Friday Puzzler

Here's another logic puzzle I've adapted directly from a recent "Futility Closet" offering. It is the sort of mind-bender that either you like... or, it drives you bonkers!:

A question on a multiple-choice exam is followed by the answer choices given below. Even without knowing the question, what must be the right answer?

a) All of the below
b) None of the below
c) All of the above
d) One of the above
e) None of the above
f) None of the above

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ANSWER: e (you can work through the logical possibilities starting at "a" and working down, or starting at "f" and working up, to arrive at the answer)

Selection Bias Rears Its Head

Save save save $$$! ....or, not. Presh Talwalkar explains why you ought not get too excited over certain insurance ads:

http://mindyourdecisions.com/blog/2012/03/29/why-you-should-ignore-car-insurance-savings-claims/

Mind Doodling... A Compulsion to Count

One individual's fascinating simple account of their compulsion to count things and it's seeming genetic origin:

http://www.lastwordonnothing.com/2012/03/27/the-compulsion-to-count/

(How common is this!?)

Andrica's Conjecture

Prime numbers are of course an unending source of fascinating mathematical work and conjecture. One prime number conundrum I learned of from Clifford Pickover's "The Math Book" is "Andrica's Conjecture" (after Dorin Andrica) which states that the square root of any nth prime number minus the square root of the (n-1)th prime will always result in a number less than 1; i.e. the difference of the square roots of any two consecutive primes will always be less than 1...

It has been successfully  tested out to beyond the 10^16th prime! In fact interestingly, the largest difference thus far comes with the 4th prime and equals only ≈ 0.670873. From there the values appear to trend asymptotically downward (thus, nowhere close to 1)… but still, the conjecture remains UNproven. More at Wikipedia here:
 

http://en.wikipedia.org/wiki/Andrica%27s_conjecture

Friday Puzzle

Another Friday puzzle drawn from the volume, "Are You Smart Enough to Work at Google?":

A national survey shows that 70% of the public likes coffee, and 80% likes tea. What then are the upper and lower bounds of people who like BOTH coffee and tea?
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ANSWER:

upper bound 70%
lower bound 50%

Book Blurbs

As is my tendency I've been reading 3 different books at once, and as I likely won't do a full review of any of them, will just do a quickie blurb on each one now:

1. I generally enjoy Amir Aczel's volumes, though they often lack some of the depth I'd prefer, and his current work "A Strange Wilderness: the lives of the great mathematicians" is similar. He covers a great many of history's accomplished mathematicians with pithy biographical sketches. If you like biography and haven't read up much on the lives of famous mathematicians this is a fine starter book. If you've already read widely in this area, this volume may not add much to your knowledge. It isn't always clear to me why space was allotted as it was between individuals -- I thought some mathematicians deserved longer treatments, and perhaps others shorter. More troublesome to me is that Bernhard Riemann and Kurt Gödel are barely mentioned in the book (HOW could you leave them out!!?). It's a fairly quick, easy, pleasant read for those who particularly like biographical sketches, but not a volume I'd be rushing out to buy.

2) A number of books have come out in recent years covering different selections of famous or important mathematical equations; it's almost a genre unto itself. "In Pursuit of the Unknown: 17 equations that changed the world" is Ian Stewart's latest effort at putting together such a compendium, and (as usual for Stewart), it is quite good. A nice (and large) selection of varied equations, important in different fields (though mostly math and physics), treated with usually interesting prose from the prolific Dr. Stewart.
 If reading the "biographies" of equations, as it were, is at least as interesting to you as reading the biographies of humans, I recommend the volume, especially as a good introduction for lay folks to the varied ideas presented.

3) Finally, and not altogether math-related, is James Gleick's "The Information." I always like reading Gleick, but am late to the table on this one, because I waited for it to appear in paperback. It was worth the wait. The first 3 or 4 chapters (100 or-so pages) seemed a bit sloggish, but after that it picks up considerably (with more mathematics tie-ins) and becomes the rich sort of a read that Gleick is known for… certainly the richest volume of the 3 I've listed here.

Infinity Computing??

