# Rotating teams through stations without repeating a topic?

I am putting together a gallery walk activity and want to rotate 6 teams through 4 unique “topics.”

This activity will take place in a rectangular room. There will be 6 “stations” set up. Each station will have a facilitator that stays at the station and is a topic expert. I was thinking of having 2 stations that repeat one of the 4 topics to make this work. Is it possible to do this without having any group see any topic twice?

I have 59 people in total for reference.

• Please don't remove vital parts of the description. Edits should be to clarify, and should never invalidate answers or remove information. Apr 13, 2023 at 2:09
• Apr 13, 2023 at 6:16

## 1 Answer

I don't think this is really a puzzle, but here goes.

Assume you have 4 topics A B C D.
You have six stations, but they are all assigned some topic that (presumably) remains constant throughout the entire event.
Let's call the stations after what their topic is. Your proposal is for, e.g., A A B B C D so we have all four topics present, but two of them repeat.
And you have six teams visiting those six stations.

It makes sense to look at 4 rounds of six teams visiting six stations and ideally each team visiting each topic.
It's easy to see that in four rounds, these are the topics that can be visited:

A 8 times
B 8 times
C 4 times
D 4 times

Clearly with six teams, only 4 of the 6 can possibly visit topic C or D in four rounds. Indeed, no matter what you do, if at least one of the topics is only available at ONE station, you would need at minimum six rounds for all teams to see all four topics. With fewer than six rounds, at least one team will miss out on at least one topic.

Crucially, this means: with four rounds, six teams, and at least one topic being at only one station,
at least one team will not visit all four topics in those four rounds,
which means at least one topic they visit in those four rounds must be a duplicate.
Unless they just sit out a round.

For things to work well, you need a number of stations that is an even multiple of the number of topics, with the topics equally represented amongst the stations.