I was just figuring out how to rotate 21 kids through 7 stations in such a way that each kid does each station with a different group. And it just dropped in to try this: 1/3 of the kids advance by 1’s. (If you’re at station 2, move to 3). 1/3 advance by 2’s (e.g. if you’re at 2, move to 4). 1/3 advance by 3’s (if you’re at 2, move to 5.) And it works! I wonder if any group reconstitutes itself along the way somewhere … and what other combinations of kids/stations this or similar patterns work for ….

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I have started to tackle this problem so many times, trying to look at symmetry groups and graph colorings, but I haven’t been able to find a model/formalization that gives me any insight.

I did manage to put together a spreadsheet to check for duplicates: http://goo.gl/wFUt1z. Feel free to change the rules used in the sheet. Playing around, I found another rule that works: 1/3 of the kids move clockwise (advance by 1) and 2/3 move counterclockwise (advance by 6). Actually, if 1/3 picks any rule, so long as the other 2/3 pick a different rule it seems to work out. Someday I hope to understand how/why…

ABCD ACDB ADBC

BADC CABD DACB

CDAB DBAC BCAD

DCBA BDCA CBDA

Each column is a student; each row is a station. This works for 12 (or 24) students in groups of 3 covering four stations.

If you just graded the first four students’ papers, you’d have a grade for each person at each station.

Your method will work for any prime number of stations due to Euclid’s lemma.