Let's say that I have a party of 48 people, sitting at tables split up into 12 groups of 4. Every few minutes, the people shuffle themselves into 12 new groups of 4.
Is it possible for each one of the 48 people to have sat at the table with each of the others, but no two people sit at the same table together twice?
If this is possible, what pattern of shuffling the people among the tables produces this outcome?
Suppose you were one of the 48 people. In order not to sit with the same person twice, you would need to sit with 3 new people at every shuffle. But there are 47 people to sit with and 47 is not divisible by 3.