As a member of boy scouts I was considering the following problem:  

suppose you're organising some kind of olympic games...  

\*You divide the boys in $2n$ teams (subsets of equal size)  

\*There are $2n-1$ types of sports which all involve two-team matches  

\*There will be $2n-1$ time slots: during every time slot every team will play one of the sports in a match against one other team.  

\*Constraint 1: Every sport can be played by only two teams simultaneously.  

\*Constraint 2: At the end, every team must have played each type of sport.  

\*Constraint 3: At the end, every team must have faced all other teams in one match. 


The problem is trivially solved for $n=1$, but
one can easily check that it cannot be solved for $n=2$ and I have some confidence that neither it can be solved for $n=3$. 


Is this problem only solvable for $n=1$? 


Is, however, the variant of... 


\*$2n$ teams, $2n$ sports and $2n$ time slots.  

\*constraint 1 unchanged  

\*constraint 2 unchanged  

\*constraint 3 being relaxed to: "At the end, every team must have faced all other teams in at least one match" 


...solvable for all $n \neq 2$ (I have found a solution for $n=1, 3, 4, 5$ but since writing them down is a lengthy business, I'll not give them here unless someone's really interested)?  

If so, can you give details of a constructive method to find a solution for a particular $n$?



two boy scouts problems
Problem 1 is just asking for a Room square of size 2n-1. On the [Wikipedia page for Room squares](http://en.wikipedia.org/wiki/Room_square) or in Anderson's book "Combinatorial designs and tournaments", for example, you'll find that the cases n=2,3 are the only ones for which construction is impossible.


