We are interested in a solution to the following scheduling problem, or any information about how to find it or its existence. This one comes from real life, so you will not only be helping a mathematician quench his thirst of knowledge!



> 
> We have 18 players playing a certain sport (let's say curling) on 3 different alleys (6 players per alley) at the same time. They play 17 games and we want that every combination of 2 players play exactly 5 times together.
> 
> 
> 


(As Douglas Zare points out in a comment below, this is known as a resolvable block design with t=2, v=18, k=6, lambda=5 (and b=51, and r=17)).


We asked around and someone came up with a near solution: almost every pair playing 5 times except for a few 6's and 4's. Brute force seemed too slow so we tried with a genetic algorithm, to no avail (being complete beginners in this, we could not even get close to the near-solution that we had, so we do not draw conclusions from our experiments).


I found the near-solution in my old files, in case anyone wants to tinker a bit.



```

{{1, 2, 3, 4, 5, 6}, {7, 8, 9, 10, 11, 12}, {13, 14, 15, 16, 17, 18}},
{{1, 6, 10, 12, 14, 16}, {2, 3, 8, 11, 15, 17}, {4, 5, 7, 9, 13, 18}},
{{1, 5, 7, 8, 15, 16}, {2, 4, 10, 11, 13, 14}, {3, 6, 9, 12, 17, 18}},
{{1, 4, 8, 9, 14, 17}, {2, 6, 7, 10, 15, 18}, {3, 5, 11, 12, 13, 16}},
{{1, 6, 8, 11, 13, 18}, {2, 4, 9, 12, 15, 16}, {3, 5, 7, 10, 14, 17}},
{{1, 2, 7, 12, 13, 17}, {3, 4, 8, 10, 16, 18}, {5, 6, 9, 11, 14, 15}},
{{1, 3, 9, 10, 13, 15}, {2, 5, 8, 12, 14, 18}, {4, 6, 7, 11, 16, 17}},
{{1, 5, 10, 11, 17, 18}, {2, 6, 8, 9, 13, 16}, {3, 4, 7, 12, 14, 15}},
{{1, 2, 9, 11, 16, 18}, {3, 6, 7, 8, 13, 14}, {4, 5, 10, 12, 15, 17}},
{{1, 4, 8, 12, 15, 18}, {2, 3, 7, 9, 11, 14}, {5, 6, 10, 13, 16, 17}},
{{1, 3, 7, 14, 16, 18}, {2, 5, 8, 9, 10, 17}, {4, 6, 11, 12, 13, 15}},
{{1, 5, 6, 9, 12, 14}, {2, 3, 10, 13, 15, 18}, {4, 7, 8, 11, 16, 17}},
{{1, 3, 10, 11, 12, 16}, {2, 4, 5, 8, 13, 14}, {6, 7, 9, 15, 17, 18}},
{{1, 2, 3, 4, 6, 17}, {5, 7, 11, 12, 13, 18}, {8, 9, 10, 14, 15, 16}},
{{1, 4, 7, 9, 10, 13}, {2, 12, 14, 16, 17, 18}, {3, 5, 6, 8, 11, 15}},
{{1, 2, 5, 7, 15, 16}, {3, 8, 9, 12, 13, 17}, {4, 6, 10, 11, 14, 18}},
{{1, 11, 13, 14, 15, 17}, {2, 6, 7, 8, 10, 12}, {3, 4, 5, 9, 16, 18}}

```


Is there a tournament schedule for 18 players, 17 rounds in groups of 6, which is balanced in pairs?
A collection of $6$-tuples on 18 points with the property that each pair is covered $5$ times is a balanced incomplete block design with $(v,k,\lambda) = (18,6,5)$ and $t=2$. The condition that you can schedule the matches to occur simultaneously in $17$ rounds is that the design is resolvable. 


[This article](http://www.iop.org/EJ/abstract/0025-5734/28/3/A05) claims to construct resolvable block designs with parameters including $(18,6,5)$.


