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I wrote an article on the Social Golfer Problem, which has questions like:

Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with each other twice? tournament

I compiled best-known solutions for pairs, triplets, and quadruplets at the Social Golfer Problem Demonstration. To compile this data, I had to read through a few hundred books and papers. Very often, an answer would be given in new-to-me difficult notation, or it would reference a previous paper, or it would just be "obvious".

Now that I am considered an "expert", I've been asked about quintets and sextets, and for larger number of groups. I don't recall seeing solutions for any of the below problems in the papers I perused, but it's possible I missed them.

  • 30 play in groups of 3 each day. No two play together more than once. How many days?
  • 33 play in groups of 3 each day. No two play together more than once. How many days?
  • 36 play in groups of 3 each day. No two play together more than once. How many days?
  • 40 play in groups of 4 each day. No two play together more than once. How many days?
  • 44 play in groups of 4 each day. No two play together more than once. How many days?
  • 48 play in groups of 4 each day. No two play together more than once. How many days?
  • 52 play in groups of 4 each day. No two play together more than once. How many days?
  • 25 play in groups of 5 each day. No two play together more than once. How many days?
  • 30 play in groups of 5 each day. No two play together more than once. How many days?
  • 35 play in groups of 5 each day. No two play together more than once. How many days?
  • 40 play in groups of 5 each day. No two play together more than once. How many days?
  • 45 play in groups of 5 each day. No two play together more than once. How many days?
  • 50 play in groups of 5 each day. No two play together more than once. How many days?
  • 55 play in groups of 5 each day. No two play together more than once. How many days?
  • 60 play in groups of 5 each day. No two play together more than once. How many days?
  • 36 play in groups of 6 each day. No two play together more than once. How many days?
  • 42 play in groups of 6 each day. No two play together more than once. How many days?
  • 48 play in groups of 6 each day. No two play together more than once. How many days?
  • 54 play in groups of 6 each day. No two play together more than once. How many days?
  • 60 play in groups of 6 each day. No two play together more than once. How many days?
  • 66 play in groups of 6 each day. No two play together more than once. How many days?
  • 72 play in groups of 6 each day. No two play together more than once. How many days?

Does anyone have any solutions to any of these?

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1  
25 players in groups of five is easy. There are six possible directions for a line in the plane $F_5^2$ over the field of five elements. Thus the points of the plane can be partitioned to five lines (five points on each) in six ways. Using that idea we can see do the groups in such a way that after 30 days each player has played in 6 groups, and exactly once with all the other 24 players. The uniqueness comes from the fact that two points (=two players) determine a unique line. –  Jyrki Lahtonen Oct 2 '11 at 20:10
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Oops. I meant that you can arrange 30 parties of five players. If each player will play every day that is only good for 6 days. Five groups playing on each day. –  Jyrki Lahtonen Oct 2 '11 at 20:13

1 Answer 1

Have you seen Ivan Dotu and Pascal Van Hentenryck, Scheduling social golfers locally? They claim

45 golfers in groups of 5 can play 7 days;

54 golfers in groups of 6 can play 6 days;

50 golfers in groups of 5 can play 8 days;

and some other results that lie outside the bounds of your question. But they don't actually present any instances of the solutions, just their algorithm and the amount of time it took to find the instances.

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Yes, that's one of the less helpful papers I looked at. –  Ed Pegg May 11 at 15:36

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