TL;DR

Tournaments for games with 6 players areâ€¦ hard to set up. We might even conjecture impossible. Letâ€™s see a sub-optimal approach.

Letâ€™s recap our wishlist for tournaments:

• each player should compete against each other exactly once
• in matches with $k$ players each
• in rounds that allow all participants to play at the same time
• with the least amount of overall players

This allows everybody to play an adequate amount of matches, provides variety (and avoids cheating!), allows playing games that are best with multiple players, avoids dead times and are easier to set up.

# 6 players matches are tricky

In Allocating games in tournaments - example we started investigating finite affine planes to build such tournaments, and ended up with the following table:

players/game rounds total players total games
$n$ $n + 1$ $n^2$ $n \cdot (n + 1)$
2 3 4 6
3 4 9 12
4 5 16 20
5 6 25 30
7 8 49 56
8 9 64 72
9 10 81 90

Alas, the affine plane approach is not applicable to 6-players games because $6$ is not a power of a prime (it is, in fact, the product of two different primes $2 \cdot 3$).

On the one hand I think itâ€™s extremely lucky that this is the only case for integers below $10$ - it could have been worse; on the flip side, many games are actually enjoyable with $6$ players!

# Relax

As it happens, we need to relax some of our requirements to cope with $6$ players matches.

The first requirement is actually quite strict and contains two constraints:

• everybody should play against everybody - i.e. every pair of players should end up in at least one match;
• everybody should not play against someone else more than once - i.e. every pair of players must compete against everybody else exactly once.

In this case, we have to give up on the exactly once constraint:

• either we make sure everybody plays with anyone else at least once, allowing for some or all pairs to compete twice against each other, OR
• we ensure that anyone competes at most once against each other player.

The second approach is the one usually considered in the Social Golfer Problem. We will take a look at one approach that allows for either way.

# Letâ€™s start bigger

Letâ€™s start with the next feasible design that addresses all requirements: the case for 7-players matches. Here is its scheduling for $49$ players:

round 1:
(  1,   2,   3,   4,   5,   6,   7)
(  8,  14,  20,  26,  32,  38,  44)
(  9,  15,  21,  27,  33,  39,  45)
( 10,  16,  22,  28,  34,  40,  46)
( 11,  17,  23,  29,  35,  41,  47)
( 12,  18,  24,  30,  36,  42,  48)
( 13,  19,  25,  31,  37,  43,  49)

round 2:
(  1,   8,   9,  10,  11,  12,  13)
(  2,  14,  21,  28,  35,  42,  49)
(  3,  15,  23,  31,  32,  40,  48)
(  4,  16,  25,  27,  36,  38,  47)
(  5,  17,  20,  30,  33,  43,  46)
(  6,  18,  22,  26,  37,  41,  45)
(  7,  19,  24,  29,  34,  39,  44)

round 3:
(  1,  14,  15,  16,  17,  18,  19)
(  2,  13,  20,  27,  34,  41,  48)
(  3,  12,  21,  29,  37,  38,  46)
(  4,  11,  22,  31,  33,  42,  44)
(  5,  10,  23,  26,  36,  39,  49)
(  6,   9,  24,  28,  32,  43,  47)
(  7,   8,  25,  30,  35,  40,  45)

round 4:
(  1,  20,  21,  22,  23,  24,  25)
(  2,  12,  19,  26,  33,  40,  47)
(  3,  10,  18,  27,  35,  43,  44)
(  4,   8,  17,  28,  37,  39,  48)
(  5,  13,  16,  29,  32,  42,  45)
(  6,  11,  15,  30,  34,  38,  49)
(  7,   9,  14,  31,  36,  41,  46)

round 5:
(  1,  26,  27,  28,  29,  30,  31)
(  2,  11,  18,  25,  32,  39,  46)
(  3,   8,  16,  24,  33,  41,  49)
(  4,  12,  14,  23,  34,  43,  45)
(  5,   9,  19,  22,  35,  38,  48)
(  6,  13,  17,  21,  36,  40,  44)
(  7,  10,  15,  20,  37,  42,  47)

round 6:
(  1,  32,  33,  34,  35,  36,  37)
(  2,  10,  17,  24,  31,  38,  45)
(  3,  13,  14,  22,  30,  39,  47)
(  4,   9,  18,  20,  29,  40,  49)
(  5,  12,  15,  25,  28,  41,  44)
(  6,   8,  19,  23,  27,  42,  46)
(  7,  11,  16,  21,  26,  43,  48)

round 7:
(  1,  38,  39,  40,  41,  42,  43)
(  2,   9,  16,  23,  30,  37,  44)
(  3,  11,  19,  20,  28,  36,  45)
(  4,  13,  15,  24,  26,  35,  46)
(  5,   8,  18,  21,  31,  34,  47)
(  6,  10,  14,  25,  29,  33,  48)
(  7,  12,  17,  22,  27,  32,  49)

round 8:
(  1,  44,  45,  46,  47,  48,  49)
(  2,   8,  15,  22,  29,  36,  43)
(  3,   9,  17,  25,  26,  34,  42)
(  4,  10,  19,  21,  30,  32,  41)
(  5,  11,  14,  24,  27,  37,  40)
(  6,  12,  16,  20,  31,  35,  39)
(  7,  13,  18,  23,  28,  33,  38)


