Tournament blocks

This article is meant as a repository for tournament and competitive blocks used for seating arragements. The content is mainly present in tabular form: reading it for the sake of reading is not recommended.

Ideal blocks

Ideal blocks require the organizer to plan according to the situation hey are planning for. Block sizes of 4 and of 5 change the dynamic completely.

Tournament

Tournaments often require a single hanchan between 4 players and then a shuffling of new opponents, aiming to complete a maximum or a set number of hanchan with minimal or no repeats between players.

League play

Leagues may require being able to shift between blocks of 4 and 5 players who end up playing 4 hanchan among each other per session.

Pitfalls and criticism of tournament blocks used for mahjong

Block repository

Tournament blocks

Simplest solution: Dutch cycles

Dutch cycles (also called ____) serve a purpose of fairly distributing players across a set number of games, all while reducing/preventing repeats between pairs of players. This system works best when prime factors are in play, but breaks down when composite factors are used. Large player bases can mitigate some issues.

The obvious flaw with this solution is that players are segregated into blocks. The method is fair as long as players draw from one pool with all entries mixed AND that the draw occurs fairly, in the presence of all players, or failing that, that the location and time of draw be published beforehand with a fair chance for all to attend. Both NMB tournaments (2011 in Utrecht) and the 2014 WRC used this method, however they both had flaws in their implementation.

The general formula is that table t consists of players [t, t + n/4, t + 2n/4, t + 3n/4] in hanchan 1, and for future hanchan h, tables consist of [t, ((t + 1*h) % n/4) + n/4, ((t + 2*h) % n/4) + 2n/4, ((t + 3*h) % n/4) + 3n/4]. Its usefulness can be demonstrated as soon as there are 44 or more participants, and preferably not a multiple of 6.

Players	4H	6H	8H	10H	12H	Comment
16 (4*4)	3 / Other	Other (5)	no	no	no	Different block solution for 5 hanchan.
20 (5*4)	Yes	Other (5)	no	no	no
24 (6*4)	3 / Other	Other	Other (7)	no	no
28 (7*4)	Yes	Yes	7	Other (9)	no
40 (10*4)	Yes	5 / unkn.	unkn.	unkn.	no
44 (11*4)	Yes	Yes	Yes	Yes	11
52 (13*4)	Yes	Yes	Yes	13
68 (17*4)	Yes	Yes	Yes	Yes	17

Notes: Mathpuzzle.com has a few non-Dutch solutions for the "Social Golfer Problem".
1. For 16P in 5H: ABCD EFGH IJKL MNOP, AEIM BFJN CGKO DHLP, AFKP BELO CHIN DGJM, AGLN BHKM CEJP DFIO, AHJO BGIP CFLM DEKN.
2. For 20P in 5H: ABCD EFGH IJKL MNOP QRST, AEIM BFJQ CGNR DKOS HLPT, AFOT BELR CIPS DGJM HKNQ, AJPR BHMS CEKT DFIN GLOQ, ALNS BGIT CHJO DEPQ FKMR.
3. For 24P in 7H: ABKU IJSE QRCM DGFX HLNO PTVW, ACLV IKTF QSDN EHGR BMOP JUWX, ADMW ILUG QTEO FBHS CJNP KRVX, AENX IMVH QUFP GCBT DJKO LRSW, AFOR INWB QVGJ HDCU EKLP MSTX, AEPS IOXC QWHK BEDV FJLM NRTU, AHJT IPRD QXBL CFEW GKMN OSUV.
4. For 28P in 9H: ABCD EFGH IJKL MNab cdef ghij klmn, AEgk BFMc Ndhl GIem HJai CKbn DLfj, AFjn BEae bfim HKcl GLMh CINk DJdg, AIci BJNn EKMj FLdm begl CGaf DHhk, AGbd BHgm ELNi achn FKfk CJej DIMl, AKeh BLbk FIag EJfl Ncjm CHMd DGin, AHNf BGjl FJbh Meik EIdn CLcg DKam, ALal BKdi GJck Mfgn HIbj CEhm DFNe, AJMm BIfh CFil DEbc GKNg HLen adjk.
5. For 32P in 10H: ABCD EFGH IJKL MNPO abcd efgh ijkl mnop, AEIm BgcH DFKp kPdf Maei ChbG jNoL lJnO, Alof Bhkm DNGi FaLO MbHK CPeJ Ejcp gIdn, AciL BjdK DkbJ FgoP fGOp CIla EhMn NeHm, AgNK BElP DIoH hdiO beLp Cjfm FMcJ kaGn, AhJp BFin DcmO INbf leGK CMod EgkL jPaH, AFde BoGJ DEaf hlNc PiKm CHLn gjbO IkMp, APbn BNap DglM koce fHiJ CEKO FhIj dGLm, AjMG BIeO DhPL gaJm cfKn CFkN Eobi ldHp, AkHO BMfL Djen Flbm IPcG Cgip ENdJ hoaK. This solution also has the property that there is a pairwise exclusion: Two pots could be used to guarantee that any two players will play with everone else but not with each other.

