Tournament blocks: Difference between revisions

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(Fresh data, it's Warwick invading on Trundle's turf. {Saving to not lose anything})
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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 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. Heavily composite numbers of players will lead to collisions when {...}
The general formula is that table t consists of players [t, t + n/4, t + 2n/4, t + 3n/4] in hanchan 0 {mathematically speaking so the formula works}, 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 3. Heavily composite numbers of players will lead to collisions when tables = h * factor. The 2014 WRC had 120 players over 30 tables: as 120 (as well as 30) is divisible by 3, it would lead to a collision when h = 10, making the 11th hanchan place naively two players who met before across all 30 tables.


Notes:
Notes:
[http://www.mathpuzzle.com/MAA/54-Golf%20Tournaments/mathgames_08_14_07.html Mathpuzzle.com] has a few non-Dutch solutions for the "Social Golfer Problem".<br />
[http://www.mathpuzzle.com/MAA/54-Golf%20Tournaments/mathgames_08_14_07.html Mathpuzzle.com] has a few non-Dutch solutions for the "Social Golfer Problem". -- As does the web archive, from [http://web.archive.org/web/20050407074608/http://www.icparc.ic.ac.uk/~wh/golf/solutions.html Warwick Harvey]<br />
 
===== Blocks of 4 =====
{| class="wikitable"
{| class="wikitable"
|-
|-
! Players !! Method !! Dutch !! Social Golfer !! Comment
! Players !! Block !! Of !! Raw data
|-
| 16 || 1 || 5 || ABCD EFGH IJKL MNOP
|-
| 16 || 2 || 5 || AEIM BFJN CGKO DHLP
|-
| 16 || 3 || 5 || AFKP BELO CHIN DGJM
|-
| 16 || 4 || 5 || AGLN BHKM CEJP DFIO
|-
| 16 || 5 || 5 || AHJO BGIP CFLM DEKN.
|-
| 20 || 1 || 5 || ABCD EFGH IJKL MNOP QRST
|-
| 20 || 2 || 5 || AEIM BFJQ CGNR DKOS HLPT
|-
| 20 || 3 || 5 || AFOT BELR CIPS DGJM HKNQ
|-
| 20 || 4 || 5 || AJPR BHMS CEKT DFIN GLOQ
|-
| 20 || 5 || 5 || ALNS BGIT CHJO DEPQ FKMR.
|-
| 24 || 1 || 7 || ABKU IJSE QRCM DGFX HLNO PTVW
|-
| 24 || 2 || 7 || ACLV IKTF QSDN EHGR BMOP JUWX
|-
| 24 || 3 || 7 || ADMW ILUG QTEO FBHS CJNP KRVX
|-
| 24 || 4 || 7 || AENX IMVH QUFP GCBT DJKO LRSW
|-
| 24 || 5 || 7 || AFOR INWB QVGJ HDCU EKLP MSTX
|-
| 24 || 6 || 7 || AEPS IOXC QWHK BEDV FJLM NRTU
|-
| 24 || 7 || 7 || AHJT IPRD QXBL CFEW GKMN OSUV.
|-
| 28 || 1 || 9 || ABCD EFGH IJKL MNab cdef ghij klmn
|-
| 28 || 1 || 9 || AEgk BFMc Ndhl GIem HJai CKbn DLfj
|-
| 28 || 1 || 9 || AFjn BEae bfim HKcl GLMh CINk DJdg
|-
| 28 || 1 || 9 || AIci BJNn EKMj FLdm begl CGaf DHhk
|-
| 28 || 1 || 9 || AGbd BHgm ELNi achn FKfk CJej DIMl
|-
| 28 || 1 || 9 || AKeh BLbk FIag EJfl Ncjm CHMd DGin
|-
| 28 || 1 || 9 || AHNf BGjl FJbh Meik EIdn CLcg DKam
|-  
|-  
| 16 (4*4) || SG || || 5 || ABCD EFGH IJKL MNOP, <br />AEIM BFJN CGKO DHLP, <br />AFKP BELO CHIN DGJM, <br />AGLN BHKM CEJP DFIO, <br />AHJO BGIP CFLM DEKN.
| 28 || 1 || 9 || ALal BKdi GJck Mfgn HIbj CEhm DFNe
|-  
|-  
| 20 (5*4) || Both || 5 || 5 || ABCD EFGH IJKL MNOP QRST, <br />AEIM BFJQ CGNR DKOS HLPT, <br />AFOT BELR CIPS DGJM HKNQ, <br />AJPR BHMS CEKT DFIN GLOQ, <br />ALNS BGIT CHJO DEPQ FKMR.
| 28 || 1 || 9 || AJMm BIfh CFil DEbc GKNg HLen adjk.  
|-  
|-  
| 24 (6*4) || SG || || 7 || ABKU IJSE QRCM DGFX HLNO PTVW, <br />ACLV IKTF QSDN EHGR BMOP JUWX, <br />ADMW ILUG QTEO FBHS CJNP KRVX, <br />AENX IMVH QUFP GCBT DJKO LRSW, <br />AFOR INWB QVGJ HDCU EKLP MSTX, <br />AEPS IOXC QWHK BEDV FJLM NRTU, <br />AHJT IPRD QXBL CFEW GKMN OSUV.
