Determining format of knockout stages from number of groups

Suggested format for 36 team tournament:

9 groups of 4 teams each

9 group winners advance to round of 16.

9 runners-up and best 5 third-placed teams advance to playoffs for round of 16.
Best 5 second-placed teams are randomly drawn against best 5 third-placed teams (same group protection applies), rest are drawn randomly. If there is a tie while determining best 5 second-placed teams, they will be randomly decided.
7 playoff winners advance to round of 16.

Round of 16 draw to be conducted:
Best 7 group winners to be randomly drawn against the 7 playoff winners, remaining 2 group winners are drawn randomly. If there is a tie while determining best 7 group winners, they will be randomly decided.
4 group winners and 4 playoff winners to vie for one finalist spot, 5 group winners and 3 playoff winners to vie for the other.
Same group protection applies (following guidelines are optional):
-If there is any group with 3 representatives in the knockout stages, they will be drawn first, and will each vie for a different semi-finalist spot. If not, skip this part.
-The rest of the teams will be randomly drawn. Same group protection will apply until the final.

Number of group matches: 9 x 6 = 54
Number of knockout matches: 16
Number of playoff matches: 7
Total: 77

OPTIONAL CHOICE: Playoffs for playoffs for the 3rd placed teams; best 3rd placed team qualifies for the playoffs automatically while the others play against each other to decide the remaining 4 teams. Either randomly drawn or drawn from worst to best is fine. This will take an extra 4 matches, for a total of 81 matches.



Suggested format for 40 team tournament:

10 groups of 4 teams each

10 group winners advance to round of 16.

10 runners-up and best 2 third-placed teams advance to playoffs for round of 16.
Best 2 second-placed teams are randomly drawn against best 2 third-placed teams (same group protection applies), rest are drawn randomly. If there is a tie while determining best 2 second-placed teams, they will be randomly decided.
6 playoff winners advance to round of 16.

Round of 16 draw to be conducted:
Best 6 group winners to be randomly drawn against the 6 playoff winners, remaining 4 group winners are drawn randomly. If there is a tie while determining best 6 group winners, they will be randomly decided.
5 group winners and 3 playoff winners to vie for each finalist spot.
Same group protection applies (following guidelines are optional):
-If there is any group with 3 representatives in the knockout stages, they will be drawn first, and will each vie for a different semi-finalist spot. If not, skip this part.
-The rest of the teams will be randomly drawn. Same group protection will apply until the final.

Number of group matches: 10 x 6 = 60
Number of knockout matches: 16
Number of playoff matches: 6
Total: 82



General formula:

Note: X, N, P, W are variables.

X tier knockout format consisting of 2^X matches total
Standard: 2^(X-1) groups of 4

Number of groups, N, required: [2^(X)]/3 ≤ N < (2^X+1)/3

Format for knockout stages depends on the number of groups. Knockout format will be 2^X tiers where X = [log(2)3N - 0.5] rounded to the nearest integer.

Why this format?
Reason: there are 3 points to consider when deciding on the tournament format.
-To maintain the hype, each group winner must be guaranteed qualification to the knockout stages.
-To minimise the number of meaningless matches, each runner-up must be guaranteed at least a chance to qualify for the knockout stages.
-To keep a semblance of fairness, each runner-up must be subject to the same trials i.e. they must either all qualify for the knockout stages OR they must all go through a playoff to qualify for the knockout stages.

This format will ensure that all 3 points are satisfied.
The side effect of this is, unless N = 2^(X-1), there will be third-placed teams acting as wildcards. Any third-placed team will undergo the same trials as the second-placed teams.

If N < 2^(X-1), best P third-placed teams advance straight to the knockout stages. Best P group winners to be randomly drawn against P third-placed teams, and best P second-placed teams to be randomly drawn against each other. The remaining group winners to be randomly drawn against the remaining second-placed teams. If there is a tie while determining best P group winners, they will be randomly decided.

If N = 2^(X-1), 2^(X-1) winners drawn against 2^(X-1) runners-up. Default format.

If N > 2^(X-1), all second-placed teams and best P third-placed teams advance to playoffs. Best P second-placed teams are randomly drawn against the third-placed teams (same group protection applies), rest are drawn randomly. If there is a tie while determining best P second-placed teams, they will be randomly decided. W playoff winners advance to knockout stages; best W group winners to be randomly drawn against W playoff winners; remaining group winners are drawn randomly. If there is a tie while determining best W group winners, they will be randomly decided.

Regardless of the format, same group protection applies (following guidelines are optional):
-If there is any group with 3 representatives in the knockout stages, they will be drawn first, and will each vie for a different semi-finalist spot. If not, skip this part.
-The rest of the spots will be randomly drawn. Same group protection will apply until the final.