League play blocks

Considering that league blocks contain 5 players, the counting mechanism has to be recalculated for Dutch cycles. These numbers are for 6-session events or seasons. These Dutch cycle blocks can drop the last fifth in order to make groups of 4, expanding the solved ranges of players from 80% to 100% of the maximal solutions. For 40, using the social golfer solution below will work for 32 to 40 players. Given the distribution of primes, there is a solution within that 20% range except for 66 and 67 players.

Players	P2	P3	P4	P5	Comment
30	0	0	0	0	2 sessions.
30	1	2	3	4
35	0	0	0	0	7 sessions (28P-35P)
35	1	2	3	4
35	2	4	6	1
35	3	6	2	5
35	4	1	5	2
35	5	3	1	6
35	6	5	4	3
45	0	0	0	0	3 sessions
45	1	2	3	4
45	2	4	6	8
50	0	0	0	0	5/6 sessions (40P-50P)
50	1	2	3	4
50	2	4	6	8
50	3	6	9	2
50	4	8	2	6
50	5	3	1	9	(non-Dutch)
55	0	0	0	0	11 sessions (44P-55P)
55	1	2	3	4
55	2	4	6	8
55	3	6	9	1
55	4	8	1	5
55	5	10	4	9
55	6	1	7	2
55	7	3	10	6
55	8	5	2	10
55	9	7	5	3
55	10	9	8	7

15-player season

Session and Table	Player 1	Player 2	Player 3	Player 4	Player 5	Deadline
Session 1 - Table 1 (+1)	P1	P2	P3	P4	P5	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 2	P6	P7	P8	P9	P10	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 3	P11	P12	P13	P14	P15	Saturday 2015-04-04 23:59 UTC
Session 2 - Table 1 (+3)	P1	P4	P7	P10	P13	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 2	P2	P5	P8	P11	P14	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 3	P3	P6	P9	P12	P15	Saturday 2015-04-18 23:59 UTC
Session 3 - Table 1 (+8)	P1	P9	P2	P10	P3	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 2	P11	P4	P12	P5	P13	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 3	P6	P14	P7	P15	P8	Saturday 2015-05-02 23:59 UTC
Session 4 - Table 1 (+11)	P1	P12	P8	P4	P15	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 2	P11	P7	P3	P14	P10	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 3	P6	P2	P13	P9	P5	Saturday 2015-05-16 23:59 UTC
Session 5 - Table 1 (+5)	P1	P6	P11	P5	P10	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 2	P15	P4	P9	P14	P3	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 3	P8	P13	P2	P7	P12	Saturday 2015-05-30 23:59 UTC
Session 6 - Table 1 (+7)	P1	P8	P15	P7	P14	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 2	P6	P13	P5	P12	P4	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 3	P11	P3	P10	P2	P9	Saturday 2015-06-13 23:59 UTC

25-player season

An ideal number of participants would be 25. 24 is easy, 21 to 23 require a minor adjustment.

Session and Table	Player 1	Player 2	Player 3	Player 4	Player 5	Deadline
Session 1 - Table 1 (+1)	P1	P2	P3	P4	P5	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 2	P6	P7	P8	P9	P10	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 3	P11	P12	P13	P14	P15	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 4	P16	P17	P18	P19	P20	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 5	P21	P22	P23	P24	P25	Saturday 2015-04-04 23:59 UTC
Session 2 - Table 1 (-4)	P1	P22	P18	P14	P10	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 2	P6	P2	P23	P19	P15	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 3	P11	P7	P3	P24	P20	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 4	P16	P12	P8	P4	P25	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 5	P21	P17	P13	P9	P5	Saturday 2015-04-18 23:59 UTC
Session 3 - Table 1 (+7)	P1	P17	P8	P24	P15	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 2	P6	P22	P13	P4	P20	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 3	P11	P2	P18	P9	P25	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 4	P16	P7	P23	P14	P5	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 5	P21	P12	P3	P19	P10	Saturday 2015-05-02 23:59 UTC
Session 4 - Table 1 (+11)	P1	P12	P23	P9	P20	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 2	P6	P17	P3	P14	P25	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 3	P11	P22	P8	P19	P5	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 4	P16	P2	P13	P24	P10	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 5	P21	P7	P18	P4	P15	Saturday 2015-05-16 23:59 UTC
Session 5 - Table 1 (+6)	P1	P7	P13	P19	P25	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 2	P6	P12	P18	P24	P5	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 3	P11	P17	P23	P4	P10	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 4	P16	P22	P3	P9	P15	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 5	P21	P2	P8	P14	P20	Saturday 2015-05-30 23:59 UTC
Session 6 - Table 1 (+5)	P1	P6	P11	P16	P21	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 2	P2	P7	P12	P17	P22	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 3	P3	P8	P13	P18	P23	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 4	P4	P9	P14	P19	P24	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 5	P5	P10	P15	P20	P25	Saturday 2015-06-13 23:59 UTC

35-player season

Solved via Dutch cycles up to 7 sessions.

