# 3-team tournaments

Below is a table, showing the number of teams in a tournament versus the number of games needed to create a "perfect" tournament grid, in which every team plays every other team the same number of times. (Click on a number of teams to see the appropriate grid. Here is an explanation of how to read a grid.)

TeamsGames TeamsGames TeamsGames TeamsGames TeamsGames TeamsGames TeamsGames
7 7 13 26 19 57 25 100 31 155 n n × (n - 1) ÷ 6
8 56 14 182 20 380 26 650 32 992 n n × (n - 1)
3 1 9 12 15 35 21 70 27 117 33 176 n n × (n - 1) ÷ 6
4 4 10 30 16 80 22 154 28 252 34 374 n n × (n - 1) ÷ 3
5 10 11 55 17 136 23 253 29 406 35 595 n n × (n - 1) ÷ 2
6 10 12 44 18 102 24 184 30 290 36 420 n n × (n - 1) ÷ 3

(Or you may be looking for league grids.)

Note that for tournaments, some numbers are much better for creating a grid that can run in a reasonable number of games than others: grids in the green rows (rows 1 and 5) are particularly good; grids in the red row (row 6) are particularly bad. This is because for the "good" grids, each team need play every other team only once for it all to work out perfectly. For other grids, teams must play each other 2, 3, or 6 times before everybody can have played everybody else an equal number of times.

(Entries in grey are ones for which I have not found a grid.)

So if your tournament is round-about a "good" number of teams, try and fiddle it to exactly the good number.

Note, too, that "good" grids still work fairly well played one team short. For example, if you have 14 teams, you can play the 15-team grid, with the 15th team as a blank. Every team will play the "blank" team exactly once; so every team has a single "head-to-head" (i.e. two-team) game.

Or you can accept a different kind of imperfection. The below grids give slightly more games to one or a few teams, and have teams playing most other teams once but a sprinkling of instances where given pairs of teams play each other twice. As a short nearly-minimalist round robin, these may be suitable to start a tournament, particularly if you're going to follow the round robin with a Cascade.

TeamsGames TeamsGames TeamsGames
7 7 13 26 19 57
8 11 14 33 20 67
9 12 15 35 21 70
10 17 16 43
11 20 17 47
12 28 18 54

After the round robin, you may wish to use a Cascade.

It's usual to finish a tournament with a Finals.

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