,00.html

Simply UNNNreal! (...Having said that though, pi has now been computed to 10 trillion digits, so HEY Akira, ya gotta ways to go yet!)

...Now, if only I could remember where I left my car keys last night.

A Little Bit On Randomness

h/t to Dan Meyer for this one:

The Depths and Intuition of Mathematics...

A long, interesting, insightful answer to a simple question posed on Quora.com:

http://www.quora.com/What-is-it-like-to-have-an-understanding-of-very-advanced-mathematics

Booklist

One blogger's list of 46 "interesting" math books from 2011 here:

http://tinyurl.com/8xmgf83

A variety of subject matter and material level is represented.

Friday Puzzle

A recent geometry puzzle from 'Futility Closet' here (with a possibly surprising answer):

http://www.futilitycloset.com/2011/11/28/rolling/

Carny Time!

Someone apparently submitted one of my year-old posts to the current Math Teachers At Play blog carnival (thank you to whoever you are!). The whole diverse Carnival is worth a look here:

http://letsplaymath.net/2011/02/18/math-teachers-at-play-35/

Infinite Trees Finitely Discussed

Keith Devlin takes on infinity via mathematical trees in this recent blog post:

http://devlinsangle.blogspot.com/2011/12/christmas-trees-from-land-of-santa.html

Of Higgs and Sigmas

What are the confidence intervals of confidence intervals, and other thoughts....

With a significant announcement from CERN on the Higgs boson presumed to be just hours away, KW Regan enters the fray with this interesting post about science, sigmas, and levels of significance:

http://rjlipton.wordpress.com/2011/12/13/the-higgs-confidence-game/

…an excerpt therefrom:

"The high rate of disappearing significance overall is still puzzling. Its extent in human sciences was detailed exactly one year ago in a disturbing article by Jonah Lehrer for The New Yorker. If 250 researchers try the same experiment, one would expect 2 or so of them to get  deviations (in either direction). The world will then see 2 or so published papers from them, but nary a peep from the 248 who failed and gave up quickly and forgot about it. Thus a significant result may appear independently confirmed when it was actually just by chance, and those failing to reproduce it will then peep up loudly. The effect is equally pernicious with 250 different experiments, especially given a fair chance of a lower-confidence positive from a test deemed related enough to corroborate the original."

Holiday Shopping For All Ye Merry Math Fans

Order early & order often ;-)

http://www.zazzle.com/sheckyr/gifts

Richard Elwes... Again

I've been aware of Brit Richard Elwes for barely over a year now, but he's already vaulted to one of my favorite math expounders. An interview with him here from Q Blog (and great to see that he has a new volume out, "The Maths Handbook," and yet another on the way! ....his first two books are two of my favorites):

http://www.quercusbooks.co.uk/blog/2011/12/07/the-quercus-couch-richard-elwes/

Another Friday Puzzle

Update: I had quickly adapted the original problem below from another site, only to realize it was far too easy as a problem of 2 equations with 2 unknowns, so have now re-stated it below in at least a slightly more challenging form:

"x" and "y" are two different 2-digit prime numbers composed of the same digits, but in reverse order. If it is true that x - y = 36, then what must the numbers x and y be:
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answer:  x=73 and y=37

Game Show Math

...or, why doing math in your head with a bazillion eyes upon you is no picnic!
(h/t to Presh Talwalkar for this)

Halting Problem In Verse

The Halting Problem, Dr. Seuss, poetry, and Geoffrey Pullum here... ohh my:

http://ebiquity.umbc.edu/blogger/2008/01/19/how-dr-suess-would-prove-the-halting-problem-undecidable/

quite cool!

Phi

The stream of popular books out on pi, the Fibonacci sequence, or the Golden Ratio seems unending… and, now we have another: Alfred Posamentier and Ingmar Lehmann are newly out with "The Glorious Golden Ratio" -- haven't read it, but for geometry fans, anything by these two is certainly worth some high anticipation!

Friday Puzzle

I adapted this problem directly from a recent NPR Car Talk puzzler, so if you heard that show you'll know the answer:

My brother Morty and I took a drive up to the mountains this weekend to see the Golden-breasted Flooglebird. Morty drove the first 40 miles and I drove the rest of the way. We saw the Flooglebird right away (or something closely resembling it) and immediately returned home on the exact same route.

On the return trip Morty again drove the first part of the trip, and I drove the final 50 miles this time.
In total who did the most driving, me or Morty, and how many more miles than the other brother did that driver do? (yes, there's enough information to reach an answer!)

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answer: I drove 20 more miles than Morty
(the Car Talk guys explain it their way here: http://www.cartalk.com/content/car-trip-trigonometry-0?answer )

Today's Statistics Lesson...

 From Professor Dilbert:

http://tinyurl.com/bowyu86

A Few Books

Haven't mentioned any books for awhile, so will just briefly cite 3 that have been around for a little while now, though I've only read the last one:

"The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy," Sharon Bertsch McGrayne's volume on Thomas Bayes and the statistical theory he formulated has received mostly favorable reviews, including this one from no less than The Lancet:

http://www.thelancet.com/journals/lancet/article/PIIS0140-6736%2811%2961728-5/fulltext

A book I received a review copy of but never got around to reading, is "Magical Mathematics" by Persi Diaconis and Ron Graham, recounting the mathematics that underlies a lot of magic tricks. Even though it didn't engage me, I won't argue with success, as it has received almost unanimous thumbs-ups in reviews I've seen, including this one from CTK Insights:

http://www.mathteacherctk.com/blog/2011/10/magical-mathematics/

(...As a sidenote, speaking of magic, a book I did love, but on neuroscience not mathematics is now out in paperback: "Sleights of Mind" by Macknick and Martinez-Conde.)