| 32 || 1 || 10 || ABCD EFGH IJKL MNPO abcd efgh ijkl mnop
|-  
|-  
| 28 (7*4) || SG || || 9 || ABCD EFGH IJKL MNab cdef ghij klmn, <br />AEgk BFMc Ndhl GIem HJai CKbn DLfj, <br />AFjn BEae bfim HKcl GLMh CINk DJdg, <br />AIci BJNn EKMj FLdm begl CGaf DHhk, <br />AGbd BHgm ELNi achn FKfk CJej DIMl, <br />AKeh BLbk FIag EJfl Ncjm CHMd DGin, <br />AHNf BGjl FJbh Meik EIdn CLcg DKam, <br />ALal BKdi GJck Mfgn HIbj CEhm DFNe, <br />AJMm BIfh CFil DEbc GKNg HLen adjk.
| 32 || 2 || 10 || AEIm BgcH DFKp kPdf Maei ChbG jNoL lJnO
|-  
|-  
| 32 (8*4) || SG || || 10 || ABCD EFGH IJKL MNPO abcd efgh ijkl mnop, <br />AEIm BgcH DFKp kPdf Maei ChbG jNoL lJnO, <br />Alof Bhkm DNGi FaLO MbHK CPeJ Ejcp gIdn, <br />AciL BjdK DkbJ FgoP fGOp CIla EhMn NeHm, <br />AgNK BElP DIoH hdiO beLp Cjfm FMcJ kaGn, <br />AhJp BFin DcmO INbf leGK CMod EgkL jPaH, <br />AFde BoGJ DEaf hlNc PiKm CHLn gjbO IkMp, <br />APbn BNap DglM koce fHiJ CEKO FhIj dGLm, <br />AjMG BIeO DhPL gaJm cfKn CFkN Eobi ldHp, <br />AkHO BMfL Djen Flbm IPcG Cgip ENdJ hoaK.  
| 32 || 3 || 10 || Alof Bhkm DNGi FaLO MbHK CPeJ Ejcp gIdn
|-
| 32 || 4 || 10 || AciL BjdK DkbJ FgoP fGOp CIla EhMn NeHm
|-
| 32 || 5 || 10 || AgNK BElP DIoH hdiO beLp Cjfm FMcJ kaGn
|-
| 32 || 6 || 10 || AhJp BFin DcmO INbf leGK CMod EgkL jPaH
|-
| 32 || 7 || 10 || AFde BoGJ DEaf hlNc PiKm CHLn gjbO IkMp
|-
| 32 || 8 || 10 || APbn BNap DglM koce fHiJ CEKO FhIj dGLm
|-
| 32 || 9 || 10 || AjMG BIeO DhPL gaJm cfKn CFkN Eobi ldHp
|-
| 32 || 10 || 10 || AkHO BMfL Djen Flbm IPcG Cgip ENdJ hoaK.  
|-
| 36 || 1 || 8 || [&#160;[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16], [17, 18, 19, 20], [21, 22, 23, 24], [25, 26, 27, 28], [29, 30, 31, 32], [33, 34, 35, 36]&#160;]
|-
| 36 || 2 || 8 || [&#160;[1, 5, 9, 13], [2, 6, 10, 17], [3, 7, 11, 21], [4, 8, 12, 25], [14, 18, 22, 29], [15, 19, 23, 33], [16, 26, 30, 34], [20, 27, 31, 35], [24, 28, 32, 36]&#160;]
|-
| 36 || 3 || 8 || [&#160;[1, 6, 11, 25], [2, 7, 9, 16], [3, 5, 10, 20], [4, 14, 21, 33], [8, 15, 18, 32], [12, 13, 17, 22], [19, 27, 30, 36], [23, 28, 31, 34], [24, 26, 29, 35]&#160;]
|-
| 36 || 4 || 8 || [&#160;[1, 10, 18, 21], [2, 8, 13, 29], [3, 14, 31, 36], [4, 9, 20, 23], [5, 15, 17, 28], [6, 12, 16, 19], [7, 26, 32, 33], [11, 24, 27, 34], [22, 25, 30, 35]&#160;]
|-
| 36 || 5 || 8 || [&#160;[1, 14, 24, 30], [2, 12, 18, 34], [3, 9, 17, 32], [4, 6, 28, 35], [5, 21, 29, 36], [7, 10, 19, 25], [8, 16, 23, 27], [11, 13, 20, 33], [15, 22, 26, 31]&#160;]
|-
| 36 || 6 || 8 || [&#160;[1, 12, 23, 26], [2, 11, 22, 36], [3, 6, 13, 24], [4, 10, 16, 32], [5, 18, 27, 33], [7, 20, 28, 30], [8, 14, 19, 35], [9, 15, 29, 34], [17, 21, 25, 31]&#160;]
|-
| 36 || 7 || 8 || [&#160;[1, 19, 28, 29], [2, 5, 23, 30], [3, 12, 15, 27], [4, 11, 18, 31], [6, 9, 26, 36], [7, 14, 17, 34], [8, 10, 22, 33], [13, 21, 32, 35], [16, 20, 24, 25]&#160;]
|-
| 36 || 8 || 8 || [&#160;[1, 7, 15, 35], [2, 14, 25, 32], [3, 16, 18, 28], [4, 17, 27, 29], [5, 11, 19, 26], [6, 20, 22, 34], [8, 9, 21, 30], [10, 13, 23, 36], [12, 24, 31, 33]&#160;]
|- style="background:#ccf"
|- style="background:#ccf"
| 40 (10*4) || Unkn. || 5/6ND || ||  
| 40 (10*4) || Unkn. || 5/6ND ||
|-  
|-  
| 44 (11*4) || Dutch || 11 || ||
| 44 (11*4) || Dutch || 11 ||
|-  
|-  
| 52 (13*4) || Dutch || 13 || ||
| 52 (13*4) || Dutch || 13 ||
|-  
|-  
| 68 (17*4) || Dutch || 17 || ||  
| 68 (17*4) || Dutch || 17 ||
|-  
|-  
