Number Digit Puzzle

Can you find a 10 digit number such that:

• the 1st digit tells how many zeros are in the number, 
• the 2nd digit tells how many 1's are in the number,
• the 3rd digit tells how many 2's are in the number,
• the 4th digit tells how many 3's are in the number, etc. etc. (10th digit tells how many 9's are in no.)

 answer below....
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answer: 6210001000

This is called a "self-descriptive number," BTW, and it is the only one in base 10.

Einstein In Context...

A couple of folks have emailed me over time with inquiries about the (somewhat famous) quotation from Albert Einstein used in the right-hand column of this blog. For anyone interested in its fuller context, it comes from this 1921 address of Einstein's to the Prussian Academy of Sciences in Berlin, entitled "Geometry and Experience."  (Einstein was essentially addressing the notion of what would later come to be known as "the unreasonable effectiveness of mathematics" (Wigner) when he made the remark):

http://tinyurl.com/2b4375e

This and That

A mathematics blogger comments on the "hoopla" around Stephen Hawking's new book here:

http://mathrising.com/?p=238

Does learning higher mathematics take away from the ability to think sharply on a problem? The question is raised here (includes a nice 'algebra' problem):

http://tinyurl.com/2v7uz7f

... and Princeton University Press is giving away a copy of the new book, "The Calculus Lifesaver," by Adrian Banner:

http://tinyurl.com/3426tfj

DOH!! Page 29...

Some more video entertainment today. A clip from back in 2008 of someone expressing the joy/passion that is evoked by a claim that the Riemann Hypothesis has been proven:



And now check out this note:

http://arxiv.org/abs/0807.0090
  

A Book, and a Crafty Law Student

An old Dover publication I only recently stumbled upon is one of the best I've seen at succinctly covering many of the most trenchant paradoxes and self-reference issues underlying mathematics:

Bryan Bunch's "Mathematical Fallacies and Paradoxes" (1982) covers a wide selection of, guess what, fallacies and paradoxes; some fairly light, others deeper and heavier. Below I've totally re-adapted one of the many self-reference paradoxes contained in the volume:

To demonstrate what a fine teacher he is, Larry the Lawyer contracts with each of his students such that they need only pay him for his individual instruction IF and ONLY IF they win their first law case. If they lose that first case they pay him no fee.
However, one of his students, Squiggy, upon completing the course, opts simply not to try any cases at all to avoid paying any fee. Perturbed, Larry feels compelled to sue Squiggy for payment (since avoiding trial cases in order to avoid payment was not intended as an option). Once the case comes to court Squiggy represents himself. IF he loses, then by the original contract he does NOT have to pay Larry! IF he wins the suit then the court will have ruled that he does NOT have to pay! Squiggy appears well on his way to being a superb lawyer....

The Bunch book definitely isn't for everyone (not even for all math buffs), but if you have an especial interest in the paradoxes and intrinsic issues that underlie uncertainty in mathematics, as well as in science and knowledge more generally, it's worth a look.

"Nerd Superbowl" & The Halting Problem

Some stuff to chew on for weekend:

Nice wrap-up to the current state of the recent P vs. NP excitement from "Math Trek" here:

http://tinyurl.com/32smtbf

And below, Mark Chu-Carroll takes his readers through the 'halting problem' and its unsolvability in a recent post:

http://scientopia.info/blogs/goodmath/2010/09/08/1069/#more-1069

The Mathematics of Sunscreen

Higher and higher SPFs....

Older NY Times article here:

http://www.nytimes.com/2009/05/14/fashion/14SKIN.html?_r=2

Visualizing Data...

This David McCandless (journalist) 18-minute TEDTalk has been bopping around the Web of late, in which he promotes the utility of 'visualizing' numbers/data/context; probably a tad oversimplified, but still interesting:




David's blog is here: http://www.informationisbeautiful.net/

Factoid...?

Read this on Twitter once:

2^86 is the longest known power that contains no "0" digits.

A Number Savant

Video clip of a savant who's friends are numbers:

Math: A Young Man's Game?

Another interesting post from Alex Bellos, this time on "maths' cult of youth" (the common notion that great mathematicians invariably do their best work in youth, and are past their prime by age 40).

http://alexbellos.com/?p=1354

Starts off focusing on now 15-year-old child prodigy Arran Fernandez, before addressing the issue more broadly.

(...Hopefully, we'll get a few more productive years out of 35-year-old Terence Tao , who some consider the world's greatest living mathematician, before he commences his downward slide. ;-))

My Fab Four

Speaking of Euler, as I did in the prior post....

