How can 12 teams rotate through 6 games without overlaps?

Question

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 proof of why a solution is impossible.

Trenin · Accepted 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.

AndyT · Answer

I have named the sports U, V, W, X, Y and Z for convenience

Moghwyn · Answer

There doesn't seem to be a valid setup. Picking team 1, team A and game U at random, team 1 will plays game U against team A during the first time slot. That will leave five other teams left that team 1 will have to play against, each of those have played one of the remaining games. Team 1 must not play its first game type again, none of the other five teams is allowed to play the first game type they played again, after playing at most four additional games no further valid match is possible.

Simplifying things a bit, suppose there are just two time slots, two number teams (1, 2), two letter teams (A, B) and two games (U, V). Team 1 will play game U against team A in the first time slot. Team 1 can't play game U against Team B in the second time slot because team 1 already played game U. Team 1 also can't play game V against team B because team B already played game V during the first time slot.

How can 12 teams rotate through 6 games without overlaps?

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