Combinations in Rotations
Date: Fall 2011
We are trying to establish a rotation for our physical education field day. We have 25 individual classes and 12 stations. The goal is to pair two classes together at each station, then rotating through each of the 12 stations. However, we do not want classes to be paired with the same class for more than one station. In other words, some classes may move through stations 1-12, then others would rotate through stations 12-1. Ideally, classes should move sequentially (based on location of the stations-- station 1 is next to station 2, which is next to station 3, etc.) Plus, with an odd number of classes we know there may be just one class at a station every so often (24 classes and 12 stations would be ideal). Is there a formula we can use to plug in our 25 classes/12 stations and come up with a workable solution? Do we need to add a 13th station? Do we need to combine classes to give us a total of 24?
I know of a method that will work well with an odd number of stations, but not an even number. With an odd number, just divide the set of classes into two parts, perhaps odds and evens. Set up the stations in a circular pattern, something that allows continuous motion through the cycle. Place the classes at the stations, one odd group and one even group. Rotate the odds clockwise and the evens counterclockwise. If you had 26 classes and 13 stations, it would work perfectly. To get the 26th class, divide the largest into two classes. Both must be in the same part, i.e. both "odd" or both "even". If you have only 12 stations and do not want another, make one station a rest station, or perhaps a discussion station. If odd and even are not useful, number the groups from 1 to 13, with two different colors. With Christmas approaching, I suggest red and green. Red 1 to Red 13 rotate clockwise. Green 1 to Green 13 rotate counterclockwise. After the start setup and then 12 shifts, every group will have experiences all 13 stations once without ever meeting the same group.
Dr. Ken Mellendorf
Illinois Central College
Click here to return to the Mathematics Archives
Update: June 2012