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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?

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    What you are asking is similar to the social golfer problem, SGP. The only difference is that SGP is maximizing the number of shufflings while avoiding repeats of players playing together, while your problem is about minimizing the number shufflings while covering all pairs. – Mike Earnest May 28 '20 at 00:07
  • Thank you very much: Using your hint, I could solve the problem (which I will post now). –  May 28 '20 at 05:31

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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/

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    Just for fun: this special case of $4$ teams of $4$ people each, or more generally $n$ teams of $n$ people each, is also a finite affine plane. These exist whenever $n$ is a prime power, and if such a plane exists, then you get the additional property that any pair of people only were teammates once. – antkam May 28 '20 at 06:27
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    Is there a general formula for number of shuffles for $n$ people and $m$ groups? – jonnybolton16 May 28 '20 at 08:52
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    @jonnybolton16 Even the case where $n=m^2$ is a hard open problem. In this case, at least $n+1$ shuffles are necessary, yet $n+1$ shuffles is only possible when there exists an affine plane of order $n$, which is an open problem when $n$ is composite and greater than $10$. – Mike Earnest May 28 '20 at 16:05