Tournament organization that avoids repetitions

Question

A card game is played individually in tables of 4 players. I want to organize a tournament for 24 people. Hence in each round the players are divided into 6 tables. I want organize the tables so that repetitions (two players playing each other more than once) are avoided as far as possible.

I have two questions:

What is the maximum number of rounds we can play without any two players playing each other more than once? Of course the answer must be smaller than 8.

Is there an algorithm to obtain such maximal disposition of tables? I find difficult to avoid repetitions heuristically past round 4.

Answer 1

Six rounds are possible:

{1,6,12,23} {5,11,18,24} {4,9,15,22} {2,10,13,19} {7,14,17,21} {3,8,16,20}
{1,16,19,22} {2,5,12,14} {3,7,11,13} {8,9,18,23} {6,15,17,24} {4,10,20,21}
{1,5,9,13} {2,11,17,23} {4,6,16,18} {3,15,19,21} {8,10,14,22} {7,12,20,24}
{1,7,10,18} {5,15,20,23} {2,9,16,21} {4,8,13,24} {6,11,14,19} {3,12,17,22}
{1,2,3,4} {5,6,7,8} {9,10,11,12} {13,14,15,16} {17,18,19,20} {21,22,23,24}
{1,8,11,15} {5,10,16,17} {12,13,18,21} {3,9,14,24} {4,7,19,23} {2,6,20,22}

Seven rounds are optimal:

{1,2,3,4} {5,6,7,8} {9,10,11,12} {13,14,15,16} {17,18,19,20} {21,22,23,24}
{1,6,12,24} {2,7,16,23} {3,11,14,19} {4,15,20,22} {5,9,18,21} {8,10,13,17}
{1,9,13,23} {2,8,11,20} {3,7,17,22} {4,6,14,21} {5,12,15,19} {10,16,18,24}
{1,11,18,22} {2,6,9,15} {3,8,12,16} {4,10,19,23} {5,14,17,24} {7,13,20,21}
{1,7,10,14} {2,12,17,21} {3,9,20,24} {4,5,11,13} {6,16,19,22} {8,15,18,23}
{1,8,19,21} {2,5,10,22} {3,6,13,18} {4,9,16,17} {7,11,15,24} {12,14,20,23}
{1,5,16,20} {2,13,19,24} {3,10,15,21} {4,7,12,18} {6,11,17,23} {8,9,14,22}

Answer 2

This is a special case of the Social Golfer Problem. Using the web page Good Enough Golfers, I found a five-round solution. I was unable to find a six-round solution, but this does not rule out the possibility of a six-round solution.

	Round 1	Round 2	Round 3	Round 4	Round 5
A	Group 6	Group 4	Group 2	Group 6	Group 2
B	Group 3	Group 5	Group 2	Group 3	Group 4
C	Group 5	Group 2	Group 3	Group 2	Group 3
D	Group 2	Group 2	Group 5	Group 5	Group 4
E	Group 1	Group 4	Group 3	Group 1	Group 6
F	Group 1	Group 5	Group 4	Group 6	Group 5
G	Group 4	Group 1	Group 6	Group 6	Group 3
H	Group 1	Group 1	Group 1	Group 2	Group 2
I	Group 3	Group 6	Group 3	Group 5	Group 2
J	Group 2	Group 6	Group 2	Group 2	Group 6
K	Group 4	Group 4	Group 5	Group 2	Group 1
L	Group 3	Group 3	Group 4	Group 4	Group 3
M	Group 4	Group 2	Group 4	Group 1	Group 2
N	Group 5	Group 6	Group 1	Group 6	Group 4
O	Group 5	Group 1	Group 4	Group 3	Group 6
P	Group 1	Group 2	Group 2	Group 4	Group 1
Q	Group 4	Group 3	Group 1	Group 5	Group 5
R	Group 2	Group 3	Group 3	Group 3	Group 1
S	Group 6	Group 6	Group 5	Group 3	Group 3
T	Group 2	Group 4	Group 6	Group 4	Group 5
U	Group 3	Group 1	Group 5	Group 1	Group 5
V	Group 6	Group 5	Group 1	Group 4	Group 6
W	Group 6	Group 3	Group 6	Group 1	Group 4
X	Group 5	Group 5	Group 6	Group 5	Group 1