Zeta Function

Matt Springer elucidates the Riemann Zeta Function here:

http://tinyurl.com/39btmur

A Royal Wedding... and Mathematics

Great piece by Matt Parker explaining P vs. NP in layman terms:

http://tinyurl.com/2d83t7v

In fact, his introductory lines are the simplist statement of P vs. NP I've ever come across:

"Can you solve a problem as fast as someone can check your answer? Can you show that this is possible for any problem at all? Then $1m (£600,000) is all yours."

Friday Pizza Puzzle

Four students order two 10-inch pizzas from two different pizzerias, to divide equally among themselves. When the pizzas arrive one is in the shape of a 10-inch (per side) square, while the other is in the shape of a 10-inch diameter circle. They plan to divide each pizza into 4 equal pieces, with each student receiving 1 piece from each pizza.
How many sq. inches of pizza will each student end up with?
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answer:  25 (pi + 4)  sq. in.
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Garrett Lisi's "Geometric Theory of Everything"

I've been intrigued in the past by the outside-the-box geometry/cosmology thinking of physicist Garrett Lisi (but then I'm not schooled enough in his subject matter to even know if he's only outside-the-box or, off-the-wall!). Peter Woit has a post today related to Lisi's latest theorizing:

http://www.math.columbia.edu/~woit/wordpress/?p=3292

... and here's an older TED talk Lisi gave:

http://tinyurl.com/yla4y52

Palindromically Speaking

An easy puzzle for today:

Palindromic numbers are those that read the same backwards as forwards; for example, 101, 3663, 40904.
Given the palindromic number a = 138831, come up with a palindromic number b composed of the same six digits (as a) but in another order, such that the sum of a + b also equals a palindromic number.
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answer below

...but first check out this (NON-palindromic) quirk from "Futility Closet":

http://www.futilitycloset.com/2010/11/15/home-again/
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.answer: 831138

Get Busy!

Release of Wolfram Mathematica 8 announced:

http://blog.wolfram.com/2010/11/15/mathematica-8/

...and this follow-up post:

http://tinyurl.com/3xjxp5p

Coffee and Cream... and Math

A puzzle appropriate for first-thing on a Monday morning:

http://tinyurl.com/25gu9fw

A Non-Math Friday...

Just a couple of notes tangential to math today:

First off, Peter Woit has a quick (and positive) review of the new book "Massive" by Ian Sample (on particle physics, and principally the Higgs Boson), up at his blog; worth a look:

http://www.math.columbia.edu/~woit/wordpress/?p=3266

On a lighter note, I'm now reading/enjoying Martin Gardner's final book, the "Colossal Book of Wordplay," a slightly odd volume. First it is NOT "colossal," and second, although he certainly dabbled in wordplay previously, it is not what Gardner is famous for.

This smallish volume is barely 150 pages with large print and smallish pages... I'm not sure if the "colossal" in the title is merely a takeoff on his previous "Colossal Book of Mathematics," or whether the title itself is intended as some sort of ironic 'wordplay' on this quite slim volume (or possibly the volume was originally to include yet more material that was never completed due to Gardner's death earlier this year?).

Nonetheless, it is a fun, jaunty book covering a wide range of wordplay with frequent intervening bits of quirky entertainment. Possibly it could be organized a little better, at times seeming to me slightly disjointed or 'thrown together,' and I wish Gardner had gone into greater depth at times (as Douglas Hofstadter has previously done on some of this material), but for nine bucks it's worth the price of admission, if you have an interest in the quirks of language (as a lot of mathematicians, and other analytical sorts, do). My expectations for Gardner are so high that this book probably falls short of them, but were it any lesser author, I'd easily give it a thumbs-up.

The content ranges from almost juvenile or goofy entries to well-known standards, to amusements that almost any reader will find new to them. There are puzzles and palindromes, poems and anagrams, brain teasers, word games, riddles, and everything in-between. If nothing on one page strikes your fancy, something on the next page likely will.
If you love words, get this book. Think of it as Gardner's final stocking stuffer gift to us. (I haven't seen it yet in a single bookstore, and had to order it online, so not sure how widely distributed it is.)

Finally, and slightly more math related, Alex Bellos covers a recent Rubik's Cube competition here:

http://alexbellos.com/?p=1429 

Some Miscellany For Wednesday

Another look at the foundations of mathematics in upcoming book:

http://tinyurl.com/33ylyvp

If you can't get enough of Pi:

http://facts.randomhistory.com/2009/07/03_pi.html

James Tanton's newly-active Twitter feed here:

http://twitter.com/#!/jamestanton

RSA Encryption

A fairly straightforward explanation of RSA encryption here:

http://inversezen.com/2010/11/the-rsa-algorithm/

Air Molecules, Neuron-firing, Carl Sagan

Mathematically-speaking, you've been breathing some mighty famous air:

http://io9.com/5635391/youve-probably-shared-the-same-air-with-galileo

(not sure if the actual mathematics of this holds up to closer scrutiny...?)