Finally, the one review volume I have had a chance to peruse, is this year's issue of "The Best Writing On Mathematics 2011," edited once again by Mircea Pitici. I almost feel I could duplicate the short review I gave of last year's initial edition for the current volume. It is again a fine compendium of great variety. If anything, I would say it has a bit more philosophical reading and other material that a layperson can readily follow (while still also including several quite technical entries). And again, whatever your own proclivities in math it will contain some contributions of interest and others probably not-so-much, but all-in-all a very worthwhile volume. I look forward to this series continuing well into the future.

Thanksgiving

no math today….



"If the only prayer you ever say in your whole life is 'thank you,' that would suffice."
-- Meister Eckhart

Of Numbers and Colors...

A recent medical study looks at synesthesia (the cognitive blending of sensory information) and numbers, and the differences in cortical activity between "synesthetes" and others:

http://www.medicalnewstoday.com/releases/237908.php

Well-known autistic savant Daniel Tammet is famous for his synesthetic experience. He expounds upon what it all means in this TEDTalk from earlier this year:



Tammet is also famous for his depiction of how he visualizes the number pi in color (a number he has recited from memory to over 22,500 digits -- no, that's NOT a misprint):

 

see here: http://www.newscientist.com/gallery/mg20227111500-savant-art

The "Dr Square Puzzle"

Visit the "math4love" blog to see a fun, interesting, iterative-type math problem, inspired by a 3rd grader!:

http://mathforlove.com/2011/11/the-dr-square-puzzle/

Food For Thought? (math instruction)

This TEDtalk by John Bennett seems a tad overly pedantic, but I do find the endpoint (last couple minutes) regarding deductive and inductive reasoning interesting, as well as his general take that middle and high school math ought NOT be mandatory for students:

Numberphile

YouTube is introducing a new channel for math buffs, called "Numberphile":

http://www.youtube.com/user/numberphile

 (...not too much up on it yet, but worth keeping an eye on; I'm adding a link to right-hand column under "Misc. Resources")

Thinking vs. Rote Knowing

Assessing a 14-yr.-old math whiz for entrance to Cambridge (...and a nice thought puzzle):

http://www.bbc.co.uk/news/magazine-15574889

"The Unplanned Impact of Mathematics"

"Time and again, pure mathematics displays an astonishing quality. A piece of mathematics is developed (or discovered) by a mathematician who is, often, following his or her curiosity without a plan for meeting some identified need or application. Then, later, perhaps decades or centuries later, this mathematics fits perfectly into some need or application."

…and Peter Rowlett looked for (and talked about) more examples of such here:

http://travelsinamathematicalworld.blogspot.com/2011/07/unplanned-impact-of-mathematics.html

Of Love and Fourier Transforms...

Jennifer Ouellette's ode to mathematics... and love (...and some physics guy) here:

http://blogs.scientificamerican.com/cocktail-party-physics/2011/09/29/love-among-the-equations/

an excerpt:

"It turns out that the world is filled with hidden connections, recurring patterns, and intricate details that can only be seen through math-colored glasses. Those abstract symbols hold meaning.  How could I ever have thought it was irrelevant?

This is what I have learned from loving a physicist. Real math isn’t some cold, dead set of rules to be memorized and blindly followed. The act of devising a calculus problem from your observations of the world around you – and then solving it – is as much a creative endeavor as writing a novel or composing a symphony."

Math Documentaries Available

In case you have a lot of free time to fill in, a nice listing of math documentaries that look very good (mostly from BBC) available on the Web here:

http://www.cosmolearning.com/mathematics/documentaries/

(and I've inserted a permament link to same in right-hand column under "Misc. Resources.")

More of What's-Math-Good-For

Here's a list of 15 diverse folks who have, or may have, used their geeky math skills to gain some lucrative money-making advantage:

http://tinyurl.com/6koc8wp

(hat tip to Steven Colyer)

Good Math, Not-So-Good Math

Mark Chu-Carroll over at 'Good Math Bad Math' has never suffered math cranks very well (...and he gets his share of them):

http://scientopia.org/blogs/goodmath/2011/11/05/yet-another-cantor-crank/

For more crankish entertainment you can visit here:

http://www.crank.net/maths.html

83rd Carnival of Mathematics

The latest Carnival of Mathematics is up-and-running here:

http://teachbesideme.blogspot.com/2011/11/cafrnival-of-mathematics-83.html

The Title of This Post is Recursive

Keith Devlin waxes not-quite-poetic on "The Recursion Principle" and its underlying importance to mathematics:

http://devlinsangle.blogspot.com/2011/11/how-multiplication-is-really-defined-in.html

From the posting:

"Though recursion is ubiquitous in modern mathematics, even at the most basic level of the analysis of the arithmetic of natural numbers, it is a subtle concept, easily misunderstood…
 
"This may all seem like a great deal of fuss about nothing. But what is going on here is really very deep. Much of modern mathematics involves finding ways to handle the infinite - calculus exclusively and spectacularly so. Mathematicians learned over many years of painful lessons that the step from the finite to the infinite is a tricky one that requires considerable finesse. In particular, you have to exercise great care to set it up correctly and do it right. The Recursion Theorem is one of those crucial bridges that allow us to go beyond the finite to the infinite, to extend human intellect from its finite physical limitations to the infinite world beyond that our minds can construct.  By getting the mathematics right, we can make that step with total confidence. Confidence both in that abstract world itself and in the concrete conclusions it allows us to reach about our lives, our science, and our technologies. That is huge for Humankind."

 (...ohh, and by the way, multiplication is NOT just repeated addition. ;-)

Loser....

Enjoy... stumbled upon this in the comments to a Richard Wiseman posting:

'Cold Hits'

Nice treatment of probability, fingerprint analysis, and the birthdays-in-a-room example via a recent John McGowan post:

http://tinyurl.com/6y5ad4o

Obviously, Pi = 4

For any who have never seen it, an oldie-but-goodie puzzle today... simple proof that pi = 4 :

http://i.imgur.com/AzeYc.jpg

From the What-Is-Math-Good-For-Anyway-Dept.

Wow, who knew:  

"Six of the 10 writers of 'The Simpsons' are high-level mathematics PhDs and for years they have been making mathematical jokes to each other in the series, such as writing formulae up on blackboards or on car numberplates. The jokes have developed a cult following among the scientific community."

