Unfortunately, the round was moved two hours later. Sorry for the inconvenience. It is my honor to announce that the Codeforces Round rated for the second division is going to take place tomorrow at

What do you get when you methodically build a Lisp on top of symbolic replacement semantics? You get the Mathematica language, Polygon of forces which Mathematica and Mathics appear to be the only incarnations. Let's say you forgot how to multiply matrices. Well, just type in some symbols and see the results empirically: In fact if you have a Tron zapper you can even zap your cat into Mathematica and have him fill up one of those matrix slots, for the advancement of science.

Our neighbor will miss him. The exponential identity for the Pascal matrix is not difficult to understand based on the series definition of the exponential function: You can see that the diagonal gets multiplied by subsequently shifted versions of itself, so the calculation ends up creating factorial products.

The binomial coefficient itself is of course directly related to Pascal's triangle.

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Also notice that every power of the matrix has its numbers on a different diagonal, so when we sum up all the powers there is no interaction to account for. Every term in the series is a distinct diagonal of Pascal's triangle.

I didn't find anything interesting along this line though. What about graphs represented by the Sierpinski matrix itself? It has some pretty symmetries. I did some tiresome work trying to figure out what polyhedron it might be.

It's the tetrakis hexahedron: And look, we can run this polyhedron grapherizer willy-nilly allabouts, like on the Archimedean solids: For one, the powers of the Sierpinski matrix are Sierpinski matrices!

This isn't necessarily interesting though. The powers of a triangular matrix are going to be triangular. But the numbers follow a curious sequence of powers.

And this sequence occurs in every column and every row of the matrix, if you hop over the zeros. We can normalize the powers to find: This power sequence appears in OEIS as the number of ones in the binary representation of namong other descriptions.

Here is a totally practical application of all of this.

A pretty array of buttons: Thus the game is ultimately a quaint philosophical remark on the roles of the sexes. But for our purposes there is a claim on the internets that the states of the game form Sierpinski triangle-like graphs: Then again, if you fiddle with the layout and you squint a bit, you can kinda see it, but it's the sort of Sierpinski triangle that Maddox would stamp a huge red F over.Hi, everyone!

Unfortunately, the round was moved two hours later. Sorry for the inconvenience. It is my honor to announce that the Codeforces Round # rated for the second division is going to take place tomorrow at urbanagricultureinitiative.com usual, participants from the first division will .

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An awesome list for competitive programming! - Codeforces