School Outing Combinatorial Design Problem

Question

So this is an actual organization problem I am dealing with right now as a high school teacher. There is a school outing, with $8$ groups of students. At the venue there are $7$ stations, where $2$ groups can compete against each other. Is there a way to organize the groupings over $7$ time intervals so that each group goes to each station exactly once, and each group plays each other group exactly once? Obviously over each interval there will be exactly $3$ stations that are unused.

One can generalize this problem to $2n$ groups, $2n-1$ stations, and $2n-1$ time intervals where $2$ groups play against each other, such that no group plays the same station twice, no group plays the same group twice, and no group gets left out in an interval. The original problem is the $n=4$ version of this.

The $n=1$ case is trivial (there are only two groups, and they play against each other once on the one station). The $n=2$ case can be fairly quickly seen to be impossible, as after the first interval, at least one of the stations must be used a second time, and there is no way to have a different pairing at that station that also doesn't have the same two groups compete.

Has this problem been worked on before? Is there any literature on it?

Edit: I have worked out a solution to the $n=4$ case:

Group: A    B    C    D    E    F    G

1:     12   34   56   78   XX   XX   XX

2:     57   68   XX   XX   13   24   XX

3:     XX   XX   14   23   58   67   XX

4:     36   XX   28   XX   47   XX   15

5:     XX   17   XX   45   26   XX   38

6:     XX   25   37   XX   XX   18   46

7:     48   XX   XX   16   XX   35   27

I'm still keen for a more general solution.

Edit 2: I am fairly certain the n=3 case is also impossible. There are enough values which can be filled in without loss of generality, after which sudokuing your way around the grid yields a contradiction. At this point I'm fairly sure that $n=4$ is the cutoff, after which there is a solution for all higher n, but that would be good to verify.

Answer 1

In terms of graph theory, you have a complete graph $K_{2n}$ on $2n$ nodes (one node per group) and want to decompose it into $2n-1$ perfect matchings (one matching per time interval). See Perfect 1-factorization of $K_{2n}$

But you also want its "transpose" to form such a decomposition. For $n=4$, here is a solution obtained via integer linear programming, as in Real World Problem - Groups Visiting Various Stations

       A       B       C       D       E       F       G
1      {}      {5,7}   {3,4}   {}      {2,6}   {}      {1,8}
2      {}      {}      {1,7}   {2,3}   {4,8}   {5,6}   {}
3      {2,4}   {3,8}   {}      {6,7}   {1,5}   {}      {}
4      {1,6}   {}      {}      {}      {3,7}   {2,8}   {4,5}
5      {}      {1,2}   {}      {5,8}   {}      {4,7}   {3,6}
6      {3,5}   {}      {6,8}   {1,4}   {}      {}      {2,7}
7      {7,8}   {4,6}   {2,5}   {}      {}      {1,3}   {}

And here's a solution for $n=5$:

       A       B       C       D       E       F       G       H       I
1      {}      {}      {5,8}   {}      {}      {6,7}   {1,4}   {3,9}   {2,10}
2      {1,2}   {}      {3,10}  {}      {5,7}   {}      {}      {4,6}   {8,9}
3      {}      {4,5}   {}      {1,8}   {6,10}  {2,9}   {3,7}   {}      {}
4      {5,9}   {2,6}   {}      {}      {}      {3,8}   {}      {1,10}  {4,7}
5      {}      {}      {2,4}   {3,6}   {}      {1,5}   {9,10}  {7,8}   {}
6      {3,4}   {7,9}   {}      {5,10}  {}      {}      {2,8}   {}      {1,6}
7      {}      {8,10}  {1,7}   {4,9}   {2,3}   {}      {5,6}   {}      {}
8      {6,8}   {}      {}      {2,7}   {1,9}   {4,10}  {}      {}      {3,5}
9      {7,10}  {1,3}   {6,9}   {}      {4,8}   {}      {}      {2,5}   {}

Answer 2

Yes. In terms of duplicate bridge you want a $4$-table Howell movement. Each set of boards is a station. The pairs playing against each other are the two groups competing at that station (set of boards). https://bridge.fandom.com/wiki/Howell_movement#4_tables