My math knowledge isn't strong enough to critique this piece from a Physics arXiv blog that claims an exact area calculation for the 'Sierpinski carpet'... but, I couldn't help having a gut feeling that 'crankery' was involved just from the way it read; and upon perusing the comments section was relieved to see many professional mathematicians seemed to agree... but, not all. An entertaining read... or, just a waste of time???

http://www.technologyreview.com/blog/arxiv/27656/?p1=blogs

There's Math & Then There's Math Interpretation

"...there are two broad interpretations of almost all real-world numbers problems — the stripped-down, mathematicians’ approach and the interpretive statisticians’ approach. And it’s in this wiggle room of interpretation where pure math hits the real world that misleading statistics are born."

From a nice blog piece over at Scientific American on doing math... and, being done in by math (starts off with the famous 'boy born on Tuesday problem'):

http://tinyurl.com/87jwmpy

"A Closer Look" at Fractals

Great treatment for the non-professional of Britain's 'infinite coastline' problem from Grey Matters Blog here (...more generally, an infinite perimeter enclosing a finite area!):

http://headinside.blogspot.com/2012/03/closer-look.html

Friday Puzzle

I adapted this Friday puzzle from one on "Futility Closet" a couple of months back...

A black bag contains 16 billiard balls, some of which are all white and the rest are all black. You randomly pull out two balls at the same time. IF it is EQUALLY likely that the two pulled balls will be the SAME color as DIFFERENT colors, then what is the proportion of the balls within the bag?
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 ANSWER: The bag must contain 6 balls of one color and 10 of the other -- that produces 60 ways of getting a mixed draw, 45 ways of drawing the more common color, and 15 ways of drawing the less common color; i.e. 60 = 45 + 15 the two outcomes are equally likely.

Pi & the Math of Buffon's Needle

Wasn't planning to do a post in honor of Pi Day (today), but may as well join the fray:
(video from Numberphile explaining the math behind pi and Buffon's needle... or in this case matches)

Sleeping Beauty...NOT Your Childhood Fairy Tale

I've been reading about the "Sleeping Beauty Problem (or Paradox)" lately. It's actually a decade-plus-old quandary that I was aware of, but had never paid much attention to until it popped up somewhere on one of my Twitter feeds last week. Loving a good paradox, it's been rattling in my brain since.
Some folks say it reminds them of "The Monty Hall Problem" in so much as people argue vigorously for different solutions. But the Monty Hall Problem has an actual correct answer, whereas (so far as I can tell) the SBP really can be argued in two different, divergent approaches (designated as "halfers" and "thirders"). The paradox is sometimes stated in slightly variable ways, which is part of the problem, but even a fairly standard statement of it can include slight semantic pitfalls, leading to some of the disagreement. Still, more than 'Monty Hall,' the SB problem reminds me of the famous "Newcomb's Paradox" where people also tend to split two ways, and there simply is no established "right" answer.

Wikipedia states the SB problem as follows:

"Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin is then tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday and Tuesday. But when she is put to sleep again on Monday, she is given a dose of an amnesia-inducing drug that ensures she cannot remember her previous awakening. In this case, the experiment ends after she is interviewed on Tuesday.

Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"

Many re-state the problem to ask Sleeping Beauty, "What is the probability now for the proposition that the coin landed heads?," and some argue this word change is significant and alters the discussion. I think most people however, indeed understand the problem in terms of probabilities (the probability of heads being either 1/3 or 1/2), and so I certainly prefer thinking about it that way.

The Wikipedia entry for the problem is here:

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

And here are some more links discussing it (warning though, they may cause your head to pound ;-)):

http://blog.tanyakhovanova.com/?p=356 (arguing for the 1/3 answer)

http://barryispuzzled.com/zbeauty.htm (arguing for the 1/2 answer)

http://meteuphoric.wordpress.com/2011/01/09/against-the-hybrid/

…and discussion from a "freethought forum" here:

http://www.freethought-forum.com/forum/showthread.php?t=3120

Finally, if that's not enough for you, you glutton-for-punishment, more links here:

http://www.anthropic-principle.com/?q=resources/preprints#pre5

Khan Academy on 60 Minutes

I'd been thinking of writing a post about the (I think overblown) controversy over Khan Academy for awhile... but for now will just let it go with this "60 Minutes" report from yesterday:

Friday Puzzle

 A book contains N pages, numbered, per usual, from 1 to N, with the total number of digits in those page numbers equal to 1095 (i.e. 1095 individual numbers). That being the case, how many pages are in the book?

