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?
.
.answer below
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.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)
.
.answer below
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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!"

Come One, Come All...

The 79th edition of the Carnival of Mathematics is up for your perusal and entertainment:

http://www.johndcook.com/blog/2011/07/05/golden-carnival-of-mathematics/

Circle-packing

Packing circles within a minimal circle... Matthen shows you how it's done:

http://blog.matthen.com/post/7060206689/given-n-identical-circles-the-problem-is-to-find

Friday Puzzle

What is the area of the largest semicircle that can be inscribed in a unit square?

answer below (...and not as simple as might first appear):
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
answer:   π (3 − 2 √2) or ~ 0.539

(if you didn't get it, you can go here for the explanation:
http://www.qbyte.org/puzzles/p153s.html