As we can see, players 1 to 7 always compete in different matches in all rounds except round 1. If we remove them, then, as well as round 1, all other rounds from 2 to 8 will contain non-overlapping matches with 6 participants each, with the additional bonus that no two players ever compete against each other more than once. Renumbering the rounds we end up with:

round 1:
(      8,  *9, *10, *11, *12, *13)
(     14,  21,  28,  35,  42,  49)
(     15,  23,  31,  32,  40,  48)
(     16,  25,  27,  36,  38,  47)
(     17,  20,  30,  33,  43,  46)
(     18,  22,  26,  37,  41,  45)
(     19,  24,  29,  34,  39,  44)

round 2:
(     14,  15,  16,  17,  18,  19)
(     13,  20,  27,  34,  41,  48)
(     12,  21,  29,  37,  38,  46)
(     11,  22,  31,  33,  42,  44)
(     10,  23,  26,  36,  39,  49)
(      9,  24,  28,  32,  43,  47)
(      8,  25,  30,  35,  40,  45)

round 3:
(     20,  21,  22,  23,  24,  25)
(     12,  19,  26,  33,  40,  47)
(     10,  18,  27,  35,  43,  44)
(      8,  17,  28,  37,  39,  48)
(     13,  16,  29,  32,  42,  45)
(     11,  15,  30,  34,  38,  49)
(      9,  14,  31,  36,  41,  46)

round 4:
(     26,  27,  28,  29,  30,  31)
(     11,  18,  25,  32,  39,  46)
(      8,  16,  24,  33,  41,  49)
(     12,  14,  23,  34,  43,  45)
(      9,  19,  22,  35,  38,  48)
(     13,  17,  21,  36,  40,  44)
(     10,  15,  20,  37,  42,  47)

round 5:
(     32,  33,  34,  35,  36,  37)
(     10,  17,  24,  31,  38,  45)
(     13,  14,  22,  30,  39,  47)
(      9,  18,  20,  29,  40,  49)
(     12,  15,  25,  28,  41,  44)
(      8,  19,  23,  27,  42,  46)
(     11,  16,  21,  26,  43,  48)

round 6:
(     38,  39,  40,  41,  42,  43)
(      9,  16,  23,  30,  37,  44)
(     11,  19,  20,  28,  36,  45)
(     13,  15,  24,  26,  35,  46)
(      8,  18,  21,  31,  34,  47)
(     10,  14,  25,  29,  33,  48)
(     12,  17,  22,  27,  32,  49)

round 7:
(     44,  45,  46,  47,  48,  49)
(      8,  15,  22,  29,  36,  43)
(      9,  17,  25,  26,  34,  42)
(     10,  19,  21,  30,  32,  41)
(     11,  14,  24,  27,  37,  40)
(     12,  16,  20,  31,  35,  39)
(     13,  18,  23,  28,  33,  38)


# A relaxed solution

We can, of course, renumber the player identifiers as well, from range $8..49$ to a more natural $1..42$, and get the following arrangement:

round 1:
(  1,   2,   3,   4,   5,   6)
(  7,  14,  21,  28,  35,  42)
(  8,  16,  24,  25,  33,  41)
(  9,  18,  20,  29,  31,  40)
( 10,  13,  23,  26,  36,  39)
( 11,  15,  19,  30,  34,  38)
( 12,  17,  22,  27,  32,  37)

round 2:
(  1,  18,  23,  28,  33,  38)
(  2,  17,  21,  25,  36,  40)
(  3,  16,  19,  29,  32,  42)
(  4,  15,  24,  26,  35,  37)
(  5,  14,  22,  30,  31,  39)
(  6,  13,  20,  27,  34,  41)
(  7,   8,   9,  10,  11,  12)

round 3:
(  1,  10,  21,  30,  32,  41)
(  2,   7,  24,  29,  34,  39)
(  3,  11,  20,  28,  36,  37)
(  4,   8,  23,  27,  31,  42)
(  5,  12,  19,  26,  33,  40)
(  6,   9,  22,  25,  35,  38)
( 13,  14,  15,  16,  17,  18)

round 4:
(  1,   9,  17,  26,  34,  42)
(  2,  12,  15,  28,  31,  41)
(  3,   8,  13,  30,  35,  40)
(  4,  11,  18,  25,  32,  39)
(  5,   7,  16,  27,  36,  38)
(  6,  10,  14,  29,  33,  37)
( 19,  20,  21,  22,  23,  24)

round 5:
(  1,  12,  16,  20,  35,  39)
(  2,  11,  13,  22,  33,  42)
(  3,  10,  17,  24,  31,  38)
(  4,   9,  14,  19,  36,  41)
(  5,   8,  18,  21,  34,  37)
(  6,   7,  15,  23,  32,  40)
( 25,  26,  27,  28,  29,  30)

round 6:
(  1,  11,  14,  24,  27,  40)
(  2,   9,  16,  23,  30,  37)
(  3,   7,  18,  22,  26,  41)
(  4,  12,  13,  21,  29,  38)
(  5,  10,  15,  20,  25,  42)
(  6,   8,  17,  19,  28,  39)
( 31,  32,  33,  34,  35,  36)

round 7:
(  1,   8,  15,  22,  29,  36)
(  2,  10,  18,  19,  27,  35)
(  3,  12,  14,  23,  25,  34)
(  4,   7,  17,  20,  30,  33)
(  5,   9,  13,  24,  28,  32)
(  6,  11,  16,  21,  26,  31)
( 37,  38,  39,  40,  41,  42)


This solution has the following characteristics:

• everybody plays the same number of rounds;
• always against different competitors;
• each player competes almost with every one else;
• never more than once, anyway.

In other terms, we preserve that no two participants ever compete against each other twice (or more), at the cost of never making some players compete against each other.