This is really the combined effort of a bunch of people over the years as we ran survivor and e-vent. Jamie instigated this problem, and was the first to solve it. Garrison, Jen, Roger, and I boldly took up the challenge and got stumped. Since that fateful night many moons ago, I have figured out most of the 3, 4, 5, and 6 activity rotations (Jamie's solution was for 10 teams, 5 events), mostly while standing around at work. So here they are, roataions for running a camp that will never have a team do an activity twice, or be paired up with another team twice.

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While we invented this all on our own, we were not first :(

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3 Activities

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\n"; //3 events and 3 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "
3 TeamsRotation #
123
EventsA132
B213
C321
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\n"; //3 events and 6 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "
6 TeamsRotation #
123
EventsA1 & 63 & 52 & 4
B2 & 51 & 43 & 6
C3 & 42 & 61 & 5
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3 activities with 4 teams is impossible. These are a form of latin squares called room squares (n+1 objects paired in a n x n square). n ≠ 3, 5

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\n"; //##### 4 Activities ##### echo "
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4 Activities

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\n"; //4 events and 4 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "
4 TeamsRotation #
1234
EventsA1234
B2341
C3412
D4123
\n"; //4 events and 8 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "
8 TeamsRotation #
1234
EventsA1 & 65 & 72 & 43 & 8
B2 & 71 & 36 & 84 & 5
C3 & 42 & 81 & 56 & 7
D5 & 84 & 63 & 71 & 2
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\n"; //4 events and 6 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "
6 TeamsRotation #
1234
EventsA1 & 4 2 & 35 & 6
B2 & 51 & 34 & 6 
C3 & 64 & 5 1 & 2
D 2 & 61 & 53 & 4
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4 activities with 4 teams paired is impossible. When you are building 2 team rotations, if the triangular number of the teams is equal to or less than the number of available spaces it won't work.

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\n"; //##### 5 Activities ##### echo "
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5 Activities

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\n"; //5 events and 5 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "
5 TeamsRotation #
12345
EventsA12345
B23451
C34512
D45123
E51234
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\n"; //5 events and 10 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "
10 TeamsRotation #
12345
EventsA1 & 64 & 107 & 82 & 35 & 9
B2 & 83 & 51 & 96 & 104 & 7
C3 & 101 & 84 & 57 & 92 & 6
D5 & 72 & 93 & 64 & 81 & 10
E4 & 96 & 72 & 101 & 53 & 8
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5 activities with 6 teams is impossible. It's another room square.

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5 activities with 8 teams is impossible. Or at least to my tiny brain it is.

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6 Activities

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\n"; //6 events and 6 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "
6 TeamsRotation #
123456
EventsA123456
B234561
C345612
D456123
E561234
F612345
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\n"; //6 events and 12 teams echo "\n"; echo "\n"; echo "\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "\n"; echo "\n\n\n\n\n\n\n"; echo "\n"; echo "
12 Teams*Rotation #
123456
EventsA2 & 31 & 116 & 95 & 127 & 104 & 8
B4 & 72 & 108 & 121 & 63 & 115 & 9
C5 & 68 & 93 & 102 & 74 & 121 & 11
D8 & 117 & 122 & 43 & 91 & 56 & 10
E9 & 103 & 51 & 74 & 116 & 82 & 12
F1 & 124 & 65 & 118 & 102 & 93 & 7
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*6 activities with 12 teams is impossible. This time we are dealing with a graeco-latin square (n ≠ 2, 6). The best you can get is 2 teams meeting twice.

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6 activities with 6 teams paired is impossible. (15 combos, 18 positions)

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I have not found solutions for 6 activities with 8 or 10 teams yet. I think they should be possible, but I've been plenty wrong before.

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7 Or More Activities

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Go away...unless you have a solution. I tried looking for these guys and ended up reading quite a few thesis' on the topic.

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