Circle Method - Part 2

Using the algorithm instead of the brain

The circle method is a systematic way to determine the matchups without having to think much. There are other things you can find online that can provide more info. The way I will show things is not the only way to employ the method. The goal is to find something that works and is easy for you.

I went a really long way and happened to stumble into the solution without really understanding the problem. Eventually I also stumbled on how the algorithm produces the result.

What you can do is layout the teams in a "circle", and rotate the circle to generate all the matchups. I will keep the numbers in order, but they can be laid out in any combination. The circle will be down one column and back up another column, and the resulting two columns represent the matchups in the rows. First the initial layout starting with team 1 in the upper left and filling in counterclockwise...

1 6
2 5
3 4

So that builds one set of matchups 1 v 6, 2 v 5, 3 v 4. The trick now is rotate the circle, but for an even number of teams keep one team locked in place. I will rotate counterclockwise and hold team 1 locked in place.

1 5
6 4
2 3

Turn another notch...

1 4
5 3
6 2

Turn another notch...

1 3
4 2
5 6

Turn another notch...

1 2
3 6
4 5

Another turn is unnecessary as that would put you back where you started. You have the 5 sets of matchups - every team is represented in every row and matches up against every other team once. Take each of those rotation blocks and turn them into a row of matchups involving all the teams.

1 v 6   2 v 5   3 v 4
1 v 5   6 v 4   2 v 3
1 v 4   5 v 3   6 v 2
1 v 3   4 v 2   5 v 6
1 v 2   3 v 6   4 v 5

So we made a set of matchups with the circle method. In the next part we will exercise that muscle more.