So I have to arrange a rotation schedule for a tournament. We're aiming for 32 teams. There will be 8 stations and 4 teams competing simultaneously at each station. We want all 32 teams to hit each station once, but not play the same teams twice. I was hoping to set up a template so if the teams go up or down it will be an easy fix.
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Assuming that two teams may never face each other twice (otherwise, it is trivial), here is a solution. Thanks to @MikeEarnest for providing me with the necessary mutually orthogonal Latin squares. Let S1, S2 ... S8 denote the stations, and R1, R2 ... R8 denote the rounds. For convenience, we number the teams A, B, C, D plus a number from 1 to 8. The following configuration would work:
S1 S2 S3 S4 S5 S6 S7 S8 R1 A7 A6 A5 A4 A3 A2 A1 A8 R2 A6 A7 A4 A5 A2 A3 A8 A1 R3 A5 A4 A7 A6 A1 A8 A3 A2 R4 A4 A5 A6 A7 A8 A1 A2 A3 R5 A3 A2 A1 A8 A7 A6 A5 A4 R6 A2 A3 A8 A1 A6 A7 A4 A5 R7 A1 A8 A3 A2 A5 A4 A7 A6 R8 A8 A1 A2 A3 A4 A5 A6 A7 R1 B7 B5 B3 B1 B4 B6 B8 B2 R2 B6 B4 B2 B8 B5 B7 B1 B3 R3 B5 B7 B1 B3 B6 B4 B2 B8 R4 B4 B6 B8 B2 B7 B5 B3 B1 R5 B3 B1 B7 B5 B8 B2 B4 B6 R6 B2 B8 B6 B4 B1 B3 B5 B7 R7 B1 B3 B5 B7 B2 B8 B6 B4 R8 B8 B2 B4 B6 B3 B1 B7 B5 R1 C7 C4 C1 C2 C8 C3 C6 C5 R2 C6 C5 C8 C3 C1 C2 C7 C4 R3 C5 C6 C3 C8 C2 C1 C4 C7 R4 C4 C7 C2 C1 C3 C8 C5 C6 R5 C3 C8 C5 C6 C4 C7 C2 C1 R6 C2 C1 C4 C7 C5 C6 C3 C8 R7 C1 C2 C7 C4 C6 C5 C8 C3 R8 C8 C3 C6 C5 C7 C4 C1 C2 R1 D7 D3 D4 D8 D1 D5 D2 D6 R2 D6 D2 D5 D1 D8 D4 D3 D7 R3 D5 D1 D6 D2 D3 D7 D8 D4 R4 D4 D8 D7 D3 D2 D6 D1 D5 R5 D3 D7 D8 D4 D5 D1 D6 D2 R6 D2 D6 D1 D5 D4 D8 D7 D3 R7 D1 D5 D2 D6 D7 D3 D4 D8 R8 D8 D4 D3 D7 D6 D2 D5 D1
Because every square is a Latin square, each team will visit each station during all the rounds. Each round, you have one A-team (pun intended) and one team from B, C and D on one station. Because they're mutually orthogonal, teams will never face each other in two different rounds. One of the most striking results of this field of combinatorics is that a slightly different configuration, with 12 teams and 6 stations cannot be solved. This is Euler's Thirty-six officers problem.
Here is an easier to read (but far less insightful) presentation of Glorinfidel's answer. The teams are numbered 1-32. Every row is a list of matches which can happen concurrently.
Station 1 Station 2 Station 3 Station 4 Station 5 Station 6 Station 7 Station 8 1, 9,17,25 │ 2,11,20,29 │ 3,13,23,28 │ 4,15,22,32 │ 5,12,24,31 │ 6,10,21,27 │ 7,16,18,30 │ 8,14,19,26 2,10,18,26 │ 1,12,19,30 │ 4,14,24,27 │ 3,16,21,31 │ 6,11,23,32 │ 5, 9,22,28 │ 8,15,17,29 │ 7,13,20,25 3,11,19,27 │ 4, 9,18,31 │ 1,15,21,26 │ 2,13,24,30 │ 7,10,22,29 │ 8,12,23,25 │ 5,14,20,32 │ 6,16,17,28 4,12,20,28 │ 3,10,17,32 │ 2,16,22,25 │ 1,14,23,29 │ 8, 9,21,30 │ 7,11,24,26 │ 6,13,19,31 │ 5,15,18,27 5,13,21,29 │ 6,15,24,25 │ 7, 9,19,32 │ 8,11,18,28 │ 1,16,20,27 │ 2,14,17,31 │ 3,12,22,26 │ 4,10,23,30 6,14,22,30 │ 5,16,23,26 │ 8,10,20,31 │ 7,12,17,27 │ 2,15,19,28 │ 1,13,18,32 │ 4,11,21,25 │ 3, 9,24,29 7,15,23,31 │ 8,13,22,27 │ 5,11,17,30 │ 6, 9,20,26 │ 3,14,18,25 │ 4,16,19,29 │ 1,10,24,28 │ 2,12,21,32 8,16,24,32 │ 7,14,21,28 │ 6,12,18,29 │ 5,10,19,25 │ 4,13,17,26 │ 3,15,20,30 │ 2, 9,23,27 │ 1,11,22,31