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Happy New Year, everyone!

Sorry, there hasn’t been much activity on this blog over the past few months, but 2011 will be the year when you can expect a lot more stuff to be posted as I find more time to do so. Todd has been busy on other sites/blogs, especially MathOverflow and n-Category Cafe, but he is still an author/admin on this blog.

I got an email from WordPress, summarizing the level of activity on our blog for the year 2010. I thought it wouldn’t be so bad if I shared the information with you all.

2010 in review

The stats helper monkeys at mulled over how this blog did in 2010, and here’s a high level summary of its overall blog health:

Healthy blog!

The Blog-Health-o-Meter™ reads Wow.

Crunchy numbers

Featured image

About 3 million people visit the Taj Mahal every year. This blog was viewed about 33,000 times in 2010. If it were the Taj Mahal, it would take about 4 days for that many people to see it.

In 2010, there were 2 new posts, growing the total archive of this blog to 122 posts.

The busiest day of the year was January 20th with 196 views. The most popular post that day was Continued fraction for e.

Where did they come from?

The top referring sites in 2010 were,,,, and

Some visitors came searching, mostly for imaginary numbers, platonic solids, euler’s formula, vertex, and challenging integrals.

Attractions in 2010

These are the posts and pages that got the most views in 2010.


Continued fraction for e August 2008


Platonic Solids and Euler’s Formula for Polyhedra March 2008


High IQ and Mathematics December 2007


Problem-Solving Hall of Fame! May 2008


Integration Bee, Challenging Integrals January 2008

The following piece of news, in my humble opinion, deserves more mention in the math blogosphere than it has garnered so far. At the center of the story is an Iranian student of mathematics, Mahmoud Vahidnia, who was invited to a meeting between Iran’s Supreme Leader Ayatollah Ali Khamenei and the country’s scientific elite on Oct 28, 2009. An excerpt from a Guardian report on what transpired during a (perhaps, routine) question and answer session:

“I don’t know why in this country it’s not allowed to make any kind of criticism of you,” he told Iran’s most powerful cleric, who has the final say in all state matters. “In the past three to five years that I have been reading newspapers, I have seen no criticism of you, not even by the assembly of experts [a clerical body with the theoretical power to sack the leader]. I feel that if this doesn’t happen this situation will lead to discord and grudge.”

Vahidnia, who achieved nationwide recognition two years ago by winning Iran’s annual mathematics Olympiad, made his remarks at a meeting between Khamenei and the country’s scientific elite. They came after the supreme leader asked at the end of a question-and-answer session if anyone else wanted to speak. He chose Vahidnia after seeing him being pushed down by officials when he stood to ask a question.

Referring to the post-election crackdown sanctioned by Khamenei, he asked: “Wouldn’t our system have a better chance of preserving itself if we were using more satisfactory methods and limited the use of violence only to essential circumstances?”

I discovered the above piece of news “accidentally” via the Colbert Nation.  The Associated Press also did a report. I wonder if our Iranian readers (if there is any!) could furnish more information on this.

This being a mathematics blog, I’m sure a lot of readers out like to play Minesweeper. I’ve just obtained a personal best today (94 seconds on expert level) which, as Minesweeper buffs know, is nowhere close to world-class levels, but which made me happy anyway, as I’d never broken into the double digits before today!

It seems to me that world-class competitors must know some tricks with the mouse which I’ve never bothered to master, particularly because my laptop doesn’t have a mouse but rather a touchpad. This being the case, I keep my right index finger on the touchpad to guide the cursor, and the left index finger to click. I always left-click: that is, in my style of play, I [practically] never flag squares for bombs; I click only on non-bomb squares. For it’s well-known, or at least it should be, that the program doesn’t care if you identify where the bombs are — you get full credit for only identifying all the numbered squares.

To play in this style well, one needs to be fluent in a number of tactical tricks, which I don’t have good names for, but which in my personal argot I call things like “1-2-1”, “1-2-2-1”, “rule of three”, to name just a few. But that’s not what I set out to discuss, really. What I’d really like to hear from good players is: what opening strategies do you use?

The personal best I just set came after deciding on a new opening strategy. What I had been doing is clicking along border squares. Under that strategy, one could of course just keep clicking until one opens up an area, but often I would add to that the observation that if one clicked on a 1, thus leading to, e.g.,

x 1 x (–> border row)
x x x

then one could then click on the non-border square just adjacent to the 1, with only a 1 in 5 chance of setting off a bomb. If one then hits another 1:

x 1 x
x 1 x
x x x

then one can immediately open up the line of squares on the third rank, leading to a configuration such as

x 1 x
x 1 x
1 1 1

or better. This is often a cheap and quick way of opening up squares or otherwise getting a tactical toehold.

