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.