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 Question from Dary: I have a 9 team pool league. Each team must play each other once only. But 2 teams cannot play at the same place location each week. I can't seem to match this up and I get stuck. Please help, Dary

Hi Dary,

I suggest making the schedule as if there were no constraints as follows:

1: 1-8, 2-7, 3-6, 4-5
2: 9-7, 1-6, 2-5, 3-4
3: 8-6, 9-5, 1-4, 2-3
4: 7-5, 8-4, 9-3, 1-2
5: 6-4, 7-3, 8-2, 9-1
6: 5-3, 6-2, 7-1, 8-9
7: 4-2, 5-1, 6-9, 7-8
8: 3-1, 4-9, 5-8, 6-7
9: 2-9, 3-8, 4-7, 5-6

Notice that every team appears once in each column.

Suppose 1 and 2 are the teams that share the same hall. For each week the home team is the team listed first in games 1 and 3, and listed second in games 2 and 4. Thus the home teams are in columns 1, 4, 5, and 8.

This gives a schedule where each team has 4 home games, and teams 1 and 2 are never home on the same night (I think).

Why does this work? If it does, it is by special construction of the schedule. The numbers in the same row that differ by 1 are always in columns 1 and 3, or 2 and 4, or 3 and 5, or 4 and 6, or 5 and 7, or 6 and 8, or 7 and 8. In each row, no two numbers in columns 1, 4, 5, and 8 differ by exactly one. Therefore 1 and 2 are never home on the same night.

Victoria

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