Saturday, February 10, 2007

Just to Scare Off the Non-Mathematicians

It's been a while since I posted anything about mathematics that wasn't a protracted whimper on my part. In the spirit of improvement, here are two interesting problems I've solved for HW this semester. (Incidentally, I'm not asking you to try these at home; the solutions turn out to be applications of some relatively hefty theorems, not methods you'd arrive at starting from scratch. I just thought it was interesting to show some concrete examples of what I've been up to.)

1. Given any string of digits (say, 1-2-3), there exists a power of 2 which begins with those digits (in this case, 2^90 ≈ 1.23794 x 10^27, 2^379 is approximately 1.23131 x 10^114, etc). (Actually, the exercise proves that there are infinitely many such powers of 2.)

2. If a knight starts on a corner of an ordinary chessboard and moves randomly thereafter (choosing with equal probability from each of its legal moves), how long on average will it take for the knight to return to its original space? (More precisely, if I were to pay you $1 for every move it took before returning to that corner, what sum would you have to pay me in advance to make it a fair gamble?)

oh man, you missed out
For those interested, the first problem can be put into a better context by taking the base 10 log of powers of 2 and always dropping the part before the decimal point (because that doesn't affect the digits, only the power of 10); then it's a question of whether these points are dense in [0,1), and an 'ergodic' theorem (proved with a little Fourier analysis) proves that they are.

And the second problem is from my Probability Theory class, which is now covering Markov chains (situations in which the future doesn't depend on the past, only the current situation). There's a neat theorem that shows that the expected time of return can be derived from a stationary probability measure on the space (in this case, we put positive weights on each square that correspond to the number of ways to arrive at that square).

Looking back, I didn't really do much of a job removing/explaining jargon, but I can't think of a better way at the moment. Anyhow, I think this stuff is pretty excellent, and I do it so you don't have to!

No comments: