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Math Help - Football Fixtures Help

  1. #1
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    Football Fixtures Help

    Hi - I'm having spme problems trying to work out some football fixtures for my 5-a-side league and wondered if any of you were up for a challange - or could suggest a way to work it out.

    The rules are as follows....

    There are 2 divisions, of 6 teams each. Assume (A,B,C,D,E,F) (h,i,j,k,l,m)
    Each team must play each other team in their division 4 times
    There are 21 booking slots (although this could change +-1 depending on how it came out)
    Each booking slot consists of 2 * 1 hour 'mini-slots'
    A 1 hr slot contains 3 * 20 minute matches
    *This is the bit I have struggled with*
    Ideally within each booking slot 3 teams should play within a 1 hr slot. So on any one booking slot you would have 6 teams playing two games each, without a gap of more than one game, i.e 20 minutes for any one team.
    1 Booking slot can just be for one division on any one night.

    Good luck - and if you can solve it - Thank You!!
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by Jevs View Post
    Ideally within each booking slot 3 teams should play within a 1 hr slot. So on any one booking slot you would have 6 teams playing two games each, without a gap of more than one game, i.e 20 minutes for any one team.
    You would need to have at least one pair of teams playing consecutively for 40 minutes. For instance

    1st 20 mins: A versus B
    2nd 20 mins: C versus D
    3rd 20 mins: A versus B
    4th 20 mins: C versus D
    5th 20 mins: E versus F
    6th 20 mins: E versus F

    Alternatively they can all play consecutively for 40 minutes.
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  3. #3
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    Thanks Abstractionist, that's what I feared!

    We don't really want the same two teams playing each other on the same night either.

    What I was hoping was something along the lines of

    1st 20 mins: A versus B
    2nd 20 mins: A versus C
    3rd 20 mins: C versus B
    4th 20 mins: D versus E
    5th 20 mins: D versus F
    6th 20 mins: F versus E

    But I get the feeling that that won't propogate out over the other slots??
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  4. #4
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    Quote Originally Posted by Jevs View Post
    ...
    We don't really want the same two teams playing each other on the same night either.

    What I was hoping was something along the lines of

    1st 20 mins: A versus B
    2nd 20 mins: A versus C
    3rd 20 mins: C versus B
    4th 20 mins: D versus E
    5th 20 mins: D versus F
    6th 20 mins: F versus E

    But I get the feeling that that won't propogate out over the other slots??
    As TheAbstractionist stated, with the 20 minute constraint for the max delay a team is not playing, two teams will have to play consecutively.
    Modifying your arrangement slightly
    1st 20 mins: A v. B
    2nd 20 mins: A v. C
    3rd 20 mins: B v. D
    4th 20 mins: C v. E
    5th 20 mins: D v. F
    6th 20 mins: E v. F

    This will propogate. Just Rotate the starting team.
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  5. #5
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    Hi - thanks ever so much for that.

    Sorry to be pain but could you get me started on the next iteration just so I can pick up the pattern. Just not 100% sure whether you mean rotate the starting team..

    Do you mean the next game would be

    1st 20 mins: B v. C?

    Many thanks again for your help.
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  6. #6
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    Quote Originally Posted by Jevs View Post
    Hi - thanks ever so much for that.

    Sorry to be pain but could you get me started on the next iteration just so I can pick up the pattern. Just not 100% sure whether you mean rotate the starting team...

    .
    You have 6 teams. ( n=6 )
    For each team to play each other you will need 15 games.

     NumberOfGames = \dfrac{n}{2} (n-1)


    1v2 , 1v3 , 1v4 , 1v5 , 1v6
    2v3 , 2v4 , 2v5 , 2v6
    3v4 , 3v5, 3v6
    4v5 , 4v6
    5v6

    This is the round robin result.

    Since you will have an odd set:
    6 6 3
    you need to rotate half of the next round robin
    with the 3 above.
    3 6 6


    Here is a Pairing setup for 30 games.
    The constraint is that NO team will have to wait
    more than twenty minutes, and each team will
    play twice in two hours.


    This is not optimal.
    Optimal being giving each team a 20 minute break between plays.

    =======
    1 :: A .v F
    1 :: C .v F
    1 :: A .v B
    1 :: C .v D
    1 :: B .v E
    1 :: D .v E
    ------------
    2 :: A .v C
    2 :: A .v D
    2 :: C .v E
    2 :: B .v D
    2 :: E .v F
    2 :: B .v F
    ------------
    3 :: A .v F
    3 :: D .v F
    3 :: A .v E
    3 :: C .v D
    3 :: B .v E
    3 :: B .v C
    ------------
    4 :: A .v B
    4 :: A .v C
    4 :: B .v D
    4 :: C .v F
    4 :: D .v E
    4 :: E .v F
    ------------
    5 :: A .v D
    5 :: A .v E
    5 :: D .v F
    5 :: C .v E
    5 :: B .v F
    5 :: B .v C
    =======

    Then use the same schedule for the 2nd 30 games.

    Then each team will have played each other team 4 times.
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  7. #7
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    Wow - thanks Aidan you are an absolute star!! This has been giving me a headache for the last couple of weeks.

    Thanks again to you and everyone else who chipped in.

    Best Regards,
    Craig
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