Compute This!

[Poolgod]

How many teams could win more than 8 times in a 12 man round Robin tournament?


Moved from Distributed Computing

AT Moderator ElFenix

 

[crashtech]

Welcome Poolgod! I'm not exactly sure what you are talking about yet, but the TeAm is always considering new challenges!

 

[Ken g6]

Sounds like homework.

 

[Orange Kid]

0

 

[Poolgod]

I need to know how many people could possibly win 8 or more matches in a 12 man round Robin tournament. I know half would lose first round and half of them would lose when they played each other, but not sure where to go from there. I'm considering having a tournament where I pay everyone who wins at least 8 matches in a 12 man round robin.

 

[Orange Kid]

Sorry thought it was one of those trick grammar sudo math problems.

 

[Fardringle]

As a round robin, are you setting it up so that every participant plays against every other participant once (so everyone plays 11 games total)?

If so, and if nobody wins more than 8 games, then (if my math is right), it is possible for six participants to have 8 wins and 3 losses, five participants to have 0 wins and 11 losses, and the final participant to have 7 wins and 4 losses. This is an unlikely scenario, but I believe it represents the maximum possible number that could have 8 wins in the tournament.

I just saw that you did ask how many could win MORE than 8 games, so the math is slightly different, but the results are actually still the same. Six participants win 9 games and lose 2, five participants have 0 wins and 11 losses, and the final participant drops to 1 win and 10 losses. Again, still highly unlikely but possible.

 

[Ken g6]

I don't think it's possible for more than one person to have zero wins in a round robin tournament. Everyone plays everyone else, so out of the two players who would have zero wins, one must have one win. (Or there can be draws.)

 

[Ken g6]

I think the maximum number of players with 8 wins is seven:

  1 2 3 4 5 6 7 8 9 10 11 12
1 X 1 1 1 1 1 1 1 1 1 1 1
2 0 X 1 1 1 1 1 1 1 1 1 1
3 0 0 X 1 1 1 1 1 1 1 1 1
4 0 0 0 X 1 1 1 1 1 1 1 1
5 0 0 0 0 X 1 1 1 1 1 1 1
6 0 0 0 0 0 X 1 1 1 0 0 0
7 0 0 0 0 0 0 X 0 0 1 1 1
8 0 0 0 0 0 0 1 X 1 0 1 0
9 0 0 0 0 0 0 1 0 X 1 0 1
10  0 0 0 0 1 0 1 0 X 1 0
11  0 0 0 0 1 0 0 1 0 X 1
12  0 0 0 0 1 0 1 0 1 0 X
  0 1 2 3 4 8 8 8 8 8 8 8

To get any of the other players more wins you have to steal from someone with more wins.

However, eight players with 8 wins might be possible.

Edit: No it's not, and I proved it above. If one player loses to everyone, then all the other players must have a win against them. That leaves at most 7 wins available for the last player.

Edit2: The number of teams that could win 9 times is probably five:

  1 2 3 4 5 6 7 8 9 10 11 12
1 X 1 1 1 1 1 1 1 1 1 1 1
2 0 X 1 1 1 1 1 1 1 1 1 1
3 0 0 X 1 1 1 1 1 1 1 1 1
4 0 0 0 X 1 1 1 1 1 1 1 1
5 0 0 0 0 X 1 1 1 1 1 1 1
6 0 0 0 0 0 X 1 1 1 1 1 1
7 0 0 0 0 0 0 X 1 1 1 1 1
8 0 0 0 0 0 0 0 X 1 0 1 0
9 0 0 0 0 0 0 0 0 X 1 0 1
10  0 0 0 0 0 0 1 0 X 1 0
11  0 0 0 0 0 0 0 1 0 X 1
12  0 0 0 0 0 0 1 0 1 0 X
  0 1 2 3 4 5 6 9 9 9 9 9

 

[Fardringle]

I don't think it's possible for more than one person to have zero wins in a round robin tournament. Everyone plays everyone else, so out of the two players who would have zero wins, one must have one win. (Or there can be draws.)

That's a good point. I forgot about that part, so if the "losers" all end up 0-11 or 1-10 (and ties/draws are not possible), a maximum of five people can win more than 8 times in the tournament.

 

[TennesseeTony]

Are you guys speaking English? I know a few words, from many languages, so I can 'read' the text you are typing.....but.....

 

[Markfw]

Just from a mod angle, if you are paying anyone in a tournament, I assume they have to pay to enter it. I think this is not allowed on our forums.

 

[crashtech]

Are you guys speaking English? I know a few words, from many languages, so I can 'read' the text you are typing.....but.....

Ha, I was in the same boat, as evidenced by my reply. I think it's in the wrong forum, but what do I know.

 

[Orange Kid]

1 and 0 are binary speak

 

[Ken g6]

In a round robin tournament everyone eventually plays against everyone else. My tables have a 1 where the person numbered in the row defeats the person numbered in the column. And a 0 for the reverse. The bottom row is the sum of the number of wins.