Simon Singh knew... see:

http://www.thebookseller.com/news/science-book-deals-frankfurt.html

[image via wikimedia commons]

Marilyn Rolls the Dice...

Marilyn Vos Savant, who brought the Monty Hall Problem to public view, stirs the probability pot again with this Sunday magazine problem asking which result is the more likely outcome from 20 throws of a die: a) 11111111111111111111 or b) 66234441536125563152

http://tinyurl.com/3zsb7tf

[Addendum: I've now gotten around to reading many of the comments to the above link, and interestingly, once again Marilyn has opened a can of worms. Her logic/math is correct, yet many misinterpret the problem and once again think she is wrong, as they did originally in the case of the Monty Hall Problem. 
The problem reminds me slightly of Newcomb's Paradox where a notion of 'backwards causation' comes into play to confuse the issues; except that Newcomb's Paradox is essentially unresolved, whereas this dice problem is clearly resolvable.]

Rumbling In Cantor's Paradise

 "No one will drive us from the paradise which Cantor created for us." -- David Hilbert

I wasn't aware there was very much serious controversy over Cantor's proof that the real number set is uncountable (versus the set of integers which is countable), but RJ Lipton is aware of the naysayers out there and takes a stab at reaching them here:

http://tinyurl.com/3mo7mt4

(Not sure Lipton will win over any doubters with his argument, but for most, Cantor probably doesn't even require a defense; at any rate some interesting comments below Lipton's post.)

A Wiseman Puzzle... and Martin

Richard Wiseman's puzzle for today might keep you busy for awhile:

http://richardwiseman.wordpress.com/2011/10/21/its-the-friday-puzzle-132/

(he'll post the answer on Mon. -- answer HERE)

p.s.... Happy Birthday to Martin Gardner today, wherever he is Recreationing In Peace.
(a 4-min. NPR remembrance at the time of his death here)

Four Circles... Really!

An optical illusion for your entertainment and delectation!:

http://i.imgur.com/RYHtQ.jpg

Open Problems...

interesting older post on some "embarrassing" open problems in math:

http://angryfaic.wordpress.com/2011/02/26/embarrassing-open-problems-in-maths/#more-42

"Problem Solving Flowsheet" ;-)

Not exactly math, but problem-solving for all you lab rats... Yesterday, in a lab, I ran across a technical ;-) "problem solving flowsheet" taped to the wall -- when I got home I looked it up on the internet and of course found lots of references to the very same sheet, so I may be among the last to have seen this (don't know how long it's been going around).
In case you've missed it too you can check it out here (adult language):

http://renivstheblog.files.wordpress.com/2011/05/problem-solving-flowsheet.gif

Reuben Hersh's Mathematical Experience...

Reuben Hersh has written often of the history, culture, and deeper nature of mathematics. And he is, famously, a NON-Platonist... one who believes mathematics is more a by-product of the human mind than a real extant part of the physical Universe. One of his classic works (with Philip Davis), that most of you have likely read, was "The Mathematical Experience," which Martin Gardner reviewed quite critically back in 1981. I enjoy reading Hersh, and I've been re-reading this particular volume on its 30th anniversary, but having said that, his writing sometimes seems to skim the surface of the material he is tackling. The content bounces back-and-forth between regurgitation of standard pedagogical material and more interesting, but not always convincing, arguments of deeper philosophy. I'm sometimes reminded of the old Wendy's commercial, "Where's the beef?" in reacting to certain subjects broached in this book that don't seem fleshed out as fully as they deserve (the second half of book though is richer than the first half, and Hersh provides plenty of good references for "further study"). Perhaps I just miss the nuance of Hersh's stance on some matters, but I usually find Gardner's arguments more persuasive and articulate. Having said that, more and more respected mathematicians these days seem to be moving toward the minority non-Platonist stance that Hersh has long expounded, so the debate is hardly settled (...indeed, it is probably more UNsettled than ever!).

I mention all of this only because of recently stumbling across this informal response, I'd not seen before, from Hersh to Gardner's original review, and it makes for interesting reading:

http://cs.nyu.edu/pipermail/fom/1997-November/000128.html

The below page links to an Edge interview where Hersh further spells out his notions:

http://edge.org/3rd_culture/hersh/hersh_p1.html

"The Mathematical Experience" remains a classic mathematical opus, with a great breadth of math subject matter (and there is a newer, updated version which I don't own, so not sure how much it differs from the original), and I'm definitely finding it worth a re-read decades later.

Benford's Law vs. Finance and Accounting Data

uhh-ohhh...

http://econerdfood.blogspot.com/2011/10/benfords-law-and-decreasing-reliability.html

"...overall, the picture looks grim."

Of Hula Hoops and Geometry, and...

http://www.futilitycloset.com/2011/09/22/johnson-circles/

...and in other news, Keith Devlin has a brand-spanking newly-launched blog here (1st post on Steve Jobs):


http://profkeithdevlin.wordpress.com

Coordinated Web Effort Proposed to Solve Riemann Hypothesis

I don't hold out a lot of optimism for this, but what do I know: Indian mathematician organizing online collaborative effort to tackle the Riemann Hypothesis (based on an approach originally proposed by Freeman Dyson highlighting quasicrystals):

http://www.microfinancemonitor.com/2011/10/10/indian-mathematician-raises-storm-over-riemann-riddle/

also, see here:

http://fadereu.posterous.com/knk103-the-crystals-of-mt-zeta

which includes this quote from Dyson: "...if we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible." 
 :-)


[Bernhard Riemann image via Wikimedia Commons]

Math at Science Online 2012 ???

Scientific American's Bora Zivkovic is one of the co-organizers of the premier annual 3-day "Science Online" conference (focusing on science blogging and science communication more generally in the digital age), held in central North Carolina every January -- this year in Raleigh. Attendees come from all over North America as well as internationally. There has always been a strong emphasis on the biological and medical sciences in the multitudinous sessions of this conference, and increasingly the physical sciences are represented as well. Mathematics has been rather less prominent, and Bora recently tweeted "Where's the math?" in regards to proposals for the coming get-together.