| 76 (19*4) || Dutch || 19 || ||  
| 76 (19*4) || Dutch || 19 ||  
|}
|}


Line 52: Line 132:
{| class="wikitable"
{| class="wikitable"
|-
|-
! Players !! P1 !! P2 !! P3 !! P4 !! P5 !! Comment
! Players !! Round !! Of !! Match-ups
|-  style="background:#ccf"
| 25 || 1st || of 6 || [&#160;[1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25]&#160;]
|-  style="background:#ccf"
| 25 || 2nd || of 6 || [&#160;[1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24], [5, 10, 15, 20, 25]&#160;]
|-  style="background:#ccf"
| 25 || 3rd || of 6 || [&#160;[1, 7, 13, 19, 25], [2, 8, 14, 20, 21], [3, 9, 15, 16, 22], [4, 10, 11, 17, 23], [5, 6, 12, 18, 24]&#160;]
|-  style="background:#ccf"
| 25 || 4th || of 6 || [&#160;[1, 8, 15, 17, 24], [2, 9, 11, 18, 25], [3, 10, 12, 19, 21], [4, 6, 13, 20, 22], [5, 7, 14, 16, 23]&#160;]
|-  style="background:#ccf"
| 25 || 5th || of 6 || [&#160;[1, 9, 12, 20, 23], [2, 10, 13, 16, 24], [3, 6, 14, 17, 25], [4, 7, 15, 18, 21], [5, 8, 11, 19, 22]&#160;]
|-  style="background:#ccf"
| 25 || 6th || of 6 || [&#160;[1, 10, 14, 18, 22], [2, 6, 15, 19, 23], [3, 7, 11, 20, 24], [4, 8, 12, 16, 25], [5, 9, 13, 17, 21]&#160;]
|-  
|-  
| 30 || 0 || 0 || 0 || 0 || 0 || 2 sessions.
| 30 || 1st || of 6 || [&#160;[1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25], [26, 27, 28, 29, 30]&#160;]
|-  
|-  
| 30 || 0 || 1 || 2 || 3 || 4 ||
| 30 || 2nd || of 6 || [&#160;[1, 6, 11, 16, 21], [2, 7, 13, 19, 30], [3, 8, 14, 24, 29], [4, 9, 20, 25, 28], [5, 15, 18, 23, 27], [10, 12, 17, 22, 26]&#160;]
|-  
|-  
| 35 || 0 || 0 || 0 || 0 || 0 || 7 sessions (28P-35P)
| 30 || 3rd || of 6 || [&#160;[1, 8, 13, 20, 22], [2, 6, 12, 23, 28], [3, 9, 11, 18, 30], [4, 15, 16, 24, 26], [5, 7, 17, 25, 29], [10, 14, 19, 21, 27]&#160;]
|-  
|-  
| 35 || 0 || 1 || 2 || 3 || 4 ||
| 30 || 4th || of 6 || [&#160;[1, 7, 18, 24, 28], [2, 10, 11, 20, 29], [3, 6, 15, 19, 22], [4, 14, 17, 23, 30], [5, 9, 13, 21, 26], [8, 12, 16, 25, 27]&#160;]
|-  
|-  
| 35 || 0 || 2 || 4 || 6 || 1 ||
| 30 || 5th || of 6 || [&#160;[1, 10, 15, 25, 30], [2, 9, 17, 24, 27], [3, 7, 12, 20, 21], [4, 6, 13, 18, 29], [5, 14, 16, 22, 28], [8, 11, 19, 23, 26]&#160;]
|-  
|-  
| 35 || 0 || 3 || 6 || 2 || 5 ||  
| 30 || 6th || of 6 || [&#160;[1, 9, 12, 19, 29], [5, 6, 20, 24, 30], [4, 7, 11, 22, 27], [8, 15, 17, 21, 28], [3, 10, 13, 16, 23], [2, 14, 18, 25, 26]&#160;]
|-  style="background:#ccf"
| 35 || all || of 7 || <colspan="4"> | Dutch cycles. {28P-35P}
|-  
|-  
| 35 || 0 || 4 || 1 || 5 || 2 ||
| 40 || 1st || of 6 || [&#160;[1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25], [26, 27, 28, 29, 30], [31, 32, 33, 34, 35], [36, 37, 38, 39, 40]&#160;]
|-  
|-  
| 35 || 0 || 5 || 3 || 1 || 6 ||
| 40 || 2nd || of 6 || [&#160;[1, 6, 11, 16, 21], [2, 7, 12, 26, 31], [3, 8, 13, 17, 36], [4, 9, 14, 27, 32], [5, 10, 15, 22, 37], [18, 23, 28, 33, 38], [19, 24, 29, 34, 39], [20, 25, 30, 35, 40]&#160;]
|-  
|-  
| 35 || 0 || 6 || 5 || 4 || 3 ||
| 40 || 3rd || of 6 || [&#160;[1, 7, 14, 17, 22], [2, 6, 13, 28, 34], [3, 10, 11, 27, 40], [4, 8, 12, 20, 33], [5, 9, 16, 25, 39], [15, 21, 29, 31, 38], [18, 24, 26, 35, 36], [19, 23, 30, 32, 37]&#160;]
|-  
|-  
| 45 || 0 || 0 || 0 || 0 || 0 || 3 sessions
| 40 || 4th || of 6 || [&#160;[1, 8, 15, 19, 35], [2, 16, 22, 27, 33], [3, 9, 12, 24, 38], [4, 6, 30, 31, 36], [5, 14, 23, 34, 40], [7, 11, 20, 28, 39], [10, 17, 21, 26, 32], [13, 18, 25, 29, 37]&#160;]