Answer below:

(...and on a sidenote, worth mentioning that Salmon Khan, founder of Khan Academy, will be the focus of a segment of "60 Minutes" this coming Sunday.)
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ANSWER:  Every page will have at least one digit, so that's N digits right off the bat. All but the first 9 pages will have 2 digits so that N - 9 will have a second digit. And finally, by the same logic, there will be N - 99 3rd digits. Thus 1095 must equal N + (N - 9) + (N - 99) or 3N - 108. Working out the equation N = 401 pgs.

Math... As a Novel Idea

"Wild About Math's" latest podcast is with Tony Gonzalez, translator of Hiroshi Yuki's Japanese math novel "Math Girl" into English… even if, like me, you don't read much fiction, many interesting points covered herein:

http://wildaboutmath.com/2012/03/04/tony-gonzalez-inspired-by-math-4/

Along the way, Sol mentions Don Knuth's "Surreal Numbers" as another example of fiction where mathematics plays a central role. And another book that comes to my mind (even though I've never read it) is Scarlett Thomas's 2007 novel, entitled "Popco" -- which Martin Gardner reviewed very favorably when it came out. Finally, the popular TV show "Numb3rs" is of course a more recent and electronic example of mathematics turned into fictional entertainment.
Amazon even has a category for "math-inspired fiction":

http://www.amazon.com/Math-Inspired-Fiction/lm/RDAULNGZQ2RCW

Martin Gardner Redux

Great recent compendium of articles on Martin Gardner's recreational mathematics from MAA's "College Mathematics Journal" here:

http://www.maa.org/pubs/cmj_jan12.html

Math Texts As 'Crap'

An editor let's loose about "crap" in your kids' math textbooks:

http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_of_your_childs_math_textbook_you_should_be

It starts off this way:

"There may be a reason you can’t figure out some of those math problems in your son or daughter’s math text and it might have nothing at all to do with you. That math homework you're trying to help your child muddle through might include problems with no possible solution. It could be that key information or steps are missing, that the problem involves a concept your child hasn’t yet been introduced to, or that the math problem is structurally unsound for a host of other reasons."

 And then later, this:

"...[the] project manager mused to me aloud, 'I want to know who buys this crap.' Crap. That was the word she used after all her exhausting efforts trying to make a silk purse out of this pig’s ear. My reply to her was, 'I want to know who buys it twice.' Because that’s the only way educational publishers make money, on repeat sales. Those books are out there now in print, on the shelves in the publisher’s warehouse, being packed and shipped to a school near you."

RSA Keys... How Weak?

Rather technical, but still very interesting discussion of the recent concerns over RSA encryption from the "Gödel's Lost Letter" blog (if RSA interests you, a must-read):

http://rjlipton.wordpress.com/2012/03/01/do-gaps-between-primes-affect-rsa-keys/

Friday Probability Conundrum

 First, here's a review of a new book that may be of interest to puzzle-solvers:

http://www.newscientist.com/blogs/culturelab/2012/03/the-secrets-of-solving-mathematical-puzzles.html

And now, on to a Friday blog puzzle: I adapted this one from another recent Car Talk puzzler, and I like it a lot as another Marilyn vos Savant-type simple (...but not quite so simple) probability dilemma. In fact I've re-written it into a 'Monty Hall'-like framework:

Monty Hall shows you 3 cards, one that is the color red on both sides, one that is green on both sides, and one that is red on 1 side and green on the other side. He puts all 3 into a black bag and shuffles them around. A blindfolded assistant randomly pulls out one card from the bag and lays it flat on the table so only the top side is observable. That side is red. You as a contestant can only win a prize if you accurately guess the color of the card's opposite unseen side. Should you guess red, green, or are your chances equal whichever you guess?

Answer will be given in the comments section…

"Dr. Euler's Fabulous Formula"

Nice recent post on possibly mathematics' most famous or awe-inducing equation, e^iπ+1=0 :

http://tetrahedral.blogspot.com/2012/02/euler.html