Scheduling algorithm for a round-robin tournament?

The classic algorithm works like this:

Number the teams 1..n. (Here I’ll take n=8.)

Write all the teams in two rows.

1 2 3 4
8 7 6 5

The columns show which teams will play in that round (1 vs 8, 2 vs 7, …).

Now, keep 1 fixed, but rotate all the other teams. In week 2, you get

1 8 2 3
7 6 5 4

and in week 3, you get

1 7 8 2
6 5 4 3

This continues through week n-1, in this case,

1 3 4 5
2 8 7 6

If n is odd, do the same thing but add a dummy team. Whoever is matched against the dummy team gets a bye that week.