Welcome to the 16th Carnival of Mathematics, the comments and opinions edition. The host of the previous edition of the carnival set up some pretty high expectations, what with providing both interesting background information on the fortnight's posts, and pretty pictures to accompany them. We'll be having none of that here. No, this is the blogosphere we're talking about, and even if the subject matter is the noble quest for mathematical knowledge, our stock and trade are comments and opinions. That high bar actually has me thinking "limbo", and I want to see just how low I can go. So without further ado, here are this edition's entries.

I have nothing but admiration for the good souls who struggle day after day with teaching our children, so we begin our list with the **mathematical educators**...

Starting us off is Denise at Let's play math!, who gets us in the mood with quotations on the joy of mathematics. This batch of quotations ends with the line from G.H.Hardy, "Beauty is the first test: there is no permanent place in this world for ugly mathematics." We'll see about that; I'm pretty sure that some ugly mathematics (or is that ugly mathematicians?) lie in wait for us a bit later. But I digress... Denise then gets to the meat of pre-algebra problem solving with posts on the tools and 2nd grade. In the comments to that first post, commenter Bill Howdle describes a (long ago) homework math problem that he was unable to help his daughter with, and after 25 years it still nags at him!

Probably everyone here has heard at some point the story of how, as a young child, Gauss discovered a formula for the sum of the first n positive integers. In Pairing up with Gauss, Math Mom describes how she teaches this to 10- to 12-year-old kids, using different arithmetic progressions to reinforce the pairing technique. While you're visiting, be sure to also check out the link she provides to an American Scientist article which dissects the Gauss myth.

For our next entry, Dave Marain of MathNotations shows how to find the Golden Ratio in the geometry of a certain isosceles triangle, in Searching for Gold in Geometry. Be sure to scroll down through the comments (this will be a recurring theme in this edition) to read Eric Jablow's nifty introduction to algebraic number theory.

If someone poses a problem about a random number between 1 and 10, I think it's safe to say that everyone would assume that the number is going to be uniformly distributed. But what if we're talking about some random mathematical object where there is not an obvious, intuitive choice for the distribution? Jonathan discusses What is a random triangle? over at jd2718. The post is short, but (you guessed it) much discussion is generated in the comments.

To end this category, Jonathan also recommends reading Pissed Off's discussion of whether partial credit should be given on math tests. Now, having spent some time grading student exams, I'm a big fan of multiple choice questions-preferably using one of those machine-readable bubble answer sheets so that one doesn't even have to look at the students' work. Pissed Off, to the contrary, seems to be basing her opinion on what might be best for the student and not the instructor!

Next up for your viewing pleasure are the **mathematicians**, either as the subjects or the authors of blog entries...

John Kemeny of A Mispelt Bog has a nice post about an obscure result relating to patterns of digits in repeating decimals, that has recently been getting renewed attention. In The Secret Theorem of M. E. Midy = Casting In Nines, he tells how Midy's theorem has recently been rediscovered and extended. John is also offering up a short post looking at some statistics on Ph.D. graduation rates and SAT scores by state.

Next, John Armstrong of The Unapologetic Mathematician takes a break from his ongoing series on category theory to respond to a question that reader posed in a comment. In A little aside on linear algebra, John discusses the relationship between inner products, norms, and the choice of a basis in vector spaces.

Hmmm, only two math submitters this edition? Well, that just won't do, and fortunately there has been some other juicy math blogging recently that we can highlight.

Let's start with the low-brow... I've often thought that the way to attract more public interest in mathematics is, like just about everything else that is "marketed" these days, to spice it up a bit. The New York Times helped out in this regard with an article describing an argument by UC Berkeley mathematician David Gale, that the commonly reported statistics on male and female love partners cannot possibly be correct. This was all over the blogosphere last week, but since no one else has submitted a relevant entry, I'll suggest taking a look at Jake Young at Pure Pedantry. Jake gives a nice recap of the arguments back and forth, and has so many updates based on reader comments that I lost count. The very first commenter gives a nice little example of how, while the means must be equal, the medians definitely might not.

For something a bit more high-brow, I'd recommend Ben Webster's post at the Secret Blogging Seminar on Zeta function relations and linearly equivalent group actions. Ben mentions that the subject of this post comes from some research he did as an undergrad, and it contains the prettiest looking math blogging I've seen recently. But what caught my attention was that the comments were devoted entirely to a discussion of using mathematicians' names as nouns versus adjectives when referring to eponymous mathematical structures.

To shift back to the low-brow for a moment, I noticed that Ben is also making a call to keep the sarong theorem archive growing. I have to admit that this is a nice picture of Ben, and math ed content is welcome, too.