And from the same blog, a story that's been making the rounds about an electronic contraption for brain stimulation that might aid mathematical thinking:

http://tinyurl.com/3695xty

Not precisely math-related, but on the heels of a day celebrating Martin Gardner, I feel I'd be remiss if I didn't note that a day celebrating Carl Sagan is coming up:

http://www.centerforinquiry.net/carlsaganday

A Look At Pascal's Triangle

In his book "Wonder of Numbers" Clifford Pickover names the following article as having the "all-time strangest title" of any published mathematical paper:

Granville, A. (1992) "Zaphod Beeblebox's brain and the fifty-ninth row of Pascal's Triangle" American Mathematical Monthly April, 99(4): 318-331.

The paper (pdf) can be found here:

http://www.gianpierobiancoli.it/wp-content/uploads/2009/10/beeb.pdf

And if you don't know who the character Zaphod Beeblebrox is (from "Hitchhiker's Guide to the Galaxy") you can check him out here:

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

In other matters, a quickie intro to the Riemann Hypothesis from Matt Parker here:


http://tinyurl.com/2arhu4r 

[includes the mention that "All prime numbers (greater than five) squared are one more than a multiple of 24."]

And revisiting Mr. Fermat:

x^n + y^n = z^n  ...NO solutions for n ≥ 3

x^n + y^n = z^(n-1) ...INFINITELY many solutions for all n ≥ 3

proof:

http://tinyurl.com/2d5qucy

Biology, Complexity, Dynamics... and Math

"Biology is not yet a predictive science, there are essentially no fundamental laws [as with physics]... biology, in terms of maturity, is at the stage that physics was 300 years ago..."

Interesting post over at plus.maths.org on the work of Thomas Fink et.al., essentially trying to mathematically model biological systems.

a bit more therefrom:

"Everyone says the standard model for evolution is mutation, selection and inheritance. Put those ingredients together in a box and you get evolution. But the reality is, when we put those things into models of evolution, or set up appropriate systems of artificial life, we just don't get life-like evolution — we don't find the evolution of complex, surprising things. Some fundamental is missing. What gives a system the capacity to evolve? What makes a system evolvable?"

...and later:

"The problem is, to be able to know what is interesting, one needs to know what is boring."

Stuff For a Tuesday

I previously posted about the intriguing and unsolved "Collatz Problem" (whether or not certain created sequences always must end in the same pattern, regardless of starting point). It is also known by the name "Hailstone Numbers" and Ben Vitale recently wrote about them here:

http://benvitale-funwithnum3ers.blogspot.com/2010/10/list-of-squares.html?spref=tw

Clifford Pickover addresses the same subject here:

http://sprott.physics.wisc.edu/pickover/hailstone.html

Meanwhile, in a different vein, I just recently discovered this fairly young blog devoted entirely to prime numbers:

http://primepatterns.wordpress.com/

Finally, I've never been much of a Sudoku fan, but I do very much enjoy Ken-Ken (are there others like me out there, and if so, why is that???)... In any event, this poster's been thinking more about Ken-Ken than I ever did:

http://bit-player.org/2010/kenken-friendly-numbers

Dr. James Tanton

Kudos again to Sol, this time for introducing me to Dr. James Tanton, a creative mathematician with his own YouTube channel of interesting videos here (okay, I'm a sucker for an Aussie accent):

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

His homepage website is here:

http://www.jamestanton.com/

...and "Math Mama" reviewed some of his work here:

http://mathmamawrites.blogspot.com/search?q=%22james+tanton%22

In other news, another recent "tweet" (from Twitter) that caught my eye:

 "It is known that e is irrational and that pi is irrational, but it is not known if their sum is irrational."

I'm wondering (maybe someone out there knows the answer) is it EVER the case that 2 irrational, transcendental numbers are known to sum to a rational???

Penney's Paradox

Want to win a few coins... The "Penney Paradox" is a very intriguing though less-discussed paradox than some of its more famous counterparts (it's named after its discoverer Walter Penney, though it is also often discussed using a penny as the working example).

If one flips a fair coin 3 separate times, there are 8 equally probable (heads/tails) triplet-results: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. In this game a first player selects one of these triplets, and then a second player chooses a different one. The coin is then flipped repeatedly until one of the selected triplets appears as a run and the player having chosen it wins the game (and coin). For example, if the chosen triplets are HTH and THT and the flips go THHHTH, the last three flips mean that HTH has won... the first triplet appearing matching a player's choice, wins.

One might first think that any one triplet is just as likely to occur as any other. However, upon reflection it will probably be clear that given a series of 4-or-more flips there are more ways for a triplet like say HTH to occur than the triplets TTT or HHH to appear. But what is far more intriguing is that NO MATTER what triplet the first player chooses, there are triplets that player #2 can select giving him/her a probabalistic edge of winning.
"Futility Closet" site mentioned this a few weeks back (and how player 2 can make his/her choice), but without elaborating much on how the mathematics of it works.

