Multi sports tournament $10$ teams $6$ sports simultaneous

Question

I'm hosting a bar sports tournament with $10$ teams and $6$ different sports (pool, darts, table tennis, foosball, beer pong and cornhole). Trying to get the fixtures as fair as possible so that each team plays each sport twice and playing against the same team multiple times is minimised. Are there any formulas to follow? There is potential for $11$ or $12$ teams to end up competing. I feel like having $12$ or $13$ would make this task a lot easier!

Edit: The plan is to have 12 rounds with each sport being played in each round. There is only one pool table, dart board etc available so the same sport cannot be played in the same round. Are there simpler solutions for say 12 teams?

Edit 2: I haven't done any mathematics beyond high school 10 years ago. I've just tried to figure it out as best I can drawing up tables and assigning teams against each other in different slots but ending up with teams playing the same sport or against each other again as there are limited playing slots. Ideally I'd like to know if there is a formula that can be applied to x amount of teams/sports/rounds for multi sports tournaments as all the fixture generators I've tried using only account for one sport being played.

Answer 1

Label the teams $0$ to $9$ and the sports $0$ to $5$. Here $k\,\%\,6$ denotes the remainder of $k$ modulo $6$.

First let each team $i$ play each other team $j$ in the sport $i+j\,\%\,6$:

$$ \matrix{&1&2&3&4&5&0&1&2&3}\\ \matrix{1&&3&4&5&0&1&2&3&4}\\ \matrix{2&3&&5&0&1&2&3&4&5}\\ \matrix{3&4&5&&1&2&3&4&5&0}\\ \matrix{4&5&0&1&&3&4&5&0&1}\\ \matrix{5&0&1&2&3&&5&0&1&2}\\ \matrix{0&1&2&3&4&5&&1&2&3}\\ \matrix{1&2&3&4&5&0&1&&3&4}\\ \matrix{2&3&4&5&0&1&2&3&&5}\\ \matrix{3&4&5&0&1&2&3&4&5&}\\ $$

The three sports that each team is missing are the ones that would have been on the diagonal and in the two columns you'd get if you'd extend the table to the right. With the diagonal in the first column and the two extension columns in the second and third column, this is:

$$ \matrix{ 0&4&5\\ 2&5&0\\ 4&0&1\\ 0&1&2\\ 2&2&3\\ 4&3&4\\ 0&4&5\\ 2&5&0\\ 4&0&1\\ 0&1&2 } $$

Now let each team $i$ for $0\le i\le8$ play team $i+1$ in sport $i+5\,\%\,6$. That takes care of the second and third columns, except for the game in sport $4$ for team $0$ and the game in sport $2$ for team $9$. These two games together with the ten missing games from the diagonal in the first column are four games each in the sports $0$, $2$ and $4$, and you can form two pairs for each of these sports in whatever way you like because none of the possible pairs have played a second game yet.

Answer 2

I've spent a good deal of time on this, and I haven't been able to do it, so in one sense, this isn't an answer at all, more of a progress report. On the other hand, someone may be able to suggest a modification to the method that will allow for solution. I tried it with both $10$ teams and $12$ teams, without success. I don't even have a viable approach in mind for an odd number of teams.

With $10$ teams each playing $12$ games, we will have $60$ matches. I determined a possible set of $60$ where each team played $9$ teams once and $3$ teams twice, and no two matches were identical. I then computed all possible rounds of $5$ matches chosen from among these $60,$ where a legitimate round includes all teams, and no game is played twice. With $10$ players, there were $172$ such rounds. Then I tried to choose $12$ rounds from among these $172$ that would partition the $60$ matches, but that was impossible. The result for $12$ teams was similar.

My program uses Knuth's dancing links algorithm for (generalized) set exact cover problems, so it's not apparent how to come up with some kind of approximate solution.

I've been wondering about trying different sets of $60$ matches, but there are so many isomorphisms! The program runs in a second or so, so trying lots of sets of matches isn't a concern, but of course, I'd like to generate non-isomorphic sets, if possible. At the moment, I don't see any quick way even to tell if two sets are isomorphic. (By isomorphic, I mean that one set of matches can be obtained from the other by permutation of the sports and of the teams.) I would welcome any suggestions about the isomorphism problem.

Failing this, there may be some acceptable relaxation of the conditions.
One thing that occurs to me is to go for as many full rounds as possible, and then hope that there aren't too many conflicts at the end. For example, suppose we could go for $11$ rounds, but not $12$. If the last $5$ games were all pool, that would be a problem, but if two were pool and the remaining ones were different games, that wouldn't be so bad. I'd like to know the OP's opinion of this approach.

EDIT

I modified the program to keep track of the maximum number of compatible rounds, and it was only $5$. Very discouraging. It is, however, possible to do this for an $8-$team tournament.

Round 1
   T0 vs T2 at table tennis
   T3 vs T6 at beer pong
   T1 vs T4 at darts
   T7 vs T5 at cornhole
Round 2
   T0 vs T1 at beer pong
   T4 vs T2 at table tennis
   T6 vs T7 at pool
   T5 vs T3 at foosball
Round 3
   T0 vs T4 at foosball
   T7 vs T3 at darts
   T2 vs T5 at beer pong
   T1 vs T6 at cornhole
Round 4
   T0 vs T3 at table tennis
   T7 vs T5 at darts
   T4 vs T2 at pool
   T1 vs T6 at beer pong
Round 5
   T0 vs T4 at pool
   T5 vs T6 at table tennis
   T1 vs T2 at darts
   T7 vs T3 at foosball
Round 6
   T0 vs T6 at darts
   T2 vs T7 at beer pong
   T1 vs T4 at cornhole
   T5 vs T3 at pool
Round 7
   T0 vs T6 at cornhole
   T4 vs T7 at table tennis
   T2 vs T3 at darts
   T5 vs T1 at pool
Round 8
   T0 vs T5 at foosball
   T7 vs T1 at pool
   T6 vs T4 at darts
   T2 vs T3 at cornhole
Round 9
   T0 vs T3 at pool
   T6 vs T7 at table tennis
   T4 vs T5 at beer pong
   T1 vs T2 at foosball
Round 10
   T0 vs T7 at beer pong
   T3 vs T1 at table tennis
   T2 vs T5 at cornhole
   T6 vs T4 at foosball
Round 11
   T0 vs T7 at cornhole
   T3 vs T4 at beer pong
   T6 vs T2 at foosball
   T5 vs T1 at table tennis
Round 12
   T0 vs T5 at darts
   T3 vs T4 at cornhole
   T6 vs T2 at pool
   T7 vs T1 at foosball

Here each team plays $5$ teams twice and $2$ teams once.