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#1 2012-05-31 19:40:19

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Ping pong tournament

9 ping pong players will participate in a tournament. There are only 3 tables where 3 games can be played simultaneoroflolusly. Two players will be playing in each game, while a third will be acting as the arbitrator. For example, the first round would be 12 3 45 6 78 9 with 3, 6 and 9 being the arbitrators and 12 45 78 playing against each other.
There are two rules for the tournament: It must be completed in 12 rounds of 3 simultaneous games, where each player will play against each of the other 8 only once, and will be arbitrating exactly 4 games. Moreover, after each player arbitrates one game, he must play at least 2 times against another athlete before being allowed to arbitrate again.
You will realize that it is impossible to have all two conditions met together. Can you write a schedule that would meet the first condition and would break the second condition for a minimum number of times? The answer must be 12 rows of 9 digits each, where the 3rd, 6th and 9th digit of each row will be the arbitrator, while all the others will be the players playing against each other, e.g. 12 3 45 6 78 9 for the first round (1 is playing against 2 and 3 arbitrates, 4 against 5 etc).

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#2 2012-05-31 20:02:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

Re: Ping pong tournament

Hi anna_gg;

It must be completed in 12 rounds of 3 simultaneous games

where each player will play against each of the other 8 only once

be arbitrating exactly 4 games

Moreover, after each player arbitrates one game, he must play at least 2 times against another athlete before being allowed to arbitrate again.

There are 4 conditions here. Which of these are not to broken and which can be?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2012-06-01 02:24:43

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Ping pong tournament

Well, you are right; actually I was considering the first 3 conditions as one smile These 3 can't be broken.

The 4th condition, which is "after each player arbitrates one game, he must play at least 2 times against another athlete before being allowed to arbitrate again", can be broken, but we request that this happens the least number of times.wave

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#4 2012-06-01 02:26:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

Re: Ping pong tournament

Hi anna_gg;

Okay, thank you. You do understand that this problem is somewhat more difficult than a progressive dinner or social golfer problem and that most of them do not have solutions.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2012-06-01 18:15:51

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Ping pong tournament

Hi Bobbym,
I didn't say it is easy smile After all, we are not here for the easy ones smile
This one, however, does have a solution because it was published it a riddles site.

Have a nice weekend!

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#6 2012-06-01 18:33:26

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Ping pong tournament

Let's start with
123456789 where 3, 6 and 9 are the arbitrators.
Then 132465798
231564897 where 1 4 and 7 have played 2 games, in order for them to be allowed to arbitrate.

That was the easy part, working on the next steps smile

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#7 2012-06-02 21:33:14

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Ping pong tournament

143286759

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#8 2012-06-02 21:36:27

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

Re: Ping pong tournament

Hi anna_gg;

My feeling is that a program will be necessary. So far none have worked.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#9 2012-06-02 22:00:02

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Ping pong tournament

You are absolutely right, but I don't have any experience in programming sad

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#10 2012-06-02 22:06:04

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

Re: Ping pong tournament

I have lots and it is not helping.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#11 2012-06-03 00:51:44

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Ping pong tournament

Am sure someone will show up with a brilliant idea smile

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