I recently bought Danica McKellar's new book, Math Doesn't Suck, for my daughter. This book has already been the subject of much discussion on the web and I won't try to add anything here. But I've been looking without much luck for a comprehensible explanation of the research paper she co-authored, "Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on Z2". So, I was delighted to see that Terry Tao has done just that in his blog entry "Math Doesn't Suck", and the Chayes-McKellar-Winn theorem. Tao gives both a high-level and a fairly detailed explanation of what the theorem means, and also provides some recollections about Danica's time as a student.

Finally, we close the mathematicians' portion of our program with Greg Muller at The Everything Seminar railing against the bad math behind a classic gambling scheme that's "guaranteed" to pay off, in Infinite Series vs. Reason. Specifically, if you are, say, flipping a coin to determine whether you win a bet, and you double your wager after each round, then you're bound to come out ahead at some point, right? What could possibly go wrong?

Finally, we come to the computer science portion of our carnival...

...and there is nothing better to make the segue from bad math to ugly computer science than the following little brouhaha. Mathematician Neal Koblitz has an article in the September Notices of the AMS in which he criticizes the cryptographic community, and it evidently has many computer scientists up in arms. None of them felt compelled to submit their entries to the carnival, unfortunately; but being the lover of soap opera that I am, I will do the honors. Luca Trevisan at in theory writes about The Swift-Boating of Modern Cryptography, and it goes without saying that you must read the comments on this one. Lots of ugly trash-talking there, but no doubt Koblitz is a bad, bad, naughty mathematician who deserves it all.

On a more positive note, Julie Rehmeyer at the Science News blog MathTrek has a couple of entries timed just perfectly for this edition. (And Julie is being promoted from math to computer science as well!)

In Squashing Worms, Julie describes research by Microsoft theoretician Jennifer Chayes into how computer worms spread through the internet, in order to calculate which hosts should get patched first to best slow the spread. There may also be implications of this research for biological organisms, and Chayes plans to work with epidemiologists to gain a better understanding of this.

In Julie's second entry for this edition, Kidney Matchmaking, we learn about new algorithms being designed to match kidney donors with patients needing transplants. An estimated 1,000 - 2,000 additional patients per year might receive transplants that otherwise wouldn't, once the techniques are implemented nationwide.

In the comments to Julie's second post, Suresh from The Geomblog mentions some additional research being done on the kidney transplant matching problem. In fact, he recently did a post on the subject himself, Saving lives with exact algorithms, which deserves its own mention here in the Carnival. The amazing take-home point from Julie's and Suresh's posts is that results from the study of algorithms are actually saving patients lives-concrete benefits that one would probably not expect from such an abstract topic.

Most of us are familiar with the history of Euclid's fifth (parallel) postulate, and how relaxing this requirement allows us to come up with different geometries in which "lines" differ from our usual intuition-for example, lines might correspond to great circles on a sphere. Now, what if we were to really relax the definition of what constitutes a line, to create something that "behaves like" a line but isn't required to be straight? According to David Eppstein at 0xDE, computational geometers do just this sort of thing, and the result is a pseudoline. However, different researchers have used different definitions for pseudolines, and in his post Was sind und was sollen die Pseudogeraden?, David attempts to clarify some of the resulting confusion about what the proper definition should be.

For our next entry, Aaron Sterling recommends an item from Scott Aaronson's blog Shtetl-Optimized. In Shor, I'll do it, Scott gives a plain-language explanation for Peter Shor's quantum factoring algorithm, which was one of the first algorithmic results exploiting the power of quantum computing. This is an older post, but one of Scott's most popular ones. Now, the comment section is always lively at Shtetl-Optimized, so it goes without saying that you want to check that out. Aaron mentions that you should particularly look for Robin Blume-Kohout's comment explaining the Fourier transform.

Finally, Aw, shucks is a very short post by Jeff Erickson at Ernie's 3D Pancakes. It seems that Jeff has a difference of opinion with Doron Zeilberger over the value and philosophical implications of computer-generated proofs, and these two gentlemen are not shy about, ahem, expressing their opinions. I can't quite decide if they love each other or hate each other's guts, but either way you've got to follow those links and read what they have to say...it seriously just doesn't get any better than this.

This concludes the 16th edition of the Carnival of Mathematics. The 17th edition will be hosted on 9/21 over at MathNotations by Dave Marain, who will undoubtedly handle the proceedings in a more timely fashion than moi.

Let me leave you with one final parting link, some words of encouragement from Professor Mark Sapir at Vanderbilt University. Be sure to turn up your speaker and click on the sound file at the bottom of the page!