I’m leaving for Croatia today, so no blog posts for a while. In the meantime here’s some random stuff I thought was interesting:
I started reading about n-in-a-row games in arbitrary dimensions after I read this post on Good Math, Bad Math, and there are tons of fascinating results:
Gomoku (five in a row) has been proven to be a win for black on anything bigger than a 15×15 grid.
For any n-in-a-row and board length at least n, there is a dimension d where if the board is filled someone has to have won. This is true even for an arbitrary number of players.
There are more, but as I write this I’m thinking other people might not find them as interesting as me :).
It’s amazing how many Simpson’s writers have a math/cs background. I always wondered who came up with that joke in episode where Bart goes to a gifted school and they take the derivative of r^3/3 and it turns out to be r^2dr, which they rewrite as “r dr r” and laugh.
Apparently Chomsky famously wrote:
(1) Colorless green ideas sleep furiously.
(2) Furiously sleep ideas green colorless.
. . . It is fair to assume that neither sentence (1) nor (2) (nor indeed any part of these sentences) has ever occurred in an English discourse. Hence, in any statistical model for grammaticalness, these sentences will be ruled out on identical grounds as equally `remote’ from English. Yet (1), though nonsensical, is grammatical, while (2) is not.
I’m not sure why, but nothing gets me more riled up than people making grandiose claims without checking them empirically. I mean it was 1957, and I’m sure he’s thinking of counting n-grams, but my god “any statistical model’ will do X??? This is just obviously wrong.
Fernando Pereira built a simple bigram model with a hidden variable proving this wrong, and wrote a nice paper about it.