A tennis club is running a mixed doubles tournament for families. the families enter one male and one female in it. they soon discover that this is a tournament with a twist (TWT), in that they NEVER PARTNER OR PLAY AGAINST THEIR OWN FAMILY MEMBER.
the TWT is arranged so that:
1. each player plays against every person of the same gender exactly once.
2. each player plays against every person of the opposite gender, except for his or her family member, exactly once.
3. each player partners every person of the opposite gender, except for his or her family member, exactly once.
using the notation M1 and F1 for the male and female from family 1, M2 and F2 for family 2 and so on. An allowable match is M3F1 v M6F4.
QUESTION: find all TWTs for four families.
(I think that tournaments that consist of the same sets of matches but just in a different order count as the same, just copies of eachother) i dont even know where to start! i really need some help! I dont think there will be that many TWTs for four families, as with all the rules and limits specified it will factor many out!
Any help is greatly appreciated! Please help me! (+POINTS!)
thanks
the TWT is arranged so that:
1. each player plays against every person of the same gender exactly once.
2. each player plays against every person of the opposite gender, except for his or her family member, exactly once.
3. each player partners every person of the opposite gender, except for his or her family member, exactly once.
using the notation M1 and F1 for the male and female from family 1, M2 and F2 for family 2 and so on. An allowable match is M3F1 v M6F4.
QUESTION: find all TWTs for four families.
(I think that tournaments that consist of the same sets of matches but just in a different order count as the same, just copies of eachother) i dont even know where to start! i really need some help! I dont think there will be that many TWTs for four families, as with all the rules and limits specified it will factor many out!
Any help is greatly appreciated! Please help me! (+POINTS!)
thanks
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Each match consists of 4 people. Since no 2 can be from the same family, each match consists of 1 member from each family.
Each player partners with every person of the opposite gender, except for his or her family member. Therefore, all pairings for teams are:
M1F2
M1F3
M1F4
M2F1
M2F3
M2F4
M3F1
M3F2
M3F4
M4F1
M4F2
M4F3
Since there are 12 pairings and none can occur more than once, there must be 6 matches
There are 2 possible TWTs
Opponents for M1 F2 is either M3 F4 or M4 F3
Once we've decided on this, all other matches fall into place
M1 F2 v M3 F4
M1 F3 v M4 F2
M1 F4 v M2 F3
M2 F1 v M4 F3
M2 F4 v M3 F1
M3 F2 v M4 F1
M1 F2 v M4 F3
M1 F3 v M2 F4
M1 F4 v M3 F2
M2 F1 v M3 F4
M2 F3 v M4 F1
M3 F1 v M4 F2
Each player partners with every person of the opposite gender, except for his or her family member. Therefore, all pairings for teams are:
M1F2
M1F3
M1F4
M2F1
M2F3
M2F4
M3F1
M3F2
M3F4
M4F1
M4F2
M4F3
Since there are 12 pairings and none can occur more than once, there must be 6 matches
There are 2 possible TWTs
Opponents for M1 F2 is either M3 F4 or M4 F3
Once we've decided on this, all other matches fall into place
M1 F2 v M3 F4
M1 F3 v M4 F2
M1 F4 v M2 F3
M2 F1 v M4 F3
M2 F4 v M3 F1
M3 F2 v M4 F1
M1 F2 v M4 F3
M1 F3 v M2 F4
M1 F4 v M3 F2
M2 F1 v M3 F4
M2 F3 v M4 F1
M3 F1 v M4 F2
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hh, I don't like.