4-player games, 3x13 games

[Mahjong]

13 individual players (not in team)
4-player game

From what I understand, I am searching for a near-resolvable NR(v=13, k=4, lambda=3) covering. One should exist with b=39 (39 games in total), r=12 (every player plays 12 games), 3 rounds, a different players sits out each round.

Something like:

round 1:
ABCD
EFGH
IJKL
(M sits out this round)

round 2:
...
...
...
(A sits out)

round 3:
...
...
...
(B sits out)

[...]

round 13:
...
...
...
(L sits out)


I've tried every single solution t=2, v=13, b=39, r=12, k=4, lambda=3 found on http://nassrat.cs.dal.ca/ddb/, but none work. They have either repeated blocks or just cannot be sliced into 13 rounds of 3 games. It's says the database is incomplete. Perhaps, you have a solution or a better source.


Thank you in advance

[Ian Wakeling]

This near resolvable design does indeed exist, there is a simple cyclic construction.

( 2  6 13  9) ( 8 10  7  5) (11 12  4  3)
( 3  7  1 10) ( 9 11  8  6) (12 13  5  4)
( 4  8  2 11) (10 12  9  7) (13  1  6  5)
( 5  9  3 12) (11 13 10  8) ( 1  2  7  6)
( 6 10  4 13) (12  1 11  9) ( 2  3  8  7)
( 7 11  5  1) (13  2 12 10) ( 3  4  9  8)
( 8 12  6  2) ( 1  3 13 11) ( 4  5 10  9)
( 9 13  7  3) ( 2  4  1 12) ( 5  6 11 10)
(10  1  8  4) ( 3  5  2 13) ( 6  7 12 11)
(11  2  9  5) ( 4  6  3  1) ( 7  8 13 12)
(12  3 10  6) ( 5  7  4  2) ( 8  9  1 13)
(13  4 11  7) ( 6  8  5  3) ( 9 10  2  1)
( 1  5 12  8) ( 7  9  6  4) (10 11  3  2)

Hope that helps.

[Mahjong]

Thank you very much!

Did you generate it yourself?

[Ian Wakeling]

It came from a file that I made over 8 years ago.  I think it was constructed using Lemma 4.1 from the following paper: Hanani, H. (1975) "Balanced Incomplete Block Designs and Related Designs", Discrete Mathematics, 11, 255-369.  The near-resolvability is a bonus as I don't think it's mentioned in the paper.

An alternative source for this schedule is a standard whist design for 13 players, you can find an example here on J D A Wiseman's web site.  Simply ignore the assignments of the four players to partners and opponents.