There are two key observations:

• toggling a light twice amounts to doing nothing,
• toggling light and then light has the same effect as toggling and then toggling .

As a result, the order in which we press the buttons is irrelevant. So to solve the n-by-n puzzle, we just need to know whether a button needs to be pressed. My old website had some pictures I made showing solutions for boards of various sizes—pictures where a white pixel meant “press” and a black pixel meant “don’t press.” I assembled these pictures into a video, showing solutions to the Lights Out puzzle for :

For as cool as that looks, there’s not much to be discovered (as far as I can tell) from watching these frames flash by. But it does look like about half the buttons have to be pressed to solve the puzzle: why is that?

The still frames of the movie are available here as PNGs in a zipped archive. Here is a solution to the 400-by-400 board:

Finding that solution involved row-reducing a -by- matrix—that’s a matrix with over 25 billion entries. On the other hand, each entry is one bit, so that matrix fits (not coincidentally) in 3 gigabytes of memory. One could surely do better, considering how sparse the matrix is: perhaps we could have a little contest for solving very large Lights Out games.

Besides the fact that all these pictures look awesome, Lights Out is a neat example to motivate some linear algebra over a finite field. It illustrates how satisfying an “easy” local condition (each light must be turned off) might require a globally complicated solution—a lesson for mathematics and for life!

I’ll be a bit more explicit: a dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.

There’s a lot of neat questions to be asked about such a process: for instance, after drawing n dots, how many components should we expect to see? As you can see in the movie, when you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color. And inbetween, something more interesting happens.

Here’s an example of “something more interesting” taken from a much larger picture than the above movie:

Kreck, Matthias An inverse to the Poincaré conjecture.Festschrift: Erich Lamprecht. Arch. Math. (Basel) 77 (2001), no. 1, 98—106.

it is pointed out that

• homology is a very basic invariant, and
• closed manifolds are very basic objects

and so a very basic question is: what sequences of abelian groups are the homology groups of a closed simply connected manifold?

It isn’t very hard to realize any sequence of abelian groups up to the middle dimension, but that middle dimension is tricky (e.g., classify -connected -manifolds).

Anyway, I was wondering: is this realization question solvable for homology with coefficients in or ?

• green arrow if A becoming more likely causes B to become more likely, and with a
• red arrow if A becoming more likely causes B to become less likely.

Shorter arrows suggest stronger relationships (technically, a lower p-value).

Running the algorithm on the market data since January 1, 2008 with a lag of two days produces the following graph:

And so, we see that the market data is encoding some

• tautologies (McCain’s nomination makes him more likely to be president, and McCain’s being president makes it more likely that a Republican is president) but also some
• conventional wisdom (a recession makes Clinton more likely to be nominated, but Obama less likely to be nominated; perhaps the perception that Obama would fare better in the general election explains the red arrows from “Democrat President” to Clinton, and the green arrows from “Democrat President” to Obama).

It’s amazing to me (and hopefully also to you) that the relationships between the prices of these Intrade contracts manages to encode popular sentiments.

There’s a library for the statistical package R to do the Granger test, and Intrade produces CSV market data. I fed the market data for various contracts since January 1, 2008 into R, and the output of that into GraphViz to make a nice-looking visualization; in particular, I connect to if Granger-causes with -value less than 0.05. Darker arrows have smaller -values. This is all an embarassing misuse of statistics and -values, but it is quick and easy to do, and the results are fun to see.

Here is the graph for a lag of one day (i.e., does yesterday’s value of predict today’s value of ):

Here is the graph for a lag of two days (i.e., can the two previous days of data for be used to forecast the next day of data for ):

And here is the graph for a lag of three days:

Don’t take this too seriously. And one word of warning: an arrow from to does not mean that if is more likely, then is more likely—rather, it ought to mean that past knowledge of can be used to forecast . I suppose it would be interesting to add some color for the direction of the relationship, and maybe I’ll do that when I have another free hour.

In particular is a torus in , and here is a quicktime movie of it spinning:

I’m particularly fond of this, as you can really see that four squares are coming together at each vertex (hence, it has zero curvature), and you can see the hole in the torus as it spins.

The complex is a genus five surface in , and here is a quicktime movie of it spinning:

I represented the extra dimensions with color—not that it helps much!

Here are a few books that I wouldn’t want to be stranded on an island with:

• Federal Income Tax: Code and Regulations Selected Sections
• A Million Random Digits with 100,000 Normal Deviates
• Government Phone Book USA 2000
• How to Build an Igloo: And Other Snow Shelters

Do you have other ideas for awful desert island reading?

Anyway, the graph is huge; and the usual visualization tool (Graphviz) doesn’t work particularly well on the whole graph, so I took a few hundred vertices around pine, apple, sauce, pan, and cake. The result was the following:

So I ran the clustering program on 15 texts written by three authors, and here is the result:

The largest eigenvalue is 25 times bigger than the next largest eigenvalue, and picks out the author pretty well. The top pile consists of Jane Austen’s books (with Emma split into three volumes). The middle pile consists of Sir Arthur Conan Doyle’s books, with the Sherlock Holmes mysteries (Valley of Fear, Sign of Four, and Hound of the Baskervilles) grouped closer than the others. The bottom pile are five of Shakespeare’s plays.

(more…)

The result was the following graph:

I know little about Shakespeare, so I can’t say too much about the above image. I’d love to know what you think: does this arrangement of his plays make any sense?

Given that modern processors are so good at vector and matrix calculations, I’m surprised that this sort of visualization tool doesn’t appear in more places. For instance,

• Your blogs and email could be organized this way. Imagine lasso-ing a bunch of similar emails to reply to them all at once!
• News could be organized into nice piles.
• Your desktop and personal files could be arranged automatically into relevant piles.

Then again, maybe the idea of piles appeals to me more than most people—just look at how I organize the papers and books on my desk!