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.
While we invented this all on our own, we were not first :(
3 Activities
| 3 Teams | Rotation # | |||
|---|---|---|---|---|
| 1 | 2 | 3 | ||
| Events | A | 1 | 3 | 2 |
| B | 2 | 1 | 3 | |
| C | 3 | 2 | 1 | |
| 6 Teams | Rotation # | |||
|---|---|---|---|---|
| 1 | 2 | 3 | ||
| Events | A | 1 & 6 | 3 & 5 | 2 & 4 |
| B | 2 & 5 | 1 & 4 | 3 & 6 | |
| C | 3 & 4 | 2 & 6 | 1 & 5 | |
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
4 Activities
| 4 Teams | Rotation # | ||||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | ||
| Events | A | 1 | 2 | 3 | 4 |
| B | 2 | 3 | 4 | 1 | |
| C | 3 | 4 | 1 | 2 | |
| D | 4 | 1 | 2 | 3 | |
| 8 Teams | Rotation # | ||||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | ||
| Events | A | 1 & 6 | 5 & 7 | 2 & 4 | 3 & 8 |
| B | 2 & 7 | 1 & 3 | 6 & 8 | 4 & 5 | |
| C | 3 & 4 | 2 & 8 | 1 & 5 | 6 & 7 | |
| D | 5 & 8 | 4 & 6 | 3 & 7 | 1 & 2 | |
| 6 Teams | Rotation # | ||||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | ||
| Events | A | 1 & 4 | 2 & 3 | 5 & 6 | |
| B | 2 & 5 | 1 & 3 | 4 & 6 | ||
| C | 3 & 6 | 4 & 5 | 1 & 2 | ||
| D | 2 & 6 | 1 & 5 | 3 & 4 | ||
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.
5 Activities
| 5 Teams | Rotation # | |||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| Events | A | 1 | 2 | 3 | 4 | 5 |
| B | 2 | 3 | 4 | 5 | 1 | |
| C | 3 | 4 | 5 | 1 | 2 | |
| D | 4 | 5 | 1 | 2 | 3 | |
| E | 5 | 1 | 2 | 3 | 4 | |
| 10 Teams | Rotation # | |||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| Events | A | 1 & 6 | 4 & 10 | 7 & 8 | 2 & 3 | 5 & 9 |
| B | 2 & 8 | 3 & 5 | 1 & 9 | 6 & 10 | 4 & 7 | |
| C | 3 & 10 | 1 & 8 | 4 & 5 | 7 & 9 | 2 & 6 | |
| D | 5 & 7 | 2 & 9 | 3 & 6 | 4 & 8 | 1 & 10 | |
| E | 4 & 9 | 6 & 7 | 2 & 10 | 1 & 5 | 3 & 8 | |
5 activities with 6 teams is impossible. It's another room square.
5 activities with 8 teams is impossible. Or at least to my tiny brain it is.
6 Activities
| 6 Teams | Rotation # | ||||||
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | ||
| Events | A | 1 | 2 | 3 | 4 | 5 | 6 |
| B | 2 | 3 | 4 | 5 | 6 | 1 | |
| C | 3 | 4 | 5 | 6 | 1 | 2 | |
| D | 4 | 5 | 6 | 1 | 2 | 3 | |
| E | 5 | 6 | 1 | 2 | 3 | 4 | |
| F | 6 | 1 | 2 | 3 | 4 | 5 | |
| 12 Teams* | Rotation # | ||||||
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | ||
| Events | A | 2 & 3 | 1 & 11 | 6 & 9 | 5 & 12 | 7 & 10 | 4 & 8 |
| B | 4 & 7 | 2 & 10 | 8 & 12 | 1 & 6 | 3 & 11 | 5 & 9 | |
| C | 5 & 6 | 8 & 9 | 3 & 10 | 2 & 7 | 4 & 12 | 1 & 11 | |
| D | 8 & 11 | 7 & 12 | 2 & 4 | 3 & 9 | 1 & 5 | 6 & 10 | |
| E | 9 & 10 | 3 & 5 | 1 & 7 | 4 & 11 | 6 & 8 | 2 & 12 | |
| F | 1 & 12 | 4 & 6 | 5 & 11 | 8 & 10 | 2 & 9 | 3 & 7 | |
*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.
6 activities with 6 teams paired is impossible. (15 combos, 18 positions)
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.
7 Or More Activities
Go away...unless you have a solution. I tried looking for these guys and ended up reading quite a few thesis' on the topic.