n + sqrt(n) = .....

James Tanton recently tweeted that for any positive integer n, the equation n + sqrt(n) rounded to the nearest integer, will never result in the square of an integer; i.e. 4 + 2 = 6 (not a square), 9 + 3 = 12, 13 + √13 = 16.605 or 17, etc.
...and Gary Davis took up the challenge to demonstrate the truth of the statement here (using proof by contradiction):

http://tinyurl.com/2bv76se

Now if I can just figure out a way to work this into my next cocktail party conversation ;-)

Self-recursion

Just recently stumbled upon this brief page on "self-recursion" over at Wolfram MathWorld:

http://mathworld.wolfram.com/Self-Recursion.html

One example of a self-recursive statement therefrom to whet your interest:

"This sentence contains ten words, eighteen syllables, and sixty-four letters."


...and, on a more humorous note:


"What is the volume of a pizza of thickness 'a' and radius 'z'?" Answer: pi z z a.

Latest From Equalis Community Blog

"Wild About Math" has a new blog entry up at the Equalis Community blog here:

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

Always includes some interesting stuff... and in this instance, introduced me to another blog I was unaware of, "Grey Matters," which I've added to the blogroll at right. Also, blogger Sol mentions he'll be doing reviews of several of James Tanton's books in the future; something to look forward to. There is a link to the latest "Math Teachers At Play" carnival, and other amusements. Good stuff for a long weekend....

Base 12

A sort of fun post from Alex Bellos today on using 12 instead of 10 as the basis of our mathematical system, as pushed by the "Dozenal Society of America" (who knew there was such an organization!). Most folks are aware of the binary system and other bases lower than 10, but employing bases greater than 10 gets less attention. There have always been many reasons why base 12 (also known as "duodecimal" or "dozenal") would make a lot of utilitarian sense, even if as a practical matter, it will never happen:
http://alexbellos.com/?p=1462

Wikipedia on base 12 here: http://en.wikipedia.org/wiki/Duodecimal

And the Dozenal Society's website here: http://www.dozenal.org/index.php?u=31

(Somehow though, asking for a baker's dozen donuts and only expecting 11, just doesn't seem as appealing... ;-)) 

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