Tonight's Random Math Question

[JoeSchmuckatelli]

In a sixty-four team double-elimination Corn Hole tournament playing 10 matches at a time... How many possible games are there before a winner will be declared? 

(In my case, the answer was 'two' to find the first loser - but theoretically, if skill wasn't an issue... How many games might we have been required to play? 

[RCgothic]

R1 is 32 matches. 32 teams lose but are not eliminated.

R2 is 32 matches. 32 teams can lose and not be eliminated.

Now every team has lost one and a standard elimination ensues.

R3 eliminates 32.

R5 eliminates 16.

R6 eliminates 8.

QFs eliminates 4

SFs eliminates 2

F eliminates 1

Total matches are 127 max. I'm not sure what relevance the 10 at a time has except to extend the time this takes.

[JoeSchmuckatelli]

5 hours ago, RCgothic said:

R1 is 32 matches. 32 teams lose but are not eliminated.

R2 is 32 matches. 32 teams can lose and not be eliminated.

Now every team has lost one and a standard elimination ensues.

R3 eliminates 32.

R5 eliminates 16.

R6 eliminates 8.

QFs eliminates 4

SFs eliminates 2

F eliminates 1

Total matches are 127 max. I'm not sure what relevance the 10 at a time has except to extend the time this takes.

Thanks!  10 pitches is what they had set up.  Probably no relevance to the overall problem but it was present, so I included it. 

... 

I know I specified removing skill - but as an aside, I learned that there are people who take Corn hole very seriously. 

Watched a couple of people warming up.  They had custom shirts and had brought their own bags.  Watched them each throw bags 12 times.  Only one failed to drop in the hole. 

Contrast this with my wife and I who were excited to keep any single bag on the board at all! 

[K^2]

So yeah, in general, if you have a standard double elimination tournament between n teams, where n is a power of two, there will be 2n-2 or 2n-1 games to be played, depending on whether the winner's bracket champion wins or loses their first try in finals. So you may need to run 127 games for 64 teams.

But distributing them to 10 pitches is not trivial. At first, you can probably just send teams to pitches as they open up. If this was always the case, and games took roughly the same duration, you'd expect about 13 rounds of games to get through these 127. But of course, when you get down to just a few teams, you have to wait for some of the games to resolve before you know who is even playing whom. So the last few games end up utilizing only a few pitches. That said, in practice, it's usually great for tension.

I used to do robotics in High School, and the competitions were always like this. You're in the pit, making final adjustments, somebody makes the run to the official brackets to see who we're playing next and at what table. Do we have another ten minutes to run a test after a battery swap? No? Well, cross fingers and hope for the best. The chaos of parallel games in double-elimination is perfect for that kind of a competition.

[farmerben]

I thought the question was: "How many possible games are there before a winner can be declared?"  In which case you need to count every possible pairing, even the ones that never play out in an actual tournament.  Unfortunately, I do not have the answer.

[JoeSchmuckatelli]

1 hour ago, farmerben said:

I thought the question was: "How many possible games are there before a winner can be declared?"  In which case you need to count every possible pairing, even the ones that never play out in an actual tournament.  Unfortunately, I do not have the answer.

I think it's been answered - taking skill out, leaving the coin flip - the answers above seem correct (given my little experience with game theory/ probability math). 

Thus I don't think 'every possible pairing' is required. 

I've seen the formulas above before.  I'm just not competent enough with probability math to figure it out on my own, and thus appreciate the help! 

[RCgothic]

Having a further thought, 127 is the maximum number of games in 8 rounds if every team loses once in the first 2 rounds, but the maximum number of rounds is greater.

2 rounds to eliminate 32 who lost twice.

2 more rounds to eliminate 16

2 more rounds to eliminate 8

2 more rounds to eliminate 4

2 more rounds to eliminate 2

Best of 3 rounds to eliminate the last 1 because neither team has lost until the final.

The maximum number of rounds to played by the winning team is 13 and the total number of matches is still 127.

Crazy that it comes out to the same number of matches, but now I think about it further I guess it has to. Everyone has to lose twice except the winners who may or may not lose once. That's 126 or 127 matches.

 

Edited August 21, 2023 by RCgothic