40-player season

With 40, it has been a problem optimally solved with the "social golfer problem", up to 8 sessions.

Session and Table	Player 1	Player 2	Player 3	Player 4	Player 5	Deadline
Session 1 - Table 1 (+8)	P1	P9	P17	P25	P33	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 2	P2	P10	P18	P26	P34	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 3	P3	P11	P19	P27	P35	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 4	P4	P12	P20	P28	P36	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 5	P5	P13	P21	P29	P37	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 6	P6	P14	P22	P30	P38	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 7	P7	P15	P23	P31	P39	Saturday 2015-04-04 23:59 UTC
Session 1 - Table 8	P8	P16	P24	P32	P40	Saturday 2015-04-04 23:59 UTC
Session 2 - Table 1	P1	P10	P19	P28	P37	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 2	P2	P9	P20	P27	P38	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 3	P3	P12	P17	P26	P39	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 4	P4	P11	P18	P25	P40	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 5	P5	P14	P23	P32	P33	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 6	P6	P13	P24	P31	P34	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 7	P7	P16	P21	P30	P35	Saturday 2015-04-18 23:59 UTC
Session 2 - Table 8	P8	P15	P22	P29	P36	Saturday 2015-04-18 23:59 UTC
Session 3 - Table 1	P1	P11	P21	P31	P36	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 2	P2	P12	P22	P32	P35	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 3	P3	P9	P23	P29	P34	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 4	P4	P10	P24	P30	P33	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 5	P5	P15	P17	P27	P40	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 6	P6	P16	P18	P28	P39	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 7	P7	P13	P19	P25	P38	Saturday 2015-05-02 23:59 UTC
Session 3 - Table 8	P8	P14	P20	P26	P37	Saturday 2015-05-02 23:59 UTC
Session 4 - Table 1	P1	P12	P23	P30	P40	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 2	P2	P11	P24	P29	P39	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 3	P3	P10	P21	P32	P38	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 4	P4	P9	P22	P31	P37	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 5	P5	P16	P19	P26	P36	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 6	P6	P15	P20	P25	P35	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 7	P7	P14	P17	P28	P34	Saturday 2015-05-16 23:59 UTC
Session 4 - Table 8	P8	P13	P18	P27	P33	Saturday 2015-05-16 23:59 UTC
Session 5 - Table 1	P1	P13	P20	P32	P39	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 2	P2	P14	P19	P31	P40	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 3	P3	P15	P18	P30	P37	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 4	P4	P16	P17	P29	P38	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 5	P5	P9	P24	P28	P35	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 6	P6	P10	P23	P27	P36	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 7	P7	P11	P22	P26	P33	Saturday 2015-05-30 23:59 UTC
Session 5 - Table 8	P8	P12	P21	P25	P34	Saturday 2015-05-30 23:59 UTC
Session 6 - Table 1	P1	P14	P18	P29	P35	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 2	P2	P13	P17	P30	P36	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 3	P3	P16	P20	P31	P33	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 4	P4	P15	P19	P32	P34	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 5	P5	P10	P22	P25	P39	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 6	P6	P9	P21	P26	P40	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 7	P7	P12	P24	P27	P37	Saturday 2015-06-13 23:59 UTC
Session 6 - Table 8	P8	P11	P23	P28	P38	Saturday 2015-06-13 23:59 UTC
Session 7 - Table 1	P1	P15	P24	P26	P38	(not played)
Session 7 - Table 2	P2	P16	P23	P25	P37	(not played)
Session 7 - Table 3	P3	P13	P22	P28	P40	(not played)
Session 7 - Table 4	P4	P14	P21	P27	P39	(not played)
Session 7 - Table 5	P5	P11	P20	P30	P34	(not played)
Session 7 - Table 6	P6	P12	P19	P29	P33	(not played)
Session 7 - Table 7	P7	P9	P18	P32	P36	(not played)
Session 7 - Table 8	P8	P10	P17	P31	P35	(not played)
Session 8 - Table 1	P1	P16	P22	P27	P34	(not played)
Session 8 - Table 2	P2	P15	P21	P28	P33	(not played)
Session 8 - Table 3	P3	P14	P24	P25	P36	(not played)
Session 8 - Table 4	P4	P13	P23	P26	P35	(not played)
Session 8 - Table 5	P5	P12	P18	P31	P38	(not played)
Session 8 - Table 6	P6	P11	P17	P32	P37	(not played)
Session 8 - Table 7	P7	P10	P20	P29	P40	(not played)
Session 8 - Table 8	P8	P9	P19	P30	P39	(not played)