If you wish then skip ahead to the in summery part of this post.

Detail:

If I understand this correctly then the nuber of games in a league should be three times:

$\displaystyle g_t = g_t_-_1 + (t - 1)$

Where g is the number of games that will occur and t is the number of teams in the league. I'm much better at algorythms so I will write one of those as well.

Code:

games = 0
for p = 1 to teams
(
games += p
)
games *= 3

According to Wolfram Alpha:

$\displaystyle g_t = ((t - 1) * t) * 0.5$

For example if we have 4 teams then $\displaystyle g_4 = ((4 - 1) * 4) * 0.5$, therefore $\displaystyle g_4 = 6$.

In our case all this must occur three times so:

$\displaystyle g_t = ((t - 1) * t) * 1.5$

and

$\displaystyle g_4 = 18$

You play 6 matches per day and there are 365 days in a year therefore you could have 38 teams per yearly league so there is no trouble with 9 time wise. However there are other problems with your set-up.

Summary:

You need an odd number of teams to satisfy having teams play there matches in sets of 2.

You need a number of teams that gives an integer when divided by 4 to satisfy playing games in daily rounds of 6.

games that will be played = $\displaystyle ((teams - 1) * teams) * 1.5$

The first two conditions wont ever be true together.

For example:

3 teams creates 9 games, thats 2 per team. Exactly 6 games per round (not ok), teams play exactly 2 games per playing week (ok).

4 teams creates 18 games, thats 9 games per team. Exactly 6 games per round (ok), teams play exactly 2 games per playing week (not ok).