If you're a blogger or other math educator/communicator and you've never been to one of these conferences I highly recommend the experience (in fact, I'd defy you to find any individual who's attended that didn't feel richly rewarded by the content, variety, and camaraderie of the meeting -- even if your interests are very narrowly 'mathematical' and not so much 'scientific' you will find very worthwhile, instructive sessions to choose from).

Session suggestions for this coming January (19th-21st) are listed at:

http://bit.ly/hhu4f2

Anyway, Bora is actively soliciting for more math-oriented content; if you have ideas/suggestions that fit into any of the above subject areas (especially if you would like to be a presenter/contributor yourself) contact him SOON at: coturnix@gmail.com

DO note that the conference is actually billed as an "Unconference" and sessions are not intended to be the typical 45-min. Powerpoint lecture format, but rather short presentations that generate active and knowledgeable audience participation/engagement. Everyone (including presenters) goes away learning from others.

If you're on Twitter you can follow the progress of Science Online 2012 at the hashtag #scio12. Even though the conference isn't until January, online planning and conversation about it will be ramping up considerably starting about now. And registration for the conference will likely close (fill up) very shortly after it opens!

Perplexing Art...

If you're a fan of optical illusions or Escher-like artwork you may enjoy this blog-site:

http://impossible-world.blogspot.com/

iMourn

Dreamer, Doer, Visionary, Sage, Wizard, Virtuoso.....

           
               (1955 - 2011)

THANKS for all the MAGIC!!!

[“Any sufficiently developed technology is indistinguishable from magic.” -- Arthur C. Clarke]

(Jobs' 2005 commencement address at Stanford HERE)

(David Pogue's tribute HERE)

"Paraconsistent Mathematics"

As a bit of followup to yesterday's post I just discovered this couple-month-old article from plus.maths.org on "paraconsistent mathematics," which addresses the sort of 'truth' issues of statements raised yesterday, and allows for certain 'logical' contradictions:

http://plus.maths.org/content/not-carrot

Just Some Classic Hofstadter...

From chapter 17 of "The Mind's I"(1981) by Douglas Hofstadter and Daniel Dennett:

"One variant is: 'Thiss sentence contains threee errors.' On reading it, one's first reaction is, 'No, no - it contains two errors. Whoever wrote the sentence can't count.' At this point, some readers simply walk away scratching their heads and wondering why anyone would write such a pointless, false remark. Other readers make a connection between the sentence's apparent falsity and its message. They think to themselves, 'Oh, it made a third error after all - namely, in counting its own errors.' A second or two later, these readers do a double-take, when they realize that if you look at it that way, it seems to have correctly counted its errors, and is thus not false, hence contains only two errors, and... 'But...wait a minute. Hey! Hmm...' The mind flips back and forth a few times and savors the bizarre sensation of a sentence undermining itself by means of an interlevel contradiction, possibly on the purpose or interest of the idea, possibly on the cause or resolution of the paradox, possibly simply to another topic entirely."

And in some further self-referential fun Tanya Khovanova recently offered these two sets of sentences that come from David Bernstein (where a sentence and its 'negation' are either both true or both false):


This sentence contains five words.
It is not true that this sentence contains five words.

This sentence contains ten words.
It is not true that this sentence contains ten words.

Questioning Peano

For those inclined toward epistemology and formal logic, another wonderful post from RJ Lipton below, this time on the possible inconsistency of Peano Arithmetic:

http://rjlipton.wordpress.com/2011/10/01/what-if-peano-is-inconsistent/

This stuff makes my head hurt... but I always enjoy watching others pursue it!

Friday Puzzle

Yeah, I'm a sucker for basic (but clever) geometry puzzles, so here's another one from "Futility Closet" from a month+ ago:

http://www.futilitycloset.com/2011/08/17/rest-stop/

(click where indicated in the posting to see the surprisingly simple answer)

New Aczel Book

The always-interesting Amir Aczel has a new volume out that should be of interest, "A Strange Wilderness: The Lives of the Great Mathematicians":

http://www.amazon.com/Strange-Wilderness-Lives-Great-Mathematicians/dp/1402785844

Here's the blurb from Publisher's Weekly:

http://www.publishersweekly.com/978-1-4027-8584-9

"Math In the Real World"

"Home School Math" has put up a nice page of resource links (entitled "Online Math Resources For Math in the Real World") for math teachers/homeschoolers/students or just math fans:

http://www.homeschoolmath.net/online/real_life_math.php

Popcorn Time... Math Movie

A movie for math buffs! Hooray!! "Julia Robinson and Hilbert's Tenth Problem," a 1 hour movie about... Julia Robinson (1 of the major female mathematicians of her day) and Hilbert's 10th problem (dealing with the algorithmic solvability of Diophantine equations) will be playing in various locales in October. See here for possible air dates in your area:

http://www.zalafilms.com/films/jrairdates.html

Wikipedia entry for Hilbert's 10th problem here:

http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem

entry for Julia Robinson here:

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

And here's the movie trailer to whet your appetite:




(David Hilbert image via Wikimedia Commons)

Gödel Simplified... sort of

I got a big kick out of this 1994 1-page (pdf) explanation of Gödel's Second Incompleteness Theorem from George Boolos, all in single-syllable words (literally)... pretty impressive!:

http://bit.ly/bWrBqz

Geometry Lovers...

