Dance Card

[Rex T]

Hi All
I can usually get my head around these sort of problems but just cannot for some reason figure this one.
If you have 7 Dance competitors is it possible for each to have exactly 3 dances each if not can it be done with 4 dances or 5.
thanks Rex

 

[Ishuda]

Hi All
I can usually get my head around these sort of problems but just cannot for some reason figure this one.
If you have 7 Dance competitors is it possible for each to have exactly 3 dances each if not can it be done with 4 dances or 5.
thanks Rex

I believe you need to be more specific. For example, if each competitor danced by themselves you could just have 21 dances so it appears to be obvious that they don't always dance alone. It also can not be a group dance of 7 or you would just hold three dances. And, assuming it were two competitors dancing together, would two competitors be allowed to dance two or three dances together? ...

 

[Ishuda]

Hi All
I can usually get my head around these sort of problems but just cannot for some reason figure this one.
If you have 7 Dance competitors is it possible for each to have exactly 3 dances each if not can it be done with 4 dances or 5.
thanks Rex

Assuming a couples dance and no two competitors are allowed to dance twice together, lets look at the dances:
In the first 3 dances (first round), competitor a get a bye.
In the second three dances (second round), competitor b gets a bye.
In the third round, competitor c gets a bye.

Now, if competitors a, b, and c are all the same competitor, everyone but that competitor has completed their three dances and competitor has no one to dance with to complete the competition.

If competitors a and b (or a and c or b and c) are the same competitor and different than competitor c (or b or a), then every one but a and c have completed the competition and competitor c needs 1 dance (having sat out once) and a needs two dances (having sat out twice).

If competitors a, b and c are distinct, then we have three competitors needing a dance each to complete the competition.

The same kind of analysis can be done for 4 dances or, in fact, for any number of dances.

 

[pka]

If you have 7 Dance competitors is it possible for each to have exactly 3 dances each if not can it be done with 4 dances or 5.

This may surprise some but this is simply a graph theory question.
The competitors are the vertices and the edges are determined by the dance pairs.
The degree of each vertex is the number of edges on that vertex.

Well known theorem: in a simple graph the is an even number of odd degree vertices.

How does that answer your question?

 

[Rex T]

Hi Thanks for the replies,
To be more specific this is actually a competition in martial arts where each competitor has to have the same amount of fights they must not fight the same person twice. We have always had everybody fight everybody in the group but as their are so many groups it makes it to time consuming so we are looking at ways to keep it fair but lesson the amount of fights.
Regards Rex

 

[Rex T]

Thanks for the reply PKA but this has gone over my head

 

[Rex T]

Thanks for the reply Ishuda
Unfortunately we are trying to avoid bye's
Regards Rex

 

[Ishuda]

Thanks for the reply Ishuda
Unfortunately we are trying to avoid bye's
Regards Rex

The final result would not be a bye for anyone, it is only when considering the first three matches. That is, in the first three matches 2*3 people compete, so someone doesn't compete in the first three matches.

EDIT: You might see if this site would help
http://www.printyourbrackets.com/