Steven greeted me with a puzzle when I entered the office today. I got it after thinking a bit, though I definitely have seen this a long time ago:

Given 51 numbers between 1 and 100, show that you can find two of them that are relatively prime

Proof:

By pigeonhole, we must get at least two numbers of the form {x, x+1}, where x is odd. These two are relatively prime.

Steven then gave me a strictly harder followup. This took me a bit longer, and afterwards I was very happy to have come up with an unorthodox solution, and hence a new post:

Given 51 numbers between 1 and 100, show that you can find two of them such that one divides the other.

Proof and analysis:

Start by putting every number between 1 and 100 into a bucket. Now, starting with 100 and moving down to 2, move *all* the numbers in the bin of every even number of the form into the bucket where is. Each time we do this, a bucket becomes empty. So at the end we have 50 buckets. By pigeonhole, we must get at least two numbers in the same bucket. The quotient of those two numbers is a power of two, so we’re done.

Note I could have just said something like: “for every odd number , put all numbers of the form into the same equivalence class” and then counted equivalence classes. However, it seems that in this language, counting equivalence classes requires at least a couple of steps of thought. For some unknown reason, by using this handwavy “bucket” talk, the number of equivalence classes is obviously 50, whereas by using the more rigorous language above, I don’t see an immediate way to get the answer at quite the same speed.

People used to competitions would probably recognize the first problem as an old chestnut; this is actually a puzzle Uncle Erdos used to test mathematical ability in young kids (‘epsilons’). I don’t know where the second problem comes from.

Happy Thanksgiving,
-Yan

Hi, my name is Steven Sam. I sit next to yanzhang here in building 2 at MIT, so now I will begin posting on this math blog too. My webpage is here.

I am a first year graduate student in the math department at MIT, and I am interested in combinatorics and its connections with representation theory, algebraic geometry, and commutative algebra. I hope to post some interesting things about that in the future.

I’m the grumpy one, so instead of starting a carnival with merriment and joy (those !’s in the title are not all exclamation marks), I’ll start with a rant.

In the previous Carnival, jd2718 mentioned that he was “all set” for 48, and I think I speak for most of us that getting to host 48 must bring some warm and fuzzy feeling to our mathematical hearts. Well, I’m going to play Simon Cowell and say that I don’t really like the number 48.

You may now wonder why I choose to give disdain to such a perfect number as 48, surely it looks to you glistening and sparkly, ready to be factored in so-many-different ways? Well, first of all, it isn’t perfect. Rather, it is semi-perfect. Also, as spam emails and telemarketers have taught me, I do not have fondness for ubiquity, as I prefer more exotic beauties such as 78, 133, or 248. 48 just appears everywhere since it is so damned composite. I had the precognition that 48 would be somewhere in the descriptions of the list of Archimidean Solids, and surely enough, it is both the number of edges of the “small rhombicuboctahedron” and the number of vertices of the “large rhombicuboctahedron” (as the reader probably guessed, the usual suspects 24, 60, 72, etc. also pollute this list). But wouldn’t our curiosity perk even perkier if they were both, say, 57 instead of 48? 48 is number theoretically a bore, an attention-seeker with lots of primes and no prime quality.

Non-number-theoretically, 48 lurks in some unexpected places: there are 48 constellations in Ptolmey’s Almagest; the 48th Proposition of Euclid’s first book of the Elements is a converse to the Pythagorean Theorem; 1848 is one of the more fun historical years for Europe, as fun as chaotic sweeping revolutions go. But by now I think we’ve given the brat much more attention than it deserves, so let’s go on.

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For the morning pre-class (yes, the carnival starts with math classes. This is how mathematicians roll) coffee, John Cook models the distribution of time customers spend in coffee shops and also gives some properties of log-concave functions ( I did not know before that they are closed under convolution).

The always unapologetic John Armstrong is currently on an exposition of the determinant, starting here and is now here. This must be about the fifth time I’m relearning the determinant, each time more “Linear Algebra Done Right” than the last and more enlightening.

For the parents and teachers, a new blog Algebra Questions Answered tackles the age-old pedagogical problem of how to teach kids new to the concept what we do with division by zero in a clear way.

From the brother realm of computer science (where 48 = ‘0′), Mark Dominus gives a lesson on monads and closure in “ Triples and Closure. I share with him both a respect for Haskell’s “mathematicality” and a fear for any book that has the Yoneda Lemma on page 10.

As a continuation of The Decimal Zoo, Brent Yorgey gives two followups on decimal expansions and their properties, which is a good starting place to study fundamental number theory and the related algebra of groups (I think that’s how I started thinking about groups myself).

Two nice good places to practice, learn, or re-learn calculus (I’ve noticed that I don’t remember any calculus) are Walking Randomly and Foxmaths. The former has another integral up, involving a function I don’t know, and the latter investigates another approximation of with a curious alternating nested square roots.

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Joining us for lunch, we have E. Kowalski’s very accessible overview of what we have to do to get to Twin Primes. Oh yeah, lunch. To feed off the starving graduate student horde, he has also prepared fractal cabbages.

Yes, fractal cabbages. It is so mathematically delectable that it gets its own line for emphasis.

(The musical accompaniment (fitting for this particular Carnival ;P) is Glenn Gould playing from The Well-tempered clavier)

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Raymond of Money Blue Book digs into those plastic cards we carry around in our wallets. I know some of you may be getting excited, but this is not about how to steal someone else’s number for your own nefarious gains, rather to explain the fairly-simple checksum system that the cards use. Try it. Mine works!

David from the not-so-Secret Blogging Seminar posted a sequence of three pieces, a really nice example of mathematical discussion “in action” about a problem in algebraic geometry. He first juggles an idea, then discussion ensues , and we have a happy ending where the Jacobi Conjecture remains at large – but we understand mathematics a little more.

Considering the number 48 is fairly well-represented in popular religions, from the 48 Jewish prophets to the 48 days Buddha meditated under the tree, it is only fair to give the more arcane mythos a visit. In a two-part act brought on by a challenge of Alan Crowe on Reddit, my friend lydianrain summons Cthulhu with power series (or at least shows that in the way you least want to do it in). Alan of course reciprocates with his own post.

To close us off, Mark Dominus delivers some fantastic history. Not only does he examine the fairly-well-known story of Archimedes and the square root of 3 (and makes an awesome point to chew over, on the role of heuristics in problem solving), he also sums up some pretty amazing developments about Godel and his role in finding a loophole in the U.S. Constitution.

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Thank you for reading, and see you at the next carnival!