I only recently discovered this great site with a great many cool geometry problems (mostly high school level):

http://gogeometry.com/problem/index.html

the more general home page for the site here:

http://gogeometry.com/

Friday Puzzle

For a Friday puzzle I'll just link you to this one from Presh Talwalkar from several weeks ago (...another geometry-oriented puzzle, involving string-cutting; with the answer included in the comments to Presh's post):

http://mindyourdecisions.com/blog/2011/08/29/geometry-puzzle-string-cutting-game/

(Math) Teaching and Twitter

Mr. Honner recently noted how much he now uses Twitter for ideas and connections to assist his math teaching activities. I suspect most of you already use Twitter, but if not, he offers a straightforward basic introduction to its use and benefits here:

http://mrhonner.com/2011/09/18/math-teachers-and-twitter/

He includes a list of some of the mathematically-inclined Twitter feeds he follows for his purposes. And if you want a much longer, broader list of math-related Tweeters you can check out this list from Listorius:

http://listorious.com/search/people?q=math&search=Search

When I Was 13 I Was...

...trying to see the significance of Venn diagrams.

Some 13 yr.-olds (like cool kid Neil Bickford) busy themselves otherwise (perhaps calculating the first 458 million terms of pi....):

http://blog.wolfram.com/2011/09/15/from-pi-to-puzzles/
[link corrected]

"How Algorithms Shape Our World"

A fine TEDTalk from Kevin Slavin on how mathematics rules the world, with or without human supervision!

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Collatz Fans

If you're really into the Collatz (3n+1) Conjecture nice new long piece on it here:

http://home.graysoncable.com/dkcox/

Moebius Noodles

The internet is the greatest distributor of information and potential educator in human history. And now Maria Droujkova has started the "Moebius Noodles Project" as an open access math education opportunity for youngsters, and even babies. Word has been spreading in math circles about the endeavor. If you're unaware of it, read more about it at these links and contribute if you can -- math education, especially for children, has never been more important than it is today:

http://www.naturalmath.com/blog/moebius-noodles-fundraiser/

http://www.naturalmath.com/blog/

Oldie-but-Goodie

I've used this old YouTube version (from the movie "Labyrinth") of one of the "liar paradoxes" before, but it's so much fun, worth showing again:



(If you need help working through the logic involved you can try this page: http://dabbycool.com/nooblingsnest/?p=100 )

Mathematics and the Aha! Moment

A post on mathematical insight from Peter Ash with a nice example from geometry:


http://tinyurl.com/44n2m7d

The Arbitrariness of Statistical Significance

Partly in response to some of the articles I've cited here earlier, physicist Chad Orzel writes this cogent worthwhile piece on the "arbitrariness" of statistical significance:

http://tinyurl.com/5v5raef

(...lots of interesting followup points/discussion in the comments section as well... be sure to peruse)

Daniel Lewin 1970 - 2001

KW Regan writes this tribute to mathematician/polynomial researcher, computer scientist, Akamai Technologies founder... and, 9/11 victim Daniel Lewin:

http://rjlipton.wordpress.com/2011/09/12/911-and-the-web/

Problems In Academia...

According to this piece approximately 50% of published studies from academic labs are not replicable. Not too surprising given the complexity of the variables involved, but not often acknowledged... (and how poor would the rate be among NON-published studies, which are the vast majority?):

http://marginalrevolution.com/marginalrevolution/2011/09/how-good-is-published-academic-research.html

In a slightly related matter, Ben Goldacre points to a widespread statistical error, dealing with "the difference in differences," that is commonplace in published (neuroscience) studies:

http://www.guardian.co.uk/commentisfree/2011/sep/09/bad-science-research-error

"Stay Hungry, Stay Foolish"

Image via Wikipedia

No math post today... just more of a Sunday sermon, courtesy of Steve Jobs, an individual of heroic proportions to many. Actually, it's courtesy of the Twittersphere where I found the link to this wonderful commencement speech Jobs gave at Stanford in 2005. No reason to wait 'til the end of a school-year though to pass such nuggets around:

http://news.stanford.edu/news/2005/june15/jobs-061505.html

"Your time is limited, so don't waste it living someone else's life. Don't be trapped by dogma — which is living with the results of other people's thinking. Don't let the noise of others' opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. They somehow already know what you truly want to become. Everything else is secondary."

...And on a separate note, can't help but in some way commemorate the incomprehensible tragedy of 10 years ago today -- Art Garfunkel sings this classic S&G in Central Park:

15,000 Pages of Calculations...

Michael Aschbacher of Caltech will be receiving the $75,000 Rolf Schock Prize for his part, among dozens of mathematicians who contributed, in the solution to the largest mathematical proof EVUH(!) involving the "Classification Theorem of Finite Simple Groups" (also, simply and appropriately known as "The Enormous Theorem").
Read all about it:

http://www.newscientist.com/article/dn20893-prize-awarded-for-largest-mathematical-proof.html

Of Squares and Circles, Pegs and Holes

Another simple Friday puzzle adapted from Richard Wiseman this week (just by guessing you have 50% chance of getting it right):

"Which is a better fit (i.e., results in less wasted space), a square peg in a round hole or a round peg in a square hole?"

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.answer:  a round peg fits into a square hole better than a square peg in a round hole

if you need an explanation you can view the answers posted at Wiseman's site:

http://richardwiseman.wordpress.com/2011/08/15/answer-to-the-friday-puzzle-117/

Testing 1 2 3

In the below post "Mr Honner" draws attention to a rather important consideration when it comes to the standardized math testing that is now prevalent for evaluating student success: Are the tests ANY GOOD?

http://mrhonner.com/2011/09/05/are-these-tests-any-good/

Noting that such tests are now used to evaluate not just students, but teachers and schools as well, he points out that mathematical errors, poor wording, and poor representation of subject matter, are more common than they ought be for quality, meaningful testing.

Fermat's Last Theorem (VIDEO)

"Math-Fail" directs readers to view this video on Andrew Wiles and Fermat's Last Theorem. It is indeed excellent and enthralling (45 mins. long). I too encourage all math-lovers who have never seen it to find the time to view it. It very well depicts the love, devotion, and elation a human being can experience toward a subject that so many others view as merely dry and tedious:



(...and what I really like is Wiles' workdesk, which actually makes mine seem almost orderly! ;-))

Statistical Puzzle

Toss a fair coin. Is the average number of coin tosses required to get a HTH pattern greater than, less than, or the same as, the number of tosses required to get a HTT pattern?

