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I’ve divided the submissions into 100 topics with at least 10 posts per topic.

 00 – Meta-topic (3’s) 10 – Pedagogy 20 – Applications 01 – Math and Culture 11 – Fun/Recreation/Puzzles 21 – More Undergraduate + 02 – Undergraduate + 12 – Elementary School & Elementary Math 22 – Middle & Secondary + School

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00 – How to find Pythagorean Triples? Alexandre Borovik discusses at Math Under the Microscope.
00 – A geometric interpretation of how to extract cube roots (for the brave) over at Blog 360. (Also, for the brave and non-brave alike, discussion of how to extract square roots).
00 – Ternary Geometry and Ternary Geometry II from Arcadian Functor, a New Zealand physics blog with an awesome photo wallpaper

01 – There is a wonderful, illustrated treatment of math and visual art at MathTrek (Julie Rehnmeyer). Read if you like that sort of math, but even if you don’t, just go and look. You won’t be disappointed.
01 – Used and abused, infinity and eternity, philosophy and mathematics, and popular culture. Alexandre Borovik supplies a special-for-the-carnival version of a post on the culture of kitsch.
01 – Math and music? We know the relationships, right? Well, how about using math to clean up a primitive recording? MathTrek tells how a good algorithm helped win a Grammy (for Woody Guthrie).
01 – Kaz of Mathematical Poetry asks Is Pure Math is Poetry? – continuing a discussion from the last carnival.
01 – And for, ahem, a different kind of poetry, Jackie’s got a link to a mathematical limerick collection.

02 – I like when I read and understand serious math. Quomodocumque publishes an accessible summary of a recent paper on the equation $A^4 + B^2 = C^n$ and its solutions. A bit reminiscent of Fermat’s Last Theorem? Yup.
02 – Charles Daney asks “Why should anyone care about unique factorization?” in his latest installment on algebraic number theory at Science and Reason. Best part? When he discusses the problems a person can get into for not caring. Like Euler…
02 – Elliptica points out that extending the chain rule to multivariate calculus gets tricky when there is a change of variables.
02 – The Universe of Discourse discusses a “most uninteresting real number” in a twist on the old chestnut “What is the first uninteresting natural number?” (Berry’s Paradox). Marc’s proposal, uninterestingly enough, is something called Liouville’s number. As the earliest submission, this one does not qualify as uninteresting!
02 – At the Geomblog Suresh discusses “Alternating Optimization,” which in practice is employed, but seems to be missing from the relevant literature.
02 – Brent continues his “Recounting the Rationals” series with a post on the Euclidean algorithm at Math Less Traveled.
02 – 11011110 (DE) plays with the combinatorics of chord diagrams (he considers those that do not form triangles), and leaves some exercises for the reader. In case you were wondering, I did them. This level of work I find challenging, but not frustrating.

10 – Sol is Wild About Math. But calculus for 4th graders? Not as wild as it sounds?
10 – H at Coffee and Graph Paper slams the notion of relevance as it is applied to teaching mathematics.
10 – Dennis DeTurck’s nutso scheme to deny teaching fraction manipulation to most public school students got a rise out of bloggers and the mainstream media. Alane speculates that we’ve been Swifted (not boat, Jonathan) in a post at Math Notes.
10 – Mathmom asks if extracting square roots by hand has any relevance today.
10 – Do you agree that Math skill = Arithmetic skill? At Killing Mind, Heath says no. I am on the fence. This is a discussion that needs to happen.
10 – Are schools “cognitively nutritious?” Alvaro presents some interesting research (and some engaging problems!) involving elementary school children at SharpBrains. And ‘engaging’ means engaging for you and me, too. Do take a look.

