Math Is Fun Forum

Galactico076

Novice

Registered: 2023-12-27

Posts: 1

Sports schedule

Hi, I don't know what the right category for my problem should be, so sorry in advance if this is not the right one.
I'd like some help with te following.
There is a sports match between team A and team B.
Each team consists of 6 players. All matches are doubles matches.
Every team member will play with each of its other team members. So the match consists of 5 rounds of 3 doubles matches = 15 matches.
A possible line up for team A could be:
Round 1
1/6
2/5
3/4
Round 2
1/5
2/4
3/6
Round 3
1/4
2/3
5/6
Round 4
1/3
2/6
4/5
Round 5
1/2
3/5
4/6
Now each player of team A has played 1 exactly match with each of its team members.
My question: what should the line up of team B look like to make sure:
-each team member of team B plays a match with every team member (like team A does).
-the number of different opponents from the other team for each player is maximised
Since each player plays 5 matches and there a 6 opponents in the other team and all matches are doubles matches (making it 10 opponents) you might think there is a schedule whereby each player plays 2 of the players of the
opposing team once and 4 players twice, making it 10. But I don't know if that's possible.

Kind regards, Carlos
p.s. Chat GPT does not come with good solutions for this problem, I've tried it.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Sports schedule

hi Carlos,

Welcome to the forum.

Sorry it's been a while. Been busy with Christmas stuff.  As you've posted a question,it would be better in the help me section, but not to worry. I think there is a way I can move it but it's not that vital.

MAJOR EDIT:

I wanted a single character for the Bs so I've used 789XYZ.

There are exactly 15 matches and 15 combinations of teammates so I decided to try and find a set of opponents to your A arrangement that had every pair of players once only.

To achieve this I made a 15 by 15 table with across 12, 13, ... 56 and down 78, 79, ... YZ.

Each time I arranged a match I put a tick in the right grid box [ eg. 1678 I put a tick in (16,78) ]
and crosses in the rest of that row and column. Here's one solution:

Round one 1678, 259X, 34YZ
Round two 159Y, 248Z, 367X
Round three 14XZ, 238Y, 5697
Round four 137Z, 2689, 45XY
Round five 128X, 357Y, 469Z

Bob
Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob