# interesting mathematics problem... I am stuck! need some help for a problem solving question please!! :)A tennis club is having a mixed doubles tournament. families enter one male and one female....

interesting mathematics problem... I am stuck! need some help for a problem solving question please!! :)

A tennis club is having a mixed doubles tournament. families enter one male and one female. they find out that it is actually 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 her 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 from family 2 and so on...

an example of an allowable match is M3F1 v M6F4

a). give an example of a TWT for *four* families, in which one of the matches is M1F4 v M2F3

b). give an example of a TWT for **five** families in which one of the matches is M1F3 v M2F5.

c). Find *ALL* TWTs for four families.

*I am completely stuck with these questions! They are very tricky! I dont even know where to start! I really need a bit of help! what method should I use to solve it? is there an equation? Thank you SO much for any assistance!! *

*-A*

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You ask several questions. I will answer the first one. You may apply the same method for the next questions.

a) Look at what is given.

You know that one game is M1F4 vs M2F3

Condition 1- for M1

implies that there will be the following games

M1?? vs M3??

M1?? vs M4??

and no other games for M1 otherwise M1 would play against a male twice.

Condition 2 implies that M1 will play against F2 F3 F4 F2 is already set in the game M1F4 vs M2F3. There is just F2 F4.

M4 can't partner F4 (rule 3)therefore M4 will partner F2 and M3 will partner F4.

The games for M1 will be

M1F4 vs M2F3

M1?? vs M3F4

M1?? vs M4F2

Now rule 3 implies that M1 will partner F2 and F3. Let's find when.

M1 cant partner F2 in the last game since F2 is in the opposite team. Therefore in the last game M1 will partner F3

and M1 will partner F2 in the second game.

So far we have determine

**M1F4 vs M2F3**

**M1F2 vs M3F4**

**M1F3 vs M4F2**

Now let's find the games for M2 using the same method.

We know M2F3 vs M1F4.

We will have using rule 1:

M2?? vs M3??

M2?? vs M4??

rule 2: M2 will play against F4 F3 F1.

He plays F4 in the first game.

He can't play against F3 in the second game since M3 doesn't partner F3. therefore he will play against F3 in the last game and play against F1 in the second game.

M2F3 vs M1F4

M2?? vs M3F1

M2?? vs M4F3

M2 will partner F3 (in the first game) and F1 F4. He can't partner F1 in the second game (F1 is already busy) therefore we will partner F1 in the last game and F4 in the second game.

We have so far

**M2F3 vs M1F4**

**M2F4 vs M3F1**

**M2F1 vs M4F3**

Now determine the games of M3.

We already have determined M3F1 vs M2F4

and M3F4 vs M1F2 in the previous steps.

M3 is already scheduled to play against M1 and M2.

The last game will be against M4.

M3?? vs M4??

M3 plays against F4 and F2 therefore M3 needs to play against F1.

M3 partners F4 and F1 therefore M3 needs to partner F2

The last game for M3 will be

**M3F2 vs M4F1**

Now let's summarize what we found.

**M1F4 vs M2F3**

**M1F2 vs M3F4**

**M1F3 vs M4F2**

**M2F4 vs M3F1**

**M2F1 vs M4F3**

**M3F2 vs M4F1**

We check that any male plays any other male. yes.

Any female plays against any other female. yes

Any male plays against any female except his family member. yes

Any female plays against any male except her family member. yes.

Any male partners any female except his family member. yes

Any female partners any male except her family member. yes.

Therefore the games we found satisfy the 3 rules. It is a solution

**M1F4 vs M2F3**

**M1F2 vs M3F4**

**M1F3 vs M4F2**

**M2F4 vs M3F1**

**M2F1 vs M4F3**

**M3F2 vs M4F1**