|-  
|-  
| 45 || 0 || 1 || 2 || 3 || 4 ||
| 40 || 5th || of 6 || [&#160;[1, 9, 13, 31, 40], [2, 15, 20, 24, 32], [3, 14, 21, 33, 39], [4, 16, 26, 34, 37], [5, 7, 27, 35, 38], [6, 12, 17, 23, 29], [8, 11, 18, 22, 30], [10, 19, 25, 28, 36]&#160;]
|-  
|-  
| 45 || 0 || 2 || 4 || 6 || 8 ||  
| 40 || 6th || of 6 || [&#160;[1, 10, 12, 18, 39], [2, 9, 11, 23, 35], [3, 6, 15, 25, 26], [4, 13, 19, 22, 38], [5, 17, 24, 30, 33], [7, 16, 29, 32, 40], [8, 14, 28, 31, 37], [20, 21, 27, 34, 36]&#160;]
|-  style="background:#ccf"
| 45 || 1st || of 6 || [&#160;[20, 24, 32, 39, 43], [6, 29, 31, 33, 42], [7, 11, 17, 34, 45], [12, 14, 15, 18, 25], [2, 5, 8, 9, 19], [1, 4, 16, 27, 28], [3, 13, 23, 30, 35], [10, 22, 26, 38, 40], [21, 36, 37, 41, 44]&#160;]
|-  style="background:#ccf"
| 45 || 2nd || of 6 || [&#160;[5, 17, 28, 40, 41], [11, 13, 22, 24, 27], [8, 25, 32, 33, 37], [9, 10, 20, 30, 45], [2, 12, 16, 29, 44], [3, 15, 36, 42, 43], [4, 6, 18, 23, 26], [1, 14, 19, 21, 34], [7, 31, 35, 38, 39]&#160;]
|-  style="background:#ccf"
| 45 || 3rd || of 6 || [&#160;[4, 21, 25, 31, 40], [13, 16, 41, 43, 45], [2, 20, 22, 33, 34], [9, 14, 23, 27, 38], [17, 18, 19, 29, 32], [7, 8, 12, 30, 36], [1, 15, 26, 37, 39], [3, 5, 6, 24, 44], [10, 11, 28, 35, 42]&#160;]
|-  style="background:#ccf"
| 45 || 4th || of 6 || [&#160;[19, 27, 40, 42, 45], [4, 12, 17, 22, 39], [8, 13, 20, 26, 44], [18, 24, 30, 37, 38], [7, 23, 28, 29, 43], [6, 11, 15, 21, 32], [3, 9, 16, 25, 34], [1, 5, 33, 35, 36], [2, 10, 14, 31, 41]&#160;]
|-  style="background:#ccf"
| 45 || 5th || of 6 || [&#160;[4, 7, 9, 15, 41], [11, 25, 26, 29, 36], [8, 16, 18, 39, 40], [27, 30, 31, 34, 44], [14, 22, 35, 37, 43], [5, 12, 13, 21, 42], [6, 19, 20, 28, 38], [1, 2, 3, 32, 45], [10, 17, 23, 24, 33]&#160;]
|-  style="background:#ccf"
| 45 || 6th || of 6 || [&#160;[4, 8, 24, 29, 45], [6, 13, 14, 36, 40], [15, 16, 19, 22, 23], [5, 7, 10, 18, 27], [3, 17, 20, 31, 37], [1, 11, 38, 43, 44], [9, 21, 26, 28, 33], [12, 32, 34, 35, 41], [2, 25, 30, 39, 42]&#160;]
|-  
|-  
| 50 || 0 || 0 || 0 || 0 || 0 || 5/6 sessions (40P-50P)
| 50 || all || of 5 || Dutch cycles work for 5 sessions only.
|-  
|-  
| 50 || 0 || 1 || 2 || 3 || 4 ||
| 50 || 1st || of 7 || [&#160;[5, 16, 18, 25, 29], [1, 7, 27, 42, 46], [4, 6, 15, 21, 41], [3, 11, 26, 30, 32], [9, 19, 24, 37, 38], [8, 13, 33, 43, 45], [35, 40, 47, 48, 49], [17, 23, 31, 34, 50], [2, 12, 14, 20, 36], [10, 22, 28, 39, 44]&#160;]
|-  
|-  
| 50 || 0 || 2 || 4 || 6 || 8 ||
| 50 || 1st || of 7 || [&#160;[8, 20, 40, 42, 44], [15, 16, 17, 22, 47], [1, 10, 23, 29, 38], [3, 6, 28, 36, 48], [2, 24, 27, 33, 39], [7, 11, 13, 21, 31], [4, 32, 35, 45, 46], [9, 14, 18, 34, 41], [25, 30, 37, 43, 50], [5, 12, 19, 26, 49]&#160;]
|-  
|-  
| 50 || 0 || 3 || 6 || 9 || 2 ||
| 50 || 1st || of 7 || [&#160;[19, 21, 29, 32, 47], [8, 24, 25, 28, 34], [3, 18, 31, 44, 46], [4, 14, 39, 43, 49], [2, 10, 15, 42, 45], [9, 11, 16, 20, 50], [5, 7, 23, 37, 40], [1, 13, 26, 41, 48], [22, 33, 35, 36, 38], [6, 12, 17, 27, 30]&#160;]
|-  
|-  
| 50 || 0 || 4 || 8 || 2 || 6 ||
| 50 || 1st || of 7 || [&#160;[10, 14, 16, 21, 26], [23, 25, 36, 39, 42], [9, 13, 32, 44, 49], [3, 5, 8, 22, 27], [4, 11, 17, 38, 40], [12, 24, 43, 46, 48], [2, 7, 19, 41, 50], [1, 18, 28, 30, 33], [6, 34, 37, 45, 47], [15, 20, 29, 31, 35]&#160;]
|-  
|-  
| 50 || 0 || 9 || 5 || 4 || X0 || (non-Dutch, one repeat for 10 pairs.)
