The Minimal Superpermutation Problem - Imagine that there is a TV series that you want to watch. The series consists of n episodes, with each episode on a single DVD. Unfortunately, however, the DVDs have become mixed up and the order of the episodes is in no way marked (and furthermore, the episodes of the TV show are not connected by any continuous storyline – there is no way to determine the order of the episodes just from watching them). Suppose that you want to watch the episodes of the TV series, consecutively, in the correct order. The question is: how many episodes must you watch in order to do this? [more inside]
Visual Patterns. Here are the first few steps. What's the equation?
Every film Pixar has produced has landed in the top fifty highest-grossing animated films of all time. What's their secret? Mathematics. Oh, and 22 Rules of Storytelling. [more inside]
Morton and Vicary on the Categorified Heisenberg Algebra - "In quantum mechanics, position times momentum does not equal momentum times position! This sounds weird, but it's connected to a very simple fact. Suppose you have a box with some balls in it, and you have the magical ability to create and annihilate balls. Then there's one more way to create a ball and then annihilate one, than to annihilate one and then create one. Huh? Yes: if there are, say, 3 balls in the box to start with, there are 4 balls you can choose to annihilate after you've created one but only 3 before you create one..." [more inside]
1,143,839,622,748,050,000,000,000,000 Sonnet Anagrams and oodles of other oddities from Mike Keith involving constrained writing, mathematics, music, and the number π.
Counting is one of the first and simplest concepts most people are taught. But when you get beyond simple 123s, counting can become an advanced subject all its own. Essentially the science of counting, combinatorics is a key component of everything from abstract algebra to probability (PDF). [more inside]
A generating function is a way to keep track of a lot of related numbers all at once... The study of generating functions is an art and a science known as 'generatingfunctionology,' and its bible is free for all to download. [more inside]
Ever wondered what comes next, and why? The On-Line Encyclopedia of Integer Sequences has the answers. (Previously.)
If you could use a great big free handbook of discrete math and algorithms, Jörg Arndt's fxtbook wants to be your friend. Plain text table of contents to whet your appetite.