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answer: greater than (on ave. taking 10 tosses to get HTH, and only 8 to get HTT)

See Peter Donnelly's TED talk if you need an explanation: http://bit.ly/qYlgHi

Number Fun & Games

An interesting simple number game to wile away some time (and thought) from "MindYourDecisions" blog:

http://mindyourdecisions.com/blog/2011/08/30/a-fun-number-elimination-game-who-will-win/

[...including an explanation of how it always plays out from different initial conditions -- i.e., the ending is automatically (mathematically) determined as soon as the initial set-up is established.]

Pluggin' A Few Books...

Been having some internet connection issues lately so a quickie post while things be workin'....

I'll just plug 3 books I've read in recent months that I've especially enjoyed, and that contain somewhat overlapping content....

Two of them I've already mentioned here previously but are so good I feel deserve a second mention, especially for the layperson math enthusiast:

1. The Big Questions: Mathematics by Tony Crilly

2. Mathematics Without the Boring Bits by Richard Elwes

Both Crilly and Elwes are Brits who really seem to have a knack for this sort of writing (making math interesting!!). I especially love the breadth of topics the Elwes volume covers and the clear, playful nature of the presentation, and the Crilly book is super as well, covering a slightly smaller, but still interesting range of subjects.

3. A much older volume, that I only recently read, is from 1988 by William Poundstone, "Labyrinths of Reason" -- an absolutely wonderful introduction to not only a few of the same mathematical notions covered in the first 2 volumes above, but with more in-depth, rich discussion of various philosophical underpinnings of logic, math, reasoning, and the like. A great, thought-provoking read; I don't know how I've missed it all these years. (Definitely makes me want to look at more of Poundstone's books.)

M&Ms in a Klein Bottle

From 10 yrs. ago, but recently tweeted by Clifford Pickover, this yummy topology puzzle:

http://www.kleinbottle.com/m%26ms_in_a_klein_bottle.htm

and the interesting, winning answer here:

http://www.kleinbottle.com/Bernie_Tao.htm

(next question: how many Snicker Bars will fit in a tesseract?... just kidding)

Vi and Sal Together

Vi Hart and Sal Khan (Khan Academy) discussing human perception of logarithmic scales:

Connecting the Towns, or Dots... With Soap Bubbles

Math-Fail directed me to this common problem elucidated by SingingBanana:

Paper Folding, a Student, and a Formula

Contrary to the popular notion that no piece of paper could be folded in half more than 8 times, high school student Britney Gallivan demonstrated (almost 10 years ago) that she could do it 9, 10, 11, AND 12 times, and developed the mathematical formula that calculates the limit to the number of possible folds for a given shape of paper:

http://pomonahistorical.org/12times.htm

Addendum: today Clifford Pickover tweeted a link to an earlier report of James Tanton's students accomplishing a 13TH FOLD of 13,000 ft. of toilet paper (tuh-duhhh!!):

http://tinyurl.com/3ffqgtl

(...hmmm, might be a good time to buy some stock in Charmin, in case this sets off a national competition!) 

Fibonacci Scores Again...

Interesting case of a 13-year-old designing a better solar array panel based on a trip to the woods and the Fibonacci sequence:

http://tinyurl.com/3wy8y5b

Simple Probability

Greg rolls a pair of dice, and the total comes up 5. If he now rolls a second pair of dice and adds that total to the first total, what is the MOST LIKELY sum he will arrive at (assume fair die):
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answer: 12

Strogatz, Dehaene in Conversation

Mr. Honner's math site led me, today, to a wonderful Princeton video of psychologist Stanislas Dehaene and mathematician Steven Strogatz in discussion of mathematics, cognition, and teaching (also touches on the relationship between subject-matter interests and neuroscientific disorders). The video is long (79 mins.) and I've only watched part of it thus far (including Strogatz's wonderful initial section starting around the 23.30 mark). Highly recommended if you have the time:

http://hulk03.princeton.edu:8080/WebMedia/flash/lectures/20110427_publect_dehaene_strogatz.shtml

N-gons in Circles, oh my...

A wonderful li'l piece of geometry from "Futility Closet" regarding regular n-gons inscribed in a unit circle:

http://www.futilitycloset.com/2011/08/04/a-side-matter/

The Difficulties of Scientific Replication

John Allen Paulos' take (from earlier this year) on the "decline effect" in scientific research -- why scientific "truths" often unravel over time (especially true in health/medical research):

http://tinyurl.com/29t39rx

P vs. NP Update

There isn't really a whole lot to say yet (although that is interesting in itself now that more than a year has passed), but RJ Lipton gives this update on Deolalikars' controversial 2010 P vs. NP proof:

http://rjlipton.wordpress.com/2011/08/11/deolalikars-claim-one-year-later/

It's A Carnival!

The 80th Carnival of Math is now up for your viewing and delectation -- good stuff!:

http://www.walkingrandomly.com/?p=3800

Friday Puzzle

A problem from YouTube:



(there are multiple answers possible, but only interested in the LOWEST answer here, and it is given below)
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answer: 49

Sounds Fascinating... (book)

A review over at plus.maths.org of a new book, What's Happening In the Mathematical Sciences? Volume 8, by Dana MacKenzie:

http://plus.maths.org/content/whats-happening-mathematical-sciences

I'm not even familiar with this series from the American Mathematical Society, but it sounds great, covering a wide range of topics that involve real-life applications of mathematics in the work-a-day world.
Check it out...

Blindness and Mathematics

Interesting post here about blind mathematicians:

http://onionesquereality.wordpress.com/2011/07/31/blind-geometers/

(can't help but wonder if some of this wouldn't relate to blind musicians/composers as well...)

Three, 5, Four, 4, Four, 4, Four...