It's of course impossible to name the four most important or influential or greatest mathematicians of all time. The arguments could go 'round-and-'round forever without resolution, and math is a surprisingly diverse, robust field where different individuals make all kinds of different contributions.

Nonetheless, one of the math T-shirts I offer at my Zazzle store is "The Fab Four" where I've designated the four I would pick out for such an honor if forced to choose. For the combined breadth, depth, and variety of their contributions they are (with their Wikipedia links):

Leonhard Euler
(1707 - 1783)

Carl Friedrich Gauss
(1777 - 1855)

David Hilbert
(1862 - 1943)

G.F. Bernhard Riemann
(1826 - 1866)

I'm leaving out both a lot of hugely important ancient and modern-day mathematicians, but these are just my subjective choices. Who might you choose for a Fab Four? Here's a list of 'greatest mathematicians' as formulated by someone else:

http://fabpedigree.com/james/greatmm.htm

e^i(π) + 1 = 0

Astrophysicist Adam Frank waxes poetic (almost) over Euler's "incomparable and glorious" Identity, involving "five magic numbers," and "beauty" (...with Justin Bieber thrown in for good measure):

http://www.npr.org/blogs/13.7/2010/09/02/129610905/best-equation-ever?ft=1&f=114424647

Frank asks, "Why are so many mathematically inclined folks sent into paroxysms of delight over this string of symbols which seem like gibberish to others," and then he proceeds to answer the question for the reader.

New Carnival of Mathematics

The 69th "Carnival of Mathematics" it is now available (with the usual variety of math-festive offerings) here:

http://jd2718.wordpress.com/2010/09/03/carnival-of-mathematics-69/

Fermat's Last Theorem Used in Proof

For any greater than is irrational...  

A demonstration of Fermat's Last Theorem being used in a very efficient ("charming") proof of the above:

http://tinyurl.com/2bof7l6

(...this is almost too simple)

Who Knew!

I just discovered there is a YouTube channel devoted to number theory. Latest offering here:

http://tinyurl.com/26unyae

...and a few problems (from elsewhere) to play with here:

http://threesixty360.wordpress.com/2010/09/02/its-a-new-newsletter/

(I suspect 4.2.3 has some shortcut solution, though I'm not seeing it?)

Oldie But Goodie

Famous old Sam Loyd 'Chinese Warrior' fun visual puzzle here:

http://alexbellos.com/?page_id=865

Cryptography Basics

Nice basic intro to cryptography (and especially RSA encryption) from Daniel Chiquito over at Equalis site (including a recommendation for Simon Singh's "The Code Book" as a good source of further learning):

http://www.equalis.com/members/blog_view.asp?id=565749&post=108495

Women In Math

Over a year ago on her blog Tanya Khovanova mentioned the numerical outcome of googling "male mathematicians" and "female mathematicians."  So I just did the same and similarly found a heavily-weighted favoritism toward females:

"male mathematician" --- 3420 hits
"female mathematician" --- 8990 hits

At first glance of course this seems peculiar given the preponderance of male mathematicians in society over females, but of course upon a moment's reflection it's clear that someone doing a search would be far more likely to "search" for find "female mathematicians" than "male mathematicans" (which is, at least subconsciously, for a lot of Americans, almost a redundancy!).

At any rate, many are working hard today, to alter this gender-bias and specifically encourage females to pursue mathematics, at least as an interest, if not even a vocation. One organization active in that endeavor is the Association For Women In Mathematics:

http://www.awm-math.org/

Check 'em out, especially if you're interested in math, and you're a possessor of two 'X' chromosomes!

Prime Numbers Always Intrigue

In 1968, a Russian mathematician conjectured that, on average, the number of prime numbers for which the sum of their digits was even was equal to the number of prime numbers for which that sum was odd.

The conjecture has now been proven true, by French mathematicians:

http://www.physorg.com/news192907929.html

Can't honestly say I understand what the significance of the finding is 8-\ but the article concludes thusly:

"The methods employed to arrive at this result, derived from combinatorial mathematics, the analytical theory of numbers and harmonic analysis, are highly groundbreaking and should pave the way to the resolution of other difficult questions concerning the representation of certain sequences of integers.
Quite apart from their theoretical interest, these questions are directly linked to the construction of sequences of pseudo-random numbers and have important applications in digital simulation and cryptography."

Pretty much any proof that involves prime numbers has a lot of potential consequences...