The new strategy I began using today is not to click along the border, but to click along the ranks or files adjacent to the border. Under this strategy, if one lands on a 1, leading to

x x x  (–> border row)
x 1 x
x x x

then one can click on the border square adjacent to the 1, with only a 1 in 8 chance of setting off a bomb. If one does not set off a bomb, that square has to be a 1:

x 1 x
x 1 x
x x x

and then one can proceed as before. So I’ve just lowered my odds of hitting a bomb, plus a very small fractional gain in processing time that comes with the certain knowledge that it’s a 1 if not a bomb. So far the strategy has paid off well!

I’d like to hear other people’s opening strategies, and also I’d like to know some statistics. For example, considered over the space of expert-level games, what is the probability of getting a 1, a 2, and so on? Does anyone know? (It seems this would be very difficult computing analytically — one is probably better off using a Monte Carlo simulation. But I don’t have the wherewithal to set that kind of thing up.)

Love is like PI – natural, irrational and very important!
– Lisa Hoffman

Happy Co-Valentines Day

Happy Co-Valentine's Day

Valentine’s Day is usually associated with romantic love, but I think such an idea although wonderful is somewhat restrictive. This time of the year, I believe, is also about letting people close and dear to you know how much you love and care about them! Keeping that in mind, I wish my parents a Happy Valentine’s Day and hope that my younger brother, Vishant, has a great Valentine’s Day too!

I also sincerely hope that Todd gets to spend a great Valentine weekend with his wife and family! And, here’s hoping that all our readers and my friends (including Aditya, Pawan and Kenji!) will today not hesitate in expressing their love to their near and dear ones.

And very importantly, here’s wishing Carolyn an unforgettable Valentine’s Day! Thanks for being my Valentine even though you are thousands of miles away!!

[I do hope Todd will forgive me for posting something completely non-mathematical. In my defense, this post has at least a reference to PI and category theory! 🙂 ]

I would like to quickly point out to our readers that Jason Dyer is currently hosting the 43rd Carnival of Mathematics and that the Carnival lists Todd’s POW-11 (Preserving Sums of Squares) post as one of its entries!

A reader brought up essentially this question: does anyone happen to know a proof that e^{x^2} does not possess an elementary antiderivative? By “elementary”, I mean a member of the class of functions which contains all constants valued in the complex numbers, the identity function, the exponential and log functions, and closed under the four basic arithmetic operations and composition.

Perhaps most people are preoccupied with the global financial crisis right now, especially with people in the US much more focused on the upcoming US presidential election in November. So, for those who haven’t been following the news in chess closely I would like to bring their kind attention to the current World Chess Championship (2008) match (Bonn, Germany) between two supreme chess grandmasters, Vishwanathan Anand (India) and Vladimir Kramnik (Russia). Technically, Anand (pronounced Aa-nand and not A-naand; well, actually it is more like Aa-nundh) is the current world chess champion, but personally I think that his winning the World Chess Championship Mexico (2007) last year was a somewhat “unsatisfactory” accomplishment, if you will, given that he won the crown by winning a tournament and not a “classical” chess match. I fervently believe that a person should be crowned world chess champion (like Fischer, Kasparov, Capablanca, Kramnik, to name a few) only after he or she has won a “proper” world championship match played under “classical time controls” (remember the  Fischer-Spassky match in 1972 and the Kasparov-Kramnik match in 2000?)  Without that, the gravitas of the chess crown is somewhat diminished. 

So, here is Anand’s chance now to silence his critics, of which there are very few really, once and for all that he is indeed the undisputed world chess champion! And judging by the result of the third game, which he just won in a dramatic fashion (woohoo!), as well as the tremendous amount of home preparation it clearly seems he has done, there is no doubt that he is on a steady path to the crown. In all the first three games, Anand has demonstrated thus far that he is the superior player. Of course, there are nine more games left and the bets are not off by any means. After all, it was Kramnik who beat Kasparov convincingly in 2000 to win the crown. 

For analyses of the first two games, click here and here

Ok, so here is a poll that I invite our readers to participate in. (Obviously, if you choose the “wrong” answer, all your future comments on this blog will be deleted! So, think hard before you vote.)

Don’t forget to download Firefox 3.0 today!

Download Day 2008

Update: Look for the ‘sexism’ video at the end of this post that essentially strengthens my argument about the media treatment meted out to a woman, who was/is as capable as any other candidate, running for a very high office in America, the world’s oldest democracy and whose founding fathers, as we learn from history, were the children of The Enlightenment!


With the Democratic primary race practically over now and knowing, as we all do, who the nominee is going to be, I just couldn’t resist writing a post on this one, having avoided writing anything about politics all this while.

Well, it was quite appalling to see/hear all these months that when it came to Hillary the discussions/commentaries in the so-called “mainstream” media were similar to ones that are generally heard in men’s locker rooms, while Obama has been treated almost like a God-like figure. And while Hillary’s “racist” remarks were dissected/analyzed with great relish, no one, it seemed to me, paid any particular attention to the disgusting misogynist remarks directed at her throughout the primary campaign season, with the result that the Democratic party has managed, or so it seems, to lose its grip over white women voters now. I have a feeling that this is going to cost the Democrats another general election. (Of course, I could be wrong; I am not a political “pundit”, after all!)

