Isn't it just 2^67?
A combinatorics problem for the group to solve
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TBentley wrote:
Isn't it just 2^67?
No, because there are 68 teams total (counting the First Four) and the First Four games are separate from the single-elimination tournament.
How is it defined? 60 teams have a guaranteed berth, and in each of the 4 "regions" there's 1 play-in game to determine which of two teams gets the 16th seed in the region? Are seedings important besides the four play-in games? A preliminary question: how many games are played during the main tournament ... that's the easy one, 63, right? because each game eliminates one team.
Cavatine wrote:
How is it defined? 60 teams have a guaranteed berth, and in each of the 4 "regions" there's 1 play-in game to determine which of two teams gets the 16th seed in the region? Are seedings important besides the four play-in games? A preliminary question: how many games are played during the main tournament ... that's the easy one, 63, right? because each game eliminates one team.
Yes, 63 games are played in the main tournament. Seedings would be important if we weren't trying to figure out every single bracket, regardless of what seed any particular game was.
I can think of several questions you might be asking here, which are quite different:
a) Given N teams with N>=68, how many different ways can the committee create the seedings and set up the tournament
b) Once the committee has set up the tournament, how many different ways can an ordinary person on the Internet guess the outcome of the 67 games? (Here I do think the answer is 2^67. All you have to do is pick one of the two teams for each branch of the binary tree, starting with the preliminary round.)
If there are more choices to be made like the point spread or the number of overtimes etc. then 2^67 is no longer the answer, right?
An interesting point is that there are more complicated ways to derive or express the answer, and then there can be an algebraic expression with 2^67 on the right hand side and some complicated thing on the left hand side, and that could turn out to be useful later in some other context.
As March Madness is currently underway (and I have filled out a bracket for the chess.com bracket pool), I though all of us intellectuals could try and solve this problem (regardless of whether or not you like college basketball): How many possible brackets are there? For those who don't follow March Madness, a preliminary 4 games are played (the First Four) to determine the 4 of the 64 teams playing in the main tournament. From there, the tournament is single-elimination. I don't even know how to start going about solving this problem, as the final number must be huge, but I hoped the group might be able to solve (or approach solving) the problem. And please, no looking up the answer. It spoils all of the fun.