Subject: Combinations of 16 couples at 4 host houses
Name: Nancy
Who are you: Teacher

This is a real-world problem. I should know the answer but I don't. A friend is starting a dinner group of 16 couples, to be distributed across 4 houses. Each month she wants to have a different set of host houses (no problem) AND she doesn't want repeats among pairs. That is, if the Smiths and Joneses are together one month, they should not be together another month until all the combinations have been exhausted. How many valid combinations are there? Is there a formula that I can convert into a computer program? I will have to get the names from my friend and give her back the combinations as she would not be able to deal with the math formulation.

Nancy,

Couples are named 1,2,..., 16; 4 weeks as follows for houses A,B,C,D, etc:

A: 1 2 3 4
B: 5 6 7 8
C: 9 10 11 12
D: 13 14 15 16

1 6 11 16
2 7 12 13
3 8 9 14
4 5 10 15

1 8 11 14
2 5 12 15
3 6 9 16
4 7 10 13

1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16

This "solution" has couples 1 and 11 at the same house in weeks 2 and 3. I don't see complete solution to your problem.

Denis