Class of 16 Participants Answering 6 Questions in Subsets of 4, Each One in A Different Combination With All Pairings Covered

Question

16 Participants are arranged into 4 groups of 4. The participants work together on a question within their groups. Next the groups are rearranged into another 4 groups of 4, where they work on the second question, and so on for six questions. Each participant must work with every other participant at least once at some point in the exercise.

Is this possible? If not, how can you prove that it is impossible. If it is possible, what are the six configurations of the participants?

The initial configuration of the 16 participants into groups A, B, C & D is:

A: 1, 2, 3, 4

B: 5, 6, 7, 8

C: 9, 10, 11, 12

D: 13, 14, 15, 16

Answer 1

This is a variant of the "social golfer problem", but there you want each pair to be together at most once.

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

So after five sessions, every pair has worked together exactly once, so you can do whatever you like for the sixth session.

This was taken from "four groups of four golfers for five weeks" at http://web.archive.org/web/20050407074608/http://www.icparc.ic.ac.uk/~wh/golf/solutions.html#4-4-5