"plus.math.org" covers the math here:

http://plus.maths.org/issue55/features/nishiyama/

Or you can check out a briefer treatment on Wikipedia here:

http://en.wikipedia.org/wiki/Penney%27s_game

I should also mention that IF you do have Martin Gardner's "Colossal Book of Mathematics" on-hand he covers the subject well in his chapter 23 on "nontransitive paradoxes" (it is the "nontransitivity" of the relationships involved that result in the differential probabilities for the triplets).

Tuesday Fun...

Just some fun stuff today:

The success of physicists and a sense of probability in poker-playing here:

http://tinyurl.com/244wukt

An entire post of math humor/goofiness here (some better than others!):

http://tetrahedral.blogspot.com/2010/10/math-jokes.html


...it includes this one, just to whet your appetitie:

And lastly just another interesting tweet from Twitter:

"Are there infinitely many prime Fibonacci numbers? Unknown, at least as of ten years ago."

Mathematics 1001

It's annoying (and expensive) when, while awaiting the arrival of certain books in a bookstore, I suddenly happen upon a volume I've never heard of that looks enticing for the money I've been saving for the other specific books!...
This weekend I stumbled upon "Mathematics 1001" by Brit Richard Elwes, a new book that does a great job of covering in a nutshell a huge number of ideas/concepts/terms cutting across a broad swath of mathematics. Several such books are already available, but this looks to be the best one I've seen yet, written at a layman's level... a great little reference source and quickie introduction to various math nuggets, for your shelf. It is a sort of glossary of mathematical ideas, organized not alphabetically, but generally from simple and basic ideas that hang together to more complex and abstract ones.

The contents are divided into the following general areas:

-- Numbers
-- Geometry
-- Algebra
-- Discrete Mathematics
-- Analysis
-- Logic
-- Metamathematics
-- Probability and statistics
-- Mathematical physics
-- Games and recreation

The Intro says that the book is "aimed at.... anyone with a curiosity about mathematics, from the novice to the informed student or enthusiast. Whatever the reader's current knowledge, I'm sure that there will be material here to enlighten and engage."

I think the author succeeds....
(It's a hardback, and I believe worth the $25 price, but of course can be gotten cheaper.)

Mandelbrot, Mobius, Mirth

Image via Wikipedia

In their tributes to the Father of Fractals, Benoit Mandelbroit (recently deceased), several sites have done posts on "infinite coastlines." One of the related notions that I've always found mind-blowing is that of a closed border or perimeter having INFINITE length, yet enclosing a region of FINITE area (very counter-intuitive!). The Koch Snowflake is probably the most oft-used example of this (though there are any number of other possible cases).
Most of you are likely well-familiar with it, but if not, or if you need a refresher, a couple of the many sites expounding on it:

http://www.cut-the-knot.org/WhatIs/Infinity/Length-Area.shtml

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

p.s... here's another one of those video zooms of the Mandelbrot Set, magnified 10^214 e.214. The poster himself calls it "mathematical porn"!:

http://vimeo.com/1908224

Further memorializing the recently-deceased, Ivars Peterson reports on yet another oddball Mobius-band trick passed along to him by Martin Gardner some years ago:

http://tinyurl.com/3484lyg

And to end the week with some chuckles, re-visit this old caption-writing contest from WildAboutMath blog, if you missed it the first time:

http://tinyurl.com/2atzumj

If I Were Stranded on a Desert Island...

...and could only have one book to read I think I know what it would be. . . .

Yes, I'll squeeze in one more bit about Martin Gardner before his day of honor tomorrow (to any who are tired of hearing about Martin Gardner by now, I apologize, but HEY! it's MY blog so deal with it ;-))
When Gardner's "The Colossal Book of Mathematics" came out in 2001, I looked at its size, price, and the many chapters covering topics I wasn't particularly interested in, and ignored it (I already had plenty of Gardner books on my shelf).  It was only years later that I checked it out from a public library and discovered not only how many chapters were of great interest to me, but also how many of the topics I wouldn't normally have found interesting were made so by Gardner's deft and insightful writing.
So again I highly recommend this volume to anyone lacking it on their shelves. If you're ever stranded on a desert island it would offer you weeks/months of mental entertainment (...and, as an alternative, just in case I got tired of math, I might take along Gardner's essay-anthology, "The Night Is Large," as well!).

Meanwhile, 'Mathematics Rising' blog has just covered the 3rd of my 'Fab Four,' Bernhard Riemann, here:

http://mathrising.com/?p=270

Finally, a couple of recent math blog carnivals here for your enjoyment and delectation:

http://tinyurl.com/2bhva6c

http://tinyurl.com/2bcdscb

And I'll end today with a factoid lifted off an old Twitter posting:

A 10,000 number gap in prime numbers: 9973! + 2 through 9973! + 10006 are all composite (non-prime).

...someone please double-check those for me over the lunch hour and get back to me to confirm.