Math CentralQuandaries & Queries


Question from Rain:

I need to have a meet and greet with 40 people and need to do small group activities so that each person meets every other person but only once.
I'm not into math, need someone to tell me how many groups I would need to have?


Hi Rain,

I don't think you're going to be too lucky with the numbers if all groups need to have the same size. For 40 people there are 780 (= 40 x 39 / 2) different pairings of people. The small numbers that are divisors of 40 are 4, 5, and 10. In each group of size 4 there are 6 pairs of people, in each group of 5 there are 10 pairs,and in each group of 10 there are 45 pairs. Eight groups of 5 account for 80 pairs, and since 80 is not a divisor of 780 there is no solution using only groups if size 5. Groups of size 10 can be ruled out in the same way. Ten groups of size 4 account for 60 pairs, so if it is possible to do this using groups of size 4 then you will need 13 rounds. That's a lot of activities! I don't know if the required combinatorial design exists though. (They usually don't.) There is a better chance of success if the groups can be of different sizes. Is that possible?


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