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