Math CentralQuandaries & Queries


Question from Denise:

We put together a game night where we have 2 teams of 6 couples each that play 6 games. We haven't been able to figure out an arrangement that allows each couple to play each game with a different couple from the opposite team (i.e. Team A couples play every game with a different couple from Team B). Is this possible? It works with 2 teams of 5 couples each.


It sounds like Euler's 36-officer problem:

IT CAN'T BE DONE! Label the "6 ranks" by the players of team A (A1 up to A6), and the 6 regiments by the players of team B (B1 to B6). The six columns represent the 6 rounds, and the six rows the 6 games. It's not possible to arrange the 36 pairs A_iB_j so that each A_i and each B_j appears exactly once in each row and each column. (The corresponding Graeco-Latin square can't be devised in the 2 by 2 case, but it's easy for 3 by 3, 4 by 4, 5 by 5, and 7 by 7.)


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