mathematics - How can 12 teams rotate through 6 games without overlaps?

Given the following:

• Six Number Teams (1 - 6)


• Six Letter Teams (A - F)


• Six Games (Basketball, Football, Baseball, Volleyball, Hockey, Rugby)


• Six Time Slots (1pm - 6pm)

Set up a game schedule that follows these rules:

• Each team must play each game once.



• Each letter team must play against each number team exactly once.


• Every game must be played once (by one letter team and one number team) during each time slot

Please provide either a solution or a mathematical prove of why a solution is impossible.

Answer

In order to do this puzzle, you'd need to create Mutually Orthogonal Latin Squares of order 6.

For example, say that instead, you had 6 teams (1-3 and A-C), 3 sports (baseball, football, hockey), 3 timeslots. Then, you could make the following schedule:

    A   B   C
1  b@1 f@2 h@3
2  h@2 b@3 f@1

3  f@3 h@1 b@2

So in this example, team A plays baseball against team 1 at 1:00pm.

This uses 2 Mutually Orthogonal Latin Squares of order 3.

1 2 3    b f h
2 3 1    h b f
3 1 2    f h b

This allows us to conform to the following rules:

• Every lettered team plays every numbered team exactly once - simply by design of the table


• Every team plays every sport exactly once


• Every team plays in every timeslot exactly once

However, it is a known impossibility to create two MOLS of order 6, so the original question is not possible.