I am trying to do as the title says. I have 10 teams in the same group. Every team must play the rest once each but each of them will always alternate home and away. This means that if they play at home their first game, the second MUST be away, the third home, away, home, and so on.
I tried like this:
1st round
0 vs. 9
1 vs. 8
2 vs. 7
3 vs. 6
4 vs. 5
2nd round
8 vs. 0
7 vs. 1
6 vs. 2
5 vs. 3
9 vs. 4
3rd round
0 vs. 7
1 vs. 6
2 vs. 5
3 vs. 9
4 vs. 8
4th round
6 vs. 0
5 vs. 1
9 vs. 2
8 vs. 3
7 vs. 4
5th round
0 vs. 5
1 vs. 9
2 vs. 8
3 vs. 7
4 vs. 6
The method works fine until the $5^{th}$ round but not for any further rounds since it would imply that $0$ must play at home against a team from $1$ to $4$, which are supposed to play at home that round as well.
Is this mathematically possible? IF so, please show me how.
Thank you.
Yes, it is possible.
By way of illustration... here's how my solution starts
$$\begin{array} \\ 1 & BvA & DvC & FvE & HvG & JvI \\ 2 & AvD & CvF & EvH & GvJ & \\ 3 & DvB & FvA & HvC & JvE & IvG \\ 4 & BvF & AvH & CvJ & EvI & \\ 5 & FvD & HvB & JvA & IvC & GvE \\ \end{array}$$
(it might be that my interest in bell-ringing is an advantage here :-) )