So much has my mother been miffed/angry at the blatant sexist remarks openly made in the media against Hillary that she has vowed now to vote for McCain this Fall. To her, the contest has “demonstrated” yet again that women still haven’t been able to break the glass ceiling in this male-dominated world. Is anyone listening to women voters like her?

A video sample:

In mathematical parlance, this is the only instance in which Left = Right, if you know what I mean.

The late great Paul Erdös was not a religious man (his take on religion seems to have been fairly ironic, referring for example to God as “The Supreme Fascist”), except of course when it came to mathematics. Ever the Platonist, he considered that when he died, he might finally get a chance to gaze upon “The Book” which, as if written by God, contains the most beautiful and enlightening proofs of all theorems. The highest form of praise from Erdös for a proof was, “It’s straight from The Book!” He also said, “You don’t have to believe in God, but you should believe in The Book!”

Do you believe in The Book? I’m not sure I do!

In fact, there is this book by Aigner and Ziegler, “Proofs from The Book”. In it they include the following one-sentence proof by Don Zagier on Fermat’s two square theorem (that a prime congruent to 1 \pmod 4 is a sum of two squares):

A One-Sentence Proof That Every Prime p congruent to 1 modulo 4 Is a Sum of Two Squares

                                        D. Zagier

      Department of Mathematics, University of Maryland, College Park, MD 20742

The involution on a finite set S = {(x,y,z) \in N^3 : x^2 +4yz = p } defined by

                      ( x+2z, z, y-x-z )  if   x < y-z
       (x,y,z) ---> { ( 2y-x, y, x-y+z )  if y-z < x < 2y
                      ( x-2y, x-y+z, y )  if   x > 2y

has exactly one fixed point, so |S| is odd and the involution defined by 

       (x,y,z) ---> (x,z,y)

also has a fixed point.

I plucked this off the Web from here; the author of the page prefaces it with a comment:

The following constitutes the essential text of a complete research article; I have
omitted only some comments at the end concerning the history of this type of argument.
The author reproves a famous result.  He builds his proof into a single sentence
as simply a tour-de-force.  In fact, he has left many straightforward steps for
the reader to verify.

1.  As an exercise in critical reading, list all the implicit claims that the
reader must verify in order to accept this argument as a proof.

2.  As an exercise in logic and algebra, supply all the details necessary to
support these claims.   Package all this as a long-winded rewrite of Zagier's
article written so that any high school algebra student could easily read it
with comprehension.

You should expect to expand Zagier's single sentence to a full page or more.

Um, yeah.

My own reaction to this proof: it is surely dazzling in its compression, although one’s first reaction is likely to be “WTF?!?” — what just happened here? The underlying idea is that the number of fixed points of an involution f: S \to S on a finite set S (i.e., a function f equal to its own inverse) has the same parity as S itself; it follows that if S has odd parity, then any involution on S has at least one fixed point; such a fixed point of the involution (x, y, z) \leftrightarrow (x, z, y) on Zagier’s set S yields a solution (x, y) to x^2 + 4y^2 = p, whence the theorem. So the bulk of the proof is in showing that S has odd parity, by showing that his nontrivial involution has exactly one fixed point.

And I guess you can see, by staring at his casewise-defined involution for a while, that its only fixed point is (x, y, z) = (1, 1, n) where p = 4n+1. It then remains to check that this really is a well-defined function from S to S, and it really is an involution. The full verification probably does take up at least a page.

It truly is a jaw-dropping proof. My problem though is that it looks like black magic. I mean, I can construct a line-by-line verification that the proof does what it purports to do, but in a deeper sense I still don’t get it. How Zagier cooked up this involution is a mystery to me, and unless I made a concerted effort to memorize it, it would remain unmemorable to me (that is, unless someone were to reveal the underlying mystery to me — I suspect that that would take a few sentences or more! Can anyone help me?).

What do you think — does it qualify as a Book proof? Me personally, I prefer proofs which are enlightening — arguments that I can really understand, proofs that stick, proofs I can take with me to the grave. Put it this way: if God were to write a proof which consumed an absolute minimum number of bytes in some optimal language, it still wouldn’t be much of a Book proof to me unless I (a limited human) could really understand it, and if it were really better in that sense than its closest competitors.

I don’t think I believe too strongly in the reality of “Book proofs”, or at least I’m skeptical that every theorem can be said to have a Book proof. Every mathematical statement and proof is embodied in some larger context or matrix of ideas, many requiring patient assimilation before a light suddenly flashes on. I tend to believe that’s the rule rather than the exception, and the idea that we should believe in a Book proof for every theorem, possessing a snappy immediacy which cries “Behold!”, is based on a dangerous and even crazy fallacy concerning the nature of mathematics.

[At the same time: we can all agree that Erdös was an absolute genius at finding Book proofs! 🙂 ]

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