Let us subdivide $n$ persons into $m$ teams. How often we must rebuild the teams that everyone worked at least once with everyone else in a team?

Question

We need to subdivide $n$ persons into $m$ teams of equal size ($m\mid n$). How often we must rebuild the teams (by shuffling the team members / reassigning them to a different team) so that everyone worked at least once with every other person in a team?

How do we (re-)form the teams in a structured manner most efficiently?

As an example consider a group of 16 persons that need to be subdivided into 4 teams. How many rounds of teamwork (with reformed teams) we need, if we want to ensure that every person at least worked with every other person?

Answer 1

Thanks to the hint of Mike Earnest, the problem could straightforwardly be solved. We need to model the problem as Social Golfer Problem (SGP) of the form $g-s-w=4-4-5$. This special case, also known as Resolvable Steiner Quadruple System (RSQS) has the solutions:

1. Round 1: ABCD EFGH IJKL MNOP
2. Round 2: AEIM BFJN CGKO DHLP (first Shuffle)
3. Round 3: AFKP BELO CHIN DGJM (second Shuffle)
4. Round 4: AGLN BHKM CEJP DFIO (third Shuffle)
5. Round 5: AHJO BGIP CFLM DEKN (fourth Shuffle)

See also

http://www.mathpuzzle.com/MAA/54-Golf%20Tournaments/mathgames_08_14_07.html or

https://demonstrations.wolfram.com/SocialGolferProblem/