Math CentralQuandaries & Queries


Question from Jennifer:

I am a sales trainer. Each month we need to observe one of our peers and give feedback. What this means is every month, not only am I observing someone, but someone is also observing me. For example, in January, Amy observes Tracy, Tracy observes Rachel, Rachel observes Ryan, and so on. The last person would then observe Amy, bringing it full circle. There are 10 trainers on my team. How do I set it up to where each trainer observes another trainer only once over the course of the year (and in return, each trainer is only being observed once by each person)? I know they will eventually repeat but would that start in the 10th month?

It cannot start any later than the 10th month, because there are only nine others to observe.

It can start that late. Arrange them (on paper) in a circle. In month N, each person observes the person N after them. This does mean that, in month 5, pairs are observing each other, and in several months there are two cycles of length 5. Is this a problem?

I can see how to get eight months of 10-cycles: for instance:

ABCDEFGHIJA and reverse;
ACHFDBJEGIA and reverse;
ADGJCFIBEHA and reverse;
AEICGBHDJFA and reverse.

But I suspect nine (or N-1 for any even N?) is impossible.
Good Hunting!

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