11 – For Valentines Day (not so long ago!) Walking Randomly investigated heart plots, surfaces, and tangrams.
11 – Terence Tao proposes a problem about some blue-eyed islanders. But who knows what their own eye color is? (logic puzzle)
11 – I have a hat-guessing (not really guessing) problem here.
11 – Zeno’s got a hot little circle problem at Halfway There.
11 – We have a mathematical magic trick (cards) with directions at TextSavvy.
11 – The Telegraph’s obituary editor says probability is hard, and proceeds to make the confusion over Monte Hall sound not so hard. Read the nice little problem at the end. Catalan?
11 – Who needs language when we have math? Il Professore at Gli studenti di oggi has a nice discussion of Steiner’s Problem, (minimizing the road distance between cities in a plane), with good diagrams, and the soap bubble trick. Read the original in Italian, or this google translation into English.

12 – Learning Games discusses a the effects of some games on learning arithmetic. Daniel discusses starts with a comparison of a physical and a mental game.
12 – There is a nice little game with lots of subtraction practice at Let’s Play Math.
12 – A well-chosen diagram can blow the lid off an elementary problem. And even some not so elementary ones. Over at Big Ideas.
12 – Praveen has a little legs problem targeted at people with little legs at Math and Logic Play. Do you agree that his count is off by two?

20 – Need help finding the perfect mate? No one here can help you. But Ben Webster turns a search for stable marriages into a combinatorial discussion at the Secret Blogging Seminar. Cool, btw, to have a real seminar blog.
20 – And, while you are there, Ben liked the marriage bit for the carnival, I kind of liked his major dis of Arrow’s Theorem (mathematics of voting, considering spoilers).
20 – Before you start, you might like to check out Pissed off teacher who has a nice introduction to fair elections. (seasonally appropriate)
20 – A bit old, but there is an annoying temperature discussion at Think Again. Just what does “average temperature” mean?

21 – Aaron Roth explores social welfare in Nash equilibrium, but the aspect that catches his attention is the “Price of Malice.”
21 – In Basics of Patch Theory Brent explores the mathematical theory behind version control systems, particularly as it applies to collaborative editing.
21 – Suresh brings a guest blogger in to the Geomblog (Ganesh Gopalakrishnan) to discuss work on model checking by Clarke, Emerson, and Difakis (won the ACM Turing award).
21 – Michi finished his thesis, and found the time to post an introduction to algebraic geometry (in two parts).

22 – There is a list of common student errors (with interesting discussion) at MathNotations.
22 – Speaking of which, Robert explains the Illini method for simplifying radicals at Casting Out Nines.
22 – And Vlorbik rails against publishers and their sloppy use of the √ (sqrt) symbol. Trust me, he takes this (-: seriously.
22 – Is Jackie’s class data normally distributed? Read about it at Continuities.
22 – Dan Myer (dy/Dan) is convinced that teaching is better with digital/2.0/power point – and here he shows off how he introduces “rates” – it’s a reasonable guess that no chalk was harmed during that lesson.
22 – Dave Marain has a rich middle school introduction to ratios.
22 – Some of us have run into conceptual problems teaching inequalities in Algebra 1. H discusses a recent experience at Coffee and Graph Paper.
22 – Jose Vilson “abstracts the concrete” as he challenges middle school kids to find missing sides. Concrete is really on his mind. Not math, but his next post discusses, concretely, what’s happening to the concrete on the squares and rectangles in his own neighborhood.

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OK, now, about this base 3 business. This is, after all Carnival 27. Carnival $3^3$, right? And it was my 3rd carnival. And I hosted 9 and 18. And I was out of ideas.

Now, some divisibility, but with bad notation. Feel free to correct. In base 3, we have some strange divisibility rules. Multiples of 10 – er, 3, end in 0. Multiples of 9 end in 00. Even number? the sum of its digits will be even (and odd? odd!). Consider a number base 3 to be a string $A = a_na_{n-1}a_{n-2}...a_2a_1a_0$. Find the difference between the sum of $a_i$ for i even, and sum of $a_i$ for j odd. If the difference is a multiple of 4, so is A.

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The next Carnival of Mathematics will revert to base 10 and appear at a blog run by the guy who used to run http://growthratenlgn.wordpress.com/ http://greedygreedyalgorithms.blogspot.com/ and he wants you to submit directly by e-mail to (tylerofmanyminds AT gmail dot com). When I know more, I’ll put it here.