| 50 || 1st || of 7 || [&#160;[13, 20, 22, 30, 34], [26, 27, 38, 45, 50], [1, 11, 36, 43, 44], [16, 32, 33, 37, 41], [4, 9, 12, 31, 47], [14, 28, 29, 40, 46], [5, 6, 24, 35, 42], [7, 15, 18, 39, 48], [2, 8, 21, 23, 49], [3, 10, 17, 19, 25]&#160;]
|-  
|-  
| 55 || 0 || 0 || 0 || 0 || 0 || 11 sessions (44P-55P)
| 50 || 1st || of 7 || [&#160;[6, 26, 31, 39, 40], [11, 14, 22, 37, 42], [2, 5, 9, 17, 48], [4, 24, 29, 44, 50], [15, 19, 34, 36, 46], [3, 7, 20, 33, 49], [10, 18, 27, 32, 43], [13, 16, 23, 28, 35], [1, 12, 21, 25, 45], [8, 30, 38, 41, 47]&#160;]
|-  
|-  
| 55 || 0 || 1 || 2 || 3 || 4 ||
| 50 || 1st || of 7 || [&#160;[11, 18, 23, 24, 47], [22, 31, 41, 45, 49], [3, 9, 21, 40, 43], [2, 6, 13, 38, 46], [14, 25, 27, 35, 44], [12, 15, 28, 32, 50], [4, 5, 10, 33, 34], [7, 8, 26, 29, 36], [1, 17, 20, 37, 39], [16, 19, 30, 42, 48]&#160;]
|-  style="background:#ccf"
| 55 || all || of 11 || Dutch cycles. {44P-55P}
|-  
|-  
| 55 || 0 || 2 || 4 || 6 || 8 ||  
| 60 || all || of 6 || G(25+35) or G(30+30).
|-  style="background:#ccf"
| 65 || all || of 13 || Dutch cycles. {52P-65P}
|-  
|-  
| 55 || 0 || 3 || 6 || 9 || 1 ||  
| 70 || all || of 7 || G(35+35).
|-  style="background:#ccf"
| 75 || all || of 6 || G(35+40), G(30+45) {both 60P-75P}; G(25+25+25) {72P-75P}.
|-  
|-  
| 55 || 0 || 4 || 8 || 1 || 5 ||  
| {5*prime} || all || of {prime} || Dutch cycles. {{4*prime}P-{5*prime}P}; prime>=5. (25 is the special case, where it gets the sixth extra session by virtue of being 25 = 5^2.)
|-
| 55 || 0 || 5 || 10 || 4 || 9 ||
|-  
| 55 || 0 || 6 || 1 || 7 || 2 ||
|-
| 55 || 0 || 7 || 3 || 10 || 6 ||
|-
| 55 || 0 || 8 || 5 || 2 || 10 ||
|-
| 55 || 0 || 9 || 7 || 5 || 3 ||
|-
| 55 || 0 || 10 || 9 || 8 || 7 ||
|}
|}


Line 141: Line 241:




==== 15-player season ====
==== X-player season ====
 
 
{| class="wikitable"
{| class="wikitable"
|-
|-
! Session and Table !! Player 1 !! Player 2 !! Player 3 !! Player 4 !! Player 5 !! Deadline
! Min. Players !! Max Players !! Mechanism !! Comment
|- style="background:#ccf"
|-  
| Session 1 - Table 1 (+1) || P1 || P2 || P3 || P4 || P5 || Saturday 2015-04-04 23:59 UTC
| 10 || 10 || 5P randomize || Repeats will forcibly occur.
|- style="background:#ccf"
|-  
| Session 1 - Table 2 || P6 || P7 || P8 || P9 || P10 || Saturday 2015-04-04 23:59 UTC
| 11 || 12 || Not possible || With thirds promotion, 10 in top group, 18 in next, top group gets 13+.
|- style="background:#ccf"
|-  
| Session 1 - Table 3 || P11 || P12 || P13 || P14 || P15 || Saturday 2015-04-04 23:59 UTC
| 13 || 15 || Mix and randomize || Repeats will forcibly occur.
|-
|-  
| Session 2 - Table 1 (+3) || P1 || P4 || P7 || P10 || P13 || Saturday 2015-04-18 23:59 UTC
| 16 || 16 || 4P perfect + Siberian or Random (6th) || 5 sessions covered, 6th pairing could be done randomly at the start or drawn later (at a pre-announced time) relative to current ranking.
