# Rotating teams through stations

How can 14 teams rotate through 7 stations (2 teams at a station) without overlaps?

NOTE:
Every team participates in every station exactly once

• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it :)
– Rubio
May 21, 2017 at 5:41

Not possible, given the conditions. In order for each of 14 teams to play each of the others once, each team will have to play 13 matches. However, no team can play more than 8 matches without appearing at least twice at at least one station. (EDIT: With the change to 7 stations, it's still not possible; this time, a team can't play more than 7 matches without appearing at least twice at at least one station.)

• Would it be possible if there were 9 stations, with the other details the same?
– Kim
May 12, 2017 at 20:51
• @Kim, no, 9 is still not equal to 13. May 12, 2017 at 23:28
• @Kim Perhaps if each station had a preferred side, then it might be possible with 7 stations, where each team plays on each side of each station, missing exactly one side of one station. May 15, 2017 at 16:28
• You really need to study this type of problem a bit more and think about your constraints. You've reached the point where you're just flailing around trying to find a combination of constraints that aren't impossible; you really should work out the constraints and a solution before posting your question/puzzle. May 18, 2017 at 11:26
• maybe i'm on the wrong kind of site. I just googled my issue. Just trying to figure out how the heck to schedule elementary kids for field day. Sorry to bother you. Thanks for the help you gave
– Kim
May 18, 2017 at 12:24

As other folks have noted in the comments, it isn't entirely clear what you want. But given your comment that this is to assign teams at a school field day, here's a possible way to assign the groups to stations that might be suitable:

Divide the teams into two groups of seven (groups A and B), and within each group, number the teams 1–7. For the first round, arrange the groups as follows:

 A1   A2   A3   A4   A5   A6   A7 ||   ||   ||   ||   ||   ||   || B1   B2   B3   B4   B5   B6   B7
Each "column" here corresponds to a station.

For the second round,

Rotate all the A groups one location to the right, and rotate all B groups one position to the left:

 A7   A1   A2   A3   A4   A5   A6   ||   ||   ||   ||   ||   ||   || B2   B3   B4   B5   B6   B7   B1

Groups that "rotate" off one end of the line come back in at the opposite end.

By continuing this pattern five more times, all 14 groups get to participate in each station once, and every team from group A plays against every team from group B exactly once. (This latter fact relies on the fact that 7 is an odd number.) However, none of the A teams ever play against each other, and none of the B teams ever play against each other; this may or may not be appropriate.