Interesting tidbit (or "curiosity") from Matthen (and I like the graphic as well):

http://blog.matthen.com/post/8554780863/pick-a-number-between-1-and-99-write-it-as-a

More on Khan Academy

The Washington Post highlights Salmon Khan and his Khan Academy here:

http://tinyurl.com/45xww75

Khan's work has come under fire in some circles of late, but I too am a big fan of his approach, while granting that it will require tweaking and refinement to meet the hype it's sometimes given. Let's put it this way: our traditional, decades-old methodologies for math instruction haven't exactly set a very high bar to be surpassed, and digital approaches like Khan's are almost certainly the wave of the future...

Please Solve It Dear Readers...

A little something different for a Friday puzzle... different simply because I don't know the answer (and it's been buggin' me!). It comes from this older "webmaths" posting, and below I quote the problem verbatim.

http://webmaths.wordpress.com/2010/08/06/thirty-numbers/

(Though I saw the puzzle back when it first appeared, I've never seen the solution, so hopefully a reader may be able to provide it! -- not even sure if it requires some involved math, or has, as I suspect, a simple, easily-overlooked solution):

"I have a list of thirty numbers where the first number is 1, the last number is 30, and each of the other numbers is one more than the average of its two neighbours. What is the largest number in the list? "

(One thing I'm not clear on: from the problem as stated, I don't know if the 30 listed numbers must be distinct, different integers, or may include numbers that repeat themselves.)

Goat, Goat, Car...

Sol Lederman is encouraging all to visit Jeremy Jones' new site explaining the Monty Hall problem. Even if you know this problem (and solution) well, it is entertaining to see how Jeremy has pieced it all together (be sure to check out all 3 sections -- play/explanation/history -- of his site):

http://www.stayorswitch.com/

While I'm on the subject I may as well again direct any who are truly enamored (obsessed?) of this classic problem to be sure and read Jason Rosenhouse's "The Monty Hall Problem" which covers it in all its nuances and variations.

The Intersection of Math and Underwear

Presh Talwalkar, who by his own admission "likes to over-analyze decisions," mixes math with common sense in this post on the number of pairs of underwear one ought own ;-) :

http://mindyourdecisions.com/blog/2011/08/03/how-many-pairs-of-underwear-should-you-own/

(His computational answer is "20," but he neglects to address the equally important question: the ratio of boxers to briefs?...)

And for the more mathematically, less fashion-inclined folks among you, Presh's prior post is a probability puzzle/conundrum for which he will be posting the answer tomorrow:

http://mindyourdecisions.com/blog/2011/08/02/puzzle-a-probability-drinking-game/

Wow! The Continuum Hypothesis Solved???...

...well, I doubt it, but what do I know: Apparently a major Berkeley mathematician/set theorist, Hugh Woodin, using a "radically stronger logical structure" known as "ultimate L," believes he has accomplished what no one has been able to do in well over a century, and demonstrate that Gödel's Cantor's Continuum Hypothesis is true (essentially, that there exist no infinite sets lying between the set of integers and the set of real numbers -- of course it still all hinges on the initial axiomatic system one adapts):

http://www.newscientist.com/article/mg21128231.400-ultimate-logic-to-infinity-and-beyond.html?full=true

(Coincidentally, this fascinating article is from Richard Elwes who I was just highlighting a few days back.)

If you're not interested in infinity or sets, skip this article; otherwise, dive in!

Mystical Path... Mystical Math?

Popularizer Clifford Pickover often writes about the mystery and even mysticism of numbers. Paul Erdos was famous for saying certain (beautiful) mathematical proofs must come from 'God's book.' Lover of numbers, Martin Gardner. regarded himself as a "Mysterian" (and also a theist/fideist) who believed, despite the reality of numbers, humans could never fully comprehend the workings of their own minds. Cantor was deeply religious, writing proofs for the existence of God, which never gained the traction his proofs involving infinity did.

In short, I've always found fascinating the link many sense between math or numbers, and the mystical or Godly realm of existence. Math is often perceived, more than any other science, to somehow be associated with a deeper reality than we can otherwise be in touch with directly.
And yet, a different school of math, views math as little more than a creation or construct of the human mind; not so much existing in the 'world out there' so much as constrained to the world inside our heads.
Such basic, fundamental notions, yet leading to such divergent, unresolved thoughts.

Here's an old Julie Rehmeyer posting that touches on the subject (in which she quotes British mathematician Brian Davies as saying that Platonism “has more in common with mystical religions than with modern science"):

http://tinyurl.com/ycsn2bl

And lastly, if you have the time, Ben Vitale recently put up this hour+ long YouTube roundtable video on "Mathematics and Religion":


http://www.quora.com/Benjamin-Vitale/Mathematics-and-Religion

Friday Puzzle... Coins Again

You flip each of four fair coins. What is the probability that you will end up with 2 heads and 2 tails?
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.answer:  3/8

Self-reference Again

Just some sentences I've culled from this KW Regan post over at RJ Lipton's blog:

http://rjlipton.wordpress.com/2011/07/23/self-defeating-sentences/

(The post once again deals with self-reference, beginning with what the poster calls "self-defeating sentences" but moving on to some other categories as well.)

"Anyone who goes to a psychoanalyst should have his head examined."

"None of my close friends has a close acquaintance."

"Ignore this sign."

"Never say 'never.' ”

"Whether you mean it or not, be sincere!"

"I saw him do it when no one was looking."

"Words are incapable of describing what I am about to tell you."

"No one goes there anymore; it’s too crowded. "

"We Scorpios don’t believe in astrology."

"Avoid clichés like the plague."

"Don’t use no double negatives."

"A preposition is something a sentence should never be ended with."

"Can I ask you a question?"


And going back to my Monday post, I may as well throw this sentence into the mix as well:

“Only idiots believe this sentence.”

