Tuesday, March 19, 2013
alt NCAA tournament
I'm not one of those people who think the NCAA tournament needs to be bigger; fairness (defined as "maximizing the likelihood that a top team wins the tournament") would dictate that it be smaller. I understand, though, that people like the spectacle of all those games, and indeed the "automatic bids" that go to the champions of conferences seem to be popular, especially when they result in upsets. The thing is, even with 31 automatic bids, that leaves 17 "at large" bids; since 9 of the top 26 teams this year were their conference champions, that means even a 48 team tournament can include all the automatic bids as well as any team that has a realistic claim on being the best team in the country.
One way to do a 48-team tournament is the way it was done 30 years ago; the bottom 32 teams play an extra round in a single-elimination tournament. Having previously presented a scheme for a 12-team sesqui-elimination tournament, I propose that that scheme be used for each of the four regions, at least for the first four rounds; the top four seeds from each region would play a single-elimination tournament the first weekend, with the "champion" of that bracket seeded into the final 8, while the other three teams are dropped into a single-elimination bracket with the other 8 teams, and would have to make it through a second weekend undefeated in order to make it into the final 8.
|
|
|
|
Thursday, March 14, 2013
geometry of child-herding
My son is walking, and running, and exhibits an independent streak. I can still run faster than he can, and, while I like to give him freedom to roam, I always want to make sure I'm positioned such that I can outrun him to the street.
He, in the above diagram, is C, and the vertical line is the curb. The larger ellipse (or semi-ellipse) has him at one focus, with the curb forming its minor axis. This particular ellipse is drawn supposing I can run twice as fast as Calvin can; a larger ratio results in a bigger, less eccentric (more circular) shape, while a ratio close to 1 would largely contain a lane between him and the curb, but 2 seems like about the right ratio, and is a good one for illustrative purposes. Thus the rule is that I have to stay within the ellipse; as long as I do, there is no point along the vertical line to which he can out-race me. As he moves around, I can envision the ellipse moving with him, shrinking when he moves toward the curb and expanding when he moves away from it.
I've placed myself (D) just inside the ellipse. If he suddenly decides he wants to maximize his chances of making it to the street, and I want to maximize my chances of catching him — suppose we're not quite certain of the 2:1 ratio — then we're both racing toward the same spot along the curb. This point (T) can be constructed by drawing the line that runs equidistant between us; that line intersects the curb at a single point, such that points on the curb to one side of that intersection are closer to me than to him, and vice versa. A circle can be drawn with this as its center and both of us along its arc; T is where the circle intersects the curb closer to him than to me. If I started exactly on the ellipse, and we both started running toward T at the same time, I would catch him exactly at the curb; if he runs in any other direction and I start running toward T, then I will be inside the new ellipse, while if he runs straight toward T and I run in any other direction, I will be outside the new ellipse. Note, however, that T was constructed without regard to the ellipse; it was constructed based only on his location, my location, and the curb. If I'm suddenly unsure about the ratio of our speeds and whether I'm in the proper ellipse or not, the most conservative thing I can do is head toward the T (assuming, of course, that this doesn't affect his behavior).