|-
|-  
| Session 2 - Table 2 || P2 || P5 || P8 || P11 || P14 || Saturday 2015-04-18 23:59 UTC
| 17 || 23 || Mix and randomize || Repeats will most likely occur.
|-
|-  
| Session 2 - Table 3 || P3 || P6 || P9 || P12 || P15 || Saturday 2015-04-18 23:59 UTC
| 24 || 25 || 5P perfect || For 24 players, ignore 25th player in grid, those sessions are 4P
|- style="background:#ccf"
|-  
| Session 3 - Table 1 (+8) || P1 || P9 || P2 || P10 || P3 || Saturday 2015-05-02 23:59 UTC
| 26 || 27 || Mix and randomize || {TBD.}
|- style="background:#ccf"
|-  
| Session 3 - Table 2 || P11 || P4 || P12 || P5 || P13 || Saturday 2015-05-02 23:59 UTC
| 28 || 28 || Use 4P ||
|- style="background:#ccf"
|-  
| Session 3 - Table 3 || P6 || P14 || P7 || P15 || P8 || Saturday 2015-05-02 23:59 UTC
| 29 || 30 || Use 5P || For 29 players, ignore 30th player in grid, those sessions are 4P
|-
|-  
| Session 4 - Table 1 (+11) || P1 || P12 || P8 || P4 || P15 || Saturday 2015-05-16 23:59 UTC
| 31 || 35 || Use 5P grid for 35 ||  
|-
|-  
| Session 4 - Table 2 || P11 || P7 || P3 || P14 || P10 || Saturday 2015-05-16 23:59 UTC
| 36 || 40 || Use 5P grid for 40 || Grid can ignore last 5 players.
|-
| Session 4 - Table 3 || P6 || P2 || P13 || P9 || P5 || Saturday 2015-05-16 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 1 (+5) || P1 || P6 || P11 || P5 || P10 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 2 || P15 || P4 || P9 || P14 || P3 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| 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.
{| class="wikitable"
|-
! Session and Table !! Player 1 !! Player 2 !! Player 3 !! Player 4 !! Player 5 !! Deadline
|- style="background:#ccf"
| Session 1 - Table 1 (+1) || P1 || P2 || P3 || P4 || P5 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 2 || P6 || P7 || P8 || P9 || P10 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 3 || P11 || P12 || P13 || P14 || P15 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 4 || P16 || P17 || P18 || P19 || P20 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| 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
|- style="background:#ccf"
| Session 3 - Table 1 (+7) || P1 || P17 || P8 || P24 || P15 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 2 || P6 || P22 || P13 || P4 || P20 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 3 || P11 || P2 || P18 || P9 || P25 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 4 || P16 || P7 || P23 || P14 || P5 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| 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
|- style="background:#ccf"
| Session 5 - Table 1 (+6) || P1 || P7 || P13 || P19 || P25 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 2 || P6 || P12 || P18 || P24 || P5 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 3 || P11 || P17 || P23 || P4 || P10 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 4 || P16 || P22 || P3 || P9 || P15 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| 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.
{| class="wikitable"
|-
! Session and Table !! Player 1 !! Player 2 !! Player 3 !! Player 4 !! Player 5 !! Deadline
|- style="background:#ccf"
| Session 1 - Table 1 (+8) || P1 || P9 || P17 || P25 || P33 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 2 || P2 || P10 || P18 || P26 || P34 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 3 || P3 || P11 || P19 || P27 || P35 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 4 || P4 || P12 || P20 || P28 || P36 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 5 || P5 || P13 || P21 || P29 || P37 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 6 || P6 || P14 || P22 || P30 || P38 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| Session 1 - Table 7 || P7 || P15 || P23 || P31 || P39 || Saturday 2015-04-04 23:59 UTC
|- style="background:#ccf"
| 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
|- style="background:#ccf"
| Session 3 - Table 1 || P1 || P11 || P21 || P31 || P36 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 2 || P2 || P12 || P22 || P32 || P35 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 3 || P3 || P9 || P23 || P29 || P34 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 4 || P4 || P10 || P24 || P30 || P33 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 5 || P5 || P15 || P17 || P27 || P40 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 6 || P6 || P16 || P18 || P28 || P39 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| Session 3 - Table 7 || P7 || P13 || P19 || P25 || P38 || Saturday 2015-05-02 23:59 UTC
|- style="background:#ccf"
| 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
|- style="background:#ccf"
| Session 5 - Table 1 || P1 || P13 || P20 || P32 || P39 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 2 || P2 || P14 || P19 || P31 || P40 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 3 || P3 || P15 || P18 || P30 || P37 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 4 || P4 || P16 || P17 || P29 || P38 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 5 || P5 || P9 || P24 || P28 || P35 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 6 || P6 || P10 || P23 || P27 || P36 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| Session 5 - Table 7 || P7 || P11 || P22 || P26 || P33 || Saturday 2015-05-30 23:59 UTC
|- style="background:#ccf"
| 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
|- style="background:#666"
| Session 7 - Table 1 || P1 || P15 || P24 || P26 || P38 || (not played)
|- style="background:#666"
| Session 7 - Table 2 || P2 || P16 || P23 || P25 || P37 || (not played)
|- style="background:#666"
| Session 7 - Table 3 || P3 || P13 || P22 || P28 || P40 || (not played)
|- style="background:#666"
| Session 7 - Table 4 || P4 || P14 || P21 || P27 || P39 || (not played)
|- style="background:#666"
| Session 7 - Table 5 || P5 || P11 || P20 || P30 || P34 || (not played)
|- style="background:#666"
| Session 7 - Table 6 || P6 || P12 || P19 || P29 || P33 || (not played)
|- style="background:#666"
| Session 7 - Table 7 || P7 || P9 || P18 || P32 || P36 || (not played)
|- style="background:#666"
| Session 7 - Table 8 || P8 || P10 || P17 || P31 || P35 || (not played)
|- style="background:#666"
| Session 8 - Table 1 || P1 || P16 || P22 || P27 || P34 || (not played)
|- style="background:#666"
| Session 8 - Table 2 || P2 || P15 || P21 || P28 || P33 || (not played)
|- style="background:#666"
| Session 8 - Table 3 || P3 || P14 || P24 || P25 || P36 || (not played)
|- style="background:#666"
| Session 8 - Table 4 || P4 || P13 || P23 || P26 || P35 || (not played)
|- style="background:#666"
| Session 8 - Table 5 || P5 || P12 || P18 || P31 || P38 || (not played)
|- style="background:#666"
| Session 8 - Table 6 || P6 || P11 || P17 || P32 || P37 || (not played)
|- style="background:#666"
| Session 8 - Table 7 || P7 || P10 || P20 || P29 || P40 || (not played)
|- style="background:#666"
| Session 8 - Table 8 || P8 || P9 || P19 || P30 || P39 || (not played)
|}


==== WAML-relevant summary ====
==== WAML-relevant summary ====
If we can get player blocks of 25♥, 35, 40, 55 (even 65) or sums of their multiples (60♥ = 25♥ + 35; 65 = 25♥ + 40; 70 = 35 + 35; 75yes; 80yes; 85♥♥yes; 90yes; 95yes; 100♥yes; ...) or a number significantly close, 80% + 4 per heart, then we can cover the whole span from <!-- [28, 35], [32,40], [44, 55], [48, 50], [52, 60], [52, 65], [56, 70]... --> 48+ if we stop forming asymmetrical groups after 4 are made. <!-- up to 55 is covered by waiting, 56 is doable with 4 groups of 4 out of 13 [56, 65] ... two 35s [60, 70] ... 35 and 40 for [64, 75] ... 40s [68, 80] --> A fifth group can be drawn at 57 with no dificulty.
If we can get player blocks of 25♥, 35, 40, 55 (even 65) or sums of their multiples (60♥ = 25♥ + 35; 65 = 25♥ + 40; 70 = 35 + 35; 75yes; 80yes; 85♥♥yes; 90yes; 95yes; 100♥yes; ...) or a number significantly close, 80% + 4 per heart, then we can cover the whole span from <!-- [28, 35], [32,40], [44, 55], [48, 50], [52, 60], [52, 65], [56, 70]... --> 48+ if we stop forming asymmetrical groups after 4 are made. <!-- up to 55 is covered by waiting, 56 is doable with 4 groups of 4 out of 13 [56, 65] ... two 35s [60, 70] ... 35 and 40 for [64, 75] ... 40s [68, 80] --> A fifth group can be drawn at 57 with no dificulty.

Revision as of 18:21, 31 January 2015

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 0 {mathematically speaking so the formula works}, 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 3. Heavily composite numbers of players will lead to collisions when tables = h * factor. The 2014 WRC had 120 players over 30 tables: as 120 (as well as 30) is divisible by 3, it would lead to a collision when h = 10, making the 11th hanchan place naively two players who met before across all 30 tables.

Notes: Mathpuzzle.com has a few non-Dutch solutions for the "Social Golfer Problem". -- As does the web archive, from Warwick Harvey

Blocks of 4
Players Block Of Raw data
16 1 5 ABCD EFGH IJKL MNOP
16 2 5 AEIM BFJN CGKO DHLP
16 3 5 AFKP BELO CHIN DGJM
16 4 5 AGLN BHKM CEJP DFIO
16 5 5 AHJO BGIP CFLM DEKN.
20 1 5 ABCD EFGH IJKL MNOP QRST
20 2 5 AEIM BFJQ CGNR DKOS HLPT
20 3 5 AFOT BELR CIPS DGJM HKNQ
20 4 5 AJPR BHMS CEKT DFIN GLOQ
20 5 5 ALNS BGIT CHJO DEPQ FKMR.
24 1 7 ABKU IJSE QRCM DGFX HLNO PTVW
24 2 7 ACLV IKTF QSDN EHGR BMOP JUWX
24 3 7 ADMW ILUG QTEO FBHS CJNP KRVX
24 4 7 AENX IMVH QUFP GCBT DJKO LRSW
24 5 7 AFOR INWB QVGJ HDCU EKLP MSTX
24 6 7 AEPS IOXC QWHK BEDV FJLM NRTU
24 7 7 AHJT IPRD QXBL CFEW GKMN OSUV.
28 1 9 ABCD EFGH IJKL MNab cdef ghij klmn
28 1 9 AEgk BFMc Ndhl GIem HJai CKbn DLfj
28 1 9 AFjn BEae bfim HKcl GLMh CINk DJdg
28 1 9 AIci BJNn EKMj FLdm begl CGaf DHhk
28 1 9 AGbd BHgm ELNi achn FKfk CJej DIMl
28 1 9 AKeh BLbk FIag EJfl Ncjm CHMd DGin
28 1 9 AHNf BGjl FJbh Meik EIdn CLcg DKam
28 1 9 ALal BKdi GJck Mfgn HIbj CEhm DFNe
28 1 9 AJMm BIfh CFil DEbc GKNg HLen adjk.
32 1 10 ABCD EFGH IJKL MNPO abcd efgh ijkl mnop
32 2 10 AEIm BgcH DFKp kPdf Maei ChbG jNoL lJnO
32 3 10 Alof Bhkm DNGi FaLO MbHK CPeJ Ejcp gIdn
32 4 10 AciL BjdK DkbJ FgoP fGOp CIla EhMn NeHm
32 5 10 AgNK BElP DIoH hdiO beLp Cjfm FMcJ kaGn
32 6 10 AhJp BFin DcmO INbf leGK CMod EgkL jPaH
32 7 10 AFde BoGJ DEaf hlNc PiKm CHLn gjbO IkMp
32 8 10 APbn BNap DglM koce fHiJ CEKO FhIj dGLm
32 9 10 AjMG BIeO DhPL gaJm cfKn CFkN Eobi ldHp
32 10 10 AkHO BMfL Djen Flbm IPcG Cgip ENdJ hoaK.
36 1 8 [ [1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16], [17, 18, 19, 20], [21, 22, 23, 24], [25, 26, 27, 28], [29, 30, 31, 32], [33, 34, 35, 36] ]
36 2 8 [ [1, 5, 9, 13], [2, 6, 10, 17], [3, 7, 11, 21], [4, 8, 12, 25], [14, 18, 22, 29], [15, 19, 23, 33], [16, 26, 30, 34], [20, 27, 31, 35], [24, 28, 32, 36] ]
36 3 8 [ [1, 6, 11, 25], [2, 7, 9, 16], [3, 5, 10, 20], [4, 14, 21, 33], [8, 15, 18, 32], [12, 13, 17, 22], [19, 27, 30, 36], [23, 28, 31, 34], [24, 26, 29, 35] ]
36 4 8 [ [1, 10, 18, 21], [2, 8, 13, 29], [3, 14, 31, 36], [4, 9, 20, 23], [5, 15, 17, 28], [6, 12, 16, 19], [7, 26, 32, 33], [11, 24, 27, 34], [22, 25, 30, 35] ]
36 5 8 [ [1, 14, 24, 30], [2, 12, 18, 34], [3, 9, 17, 32], [4, 6, 28, 35], [5, 21, 29, 36], [7, 10, 19, 25], [8, 16, 23, 27], [11, 13, 20, 33], [15, 22, 26, 31] ]
36 6 8 [ [1, 12, 23, 26], [2, 11, 22, 36], [3, 6, 13, 24], [4, 10, 16, 32], [5, 18, 27, 33], [7, 20, 28, 30], [8, 14, 19, 35], [9, 15, 29, 34], [17, 21, 25, 31] ]
36 7 8 [ [1, 19, 28, 29], [2, 5, 23, 30], [3, 12, 15, 27], [4, 11, 18, 31], [6, 9, 26, 36], [7, 14, 17, 34], [8, 10, 22, 33], [13, 21, 32, 35], [16, 20, 24, 25] ]
36 8 8 [ [1, 7, 15, 35], [2, 14, 25, 32], [3, 16, 18, 28], [4, 17, 27, 29], [5, 11, 19, 26], [6, 20, 22, 34], [8, 9, 21, 30], [10, 13, 23, 36], [12, 24, 31, 33] ]
40 (10*4) Unkn. 5/6ND
44 (11*4) Dutch 11
52 (13*4) Dutch 13
68 (17*4) Dutch 17
76 (19*4) Dutch 19

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. A way around it would be to have two "separable" leagues running up to 35 players per side, thus covering 66 and 67.

Players Round Of Match-ups
25 1st of 6 [ [1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25] ]
25 2nd of 6 [ [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24], [5, 10, 15, 20, 25] ]
25 3rd of 6 [ [1, 7, 13, 19, 25], [2, 8, 14, 20, 21], [3, 9, 15, 16, 22], [4, 10, 11, 17, 23], [5, 6, 12, 18, 24] ]
25 4th of 6 [ [1, 8, 15, 17, 24], [2, 9, 11, 18, 25], [3, 10, 12, 19, 21], [4, 6, 13, 20, 22], [5, 7, 14, 16, 23] ]
25 5th of 6 [ [1, 9, 12, 20, 23], [2, 10, 13, 16, 24], [3, 6, 14, 17, 25], [4, 7, 15, 18, 21], [5, 8, 11, 19, 22] ]
25 6th of 6 [ [1, 10, 14, 18, 22], [2, 6, 15, 19, 23], [3, 7, 11, 20, 24], [4, 8, 12, 16, 25], [5, 9, 13, 17, 21] ]
30 1st of 6 [ [1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25], [26, 27, 28, 29, 30] ]
30 2nd of 6 [ [1, 6, 11, 16, 21], [2, 7, 13, 19, 30], [3, 8, 14, 24, 29], [4, 9, 20, 25, 28], [5, 15, 18, 23, 27], [10, 12, 17, 22, 26] ]
30 3rd of 6 [ [1, 8, 13, 20, 22], [2, 6, 12, 23, 28], [3, 9, 11, 18, 30], [4, 15, 16, 24, 26], [5, 7, 17, 25, 29], [10, 14, 19, 21, 27] ]
30 4th of 6 [ [1, 7, 18, 24, 28], [2, 10, 11, 20, 29], [3, 6, 15, 19, 22], [4, 14, 17, 23, 30], [5, 9, 13, 21, 26], [8, 12, 16, 25, 27] ]