A Math Teacher Calls It Quits

A sign of the times -- someone who loves what they do, must decide to do something that pays the bills:


http://tinyurl.com/3uyyu9f

Richard Elwes... Never Boring

Some time ago I stumbled across Richard Elwes' "Mathematics 1001" volume in a bookstore (having never heard of either Elwes or the book) and was quite delighted with that encyclopedic compendium of mathematical information. This weekend, another Elwes volume, "Mathematics, Without the Boring Bits," was my lucky, accidental find in a bookstore, and it too looks to delight. My sense is that Elwes' books, coming from Britain, don't get as wide a distribution and publicity as they deserve here (in US).

I'm barely into this volume but it looks to be another wonderful, what-I-call 'nugget' book -- even at 200 pages it serves up math in very palatable bite-size nuggets, often introducing some topic in a page or less... and it only brings up the sort of topics a non-professional math person will find fun or interesting. The selection is excellent, the format attractive, and the writing entertaining and engaging. Having said that, some of the topics are covered so briefly I'm not sure a mathematical novice will always get the point or fully appreciate the significance, and a professional mathematician, on-the-other-hand, may find little new here, he/she isn't quite familiar with. So the intended audience (I think) for the book may be those who already have some background and inclination toward math, but not enough academic training to make these particular topics old hat. In any event, nice to see this array of mind-bending topics brought together succinctly in the pages of a single breezy volume.

Elwes also has a blog here:

http://richardelwes.co.uk/blog/

...and here's a fun post he did a bit ago on one of my favorite topics, self-reference/recursion:

http://richardelwes.co.uk/2011/06/06/an-idiotic-paradox/

Friday Puzzle... On the Farm

I've copied this straightforward puzzler from a Ben Vitale posting:

There is a field with sheep and cows.
Each sheep can see twice as many cows as it can see sheep.
Each cow can see the same number of sheep as it can see cows.
How many cows and how many sheep are there?
(the working assumption is that an animal can "see" all others but not him/herself)
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answer:  3 sheep and 4 cows

"The Philosophy of Applied Math"

From plus.maths.org a piece on platonism, formalism, logicism, intuitionism, and applied math:

http://plus.maths.org/content/philosophy-applied-mathematics

And, speaking of applied math, this from XKCD:

http://xkcd.com/435/

Diagnosing Logicians

Fun post from Bill Gasarch on the sometimes popular (and paradoxical) notion that professional logicians tend to be CRAZY ("a few axioms short of a complete set")... or, NOT! :

http://blog.computationalcomplexity.org/2011/07/disproofing-myth-that-many-early.html

Orgasmic Math!

Tanya Khovanova recently reflected on jealousy and "math as an aphrodisiac" from her own experience as a female mathematician, in this off-the-beaten-path blog post:

http://blog.tanyakhovanova.com/?p=345

Conrad Wolfram on Math Education

"WildAboutMath" blog highlights an interesting 9-minute interview with Conrad Wolfram on the future of math education here:

http://wildaboutmath.com/2011/07/16/conrad-wolfram-on-teaching-math/

(...he argues for much more emphasis on programming/coding skills than on calculation/computational skills in the future.)

Friday Puzzler

adapted from an earlier Futility Closet posting:

You are given 9 coins, numbered 1 through 9, 8 of which weigh exactly the same, and 1 which weighs slightly less than the others. Given an accurate balance, how can you now identify the lighter coin in only two weighings?

...for the solution, see this posting at Futility Closet where it is noted that, "J.E. Littlewood observes that a similar puzzle wasted 10,000 scientist-hours of work during World War II. 'There was a proposal to drop it over Germany.' ”):

http://www.futilitycloset.com/2011/07/08/the-counterfeit-coin-solution/

Data Without Borders

Not sure if this project will ultimately succeed or not, but the intent seems worthy... "Data Without Borders" is trying to bring together a team of qualified individuals who can assist, at low or no cost, non-profit organizations with their data-analysis needs.

From their webpage:

"...Data Without Borders seeks to match non-profits in need of data analysis with freelance and pro bono data scientists who can work to help them with data collection, analysis, visualization, or decision support."

further, "Data Without Borders aims to close that gap through a data scientist exchange, bringing exciting new problems to the data community and helping to solve social, environmental, and community problems alongside non-profits and NGOs."

Check them out here:

http://www.jakeporway.com/datawithoutborders/

They have Twitter and Facebook pages, as well.

Gödel In a Nutshell

A short, descriptive bio of Kurt Gödel here:

http://drvitelli.typepad.com/providentia/2011/07/deconstructing-godel.html

Infinity for Students and Teachers

Interested in infinity, or looking for good resources for students? This "teaching package" from plus.maths.org has lots of good links on infinity and infinite series:

http://plus.maths.org/content/teacher-package-series

Fibonacci Redux

What is it about mathematicians with 4-syllable names...? Books just keep appearing related to Pythagorus and Fibonacci. I haven't read it yet, but worth noting that math-popularizer Keith Devlin's newest offering is "The Man of Numbers: Fibonacci's Arithmetic Revolution":

http://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127

Here's one review of Devlin's new book (...from an author who wrote a book on Ar-chi-me-des ;-):

http://online.wsj.com/article/SB10001424052702304569504576403860686782154.html

Friday Puzzle

Hmmm... odd versus even-numbered house addresses:

What is the probability that any given individual lives in an odd-numbered (versus even-numbered) house address? Not exactly what you might think or assume:

http://singingbanana.tumblr.com/post/6629627960/whats-the-probability-you-live-in-an-odd-numbered

Chaos, Probability, and the Economy

A couple-year-old essay from plus.maths.org that remains relevant today:

http://plus.maths.org/content/chaos-chance-and-money

It ends as follows: 

"I hope I've managed to convince you that seemingly predictable, regular behaviour does not necessarily have a deterministic cause — it's perfectly possible for it to arise from chance events. Conversely, seemingly wild, erratic and downright chaotic behaviour can emerge from a system as simple as the interacting gravity of the Earth, the Moon, and a spaceship. The stock market displays both types of behaviour. The bad news for the government is that in some systems, like our table of coffee cups, almost all actions lead to the same result. The good news is that in chaotic systems, a very small action can change a huge and complex set of interacting behaviours. Fingers crossed that it works!"