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| Math Central | Quandaries & Queries |
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Question from Joel: We are having a youth activity where we have eight teams and four stations. At each station there will be two teams competing against each other. We need each team to go through all four stations, but without ever competing against the same team, and, of course, without ever doing the same station twice. To put it another way, each team has a color assigned, and we have stations 1-4. We want each color to go to each station only one time, and to go through all stations (the order does not matter), but none of the colors can be at the same station - any station - more than once. If this possible? Thank you! |
Joel,
This is a bit complicated. That’s because a schedule is a bit intricate to construct.
Let’s start with the two 4x4 arrays below. In reality they are mutually orthogonal Latin Squares of order 4, in case you want to look them up. Because the squares are orthogonal, using the scheme below every pair of teams who play together will be assigned a unique pair (round, station) so that each team plays in all 4 rounds and at all 4 stations. Teams A, B, C, D never play together, nor do teams E, F, G, H.
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 11 2 3 4
3 4 1 2
4 3 2 1
2 1 4 3
Label the rows of each square by A, B, C, D, and the columns by E, F, G, H. The numbers from the first square will tell you the round number, the numbers from the second square will tell you the station number, and the row and column labels will tell you the teams which play at that station in that round.
Going across the top rows of the two squares, Team A plays
Team E in round 1 at station 1
Team F in round 2 at station 2
Team G in round 3 at station 3
Team H in round 4 at station 4.Going across the second rows of the two squares, Team B plays
Team E in round 2 at station 3
Team F in round 1 at station 4
Team G in round 4 at station 1
Team H in round 2 at station 2.
The schedules for teams C and D arise from reading rows 3 and 4 similarly. The schedules for teams E, F, G, H arise from doing the same thing except reading down columns rather than across rows.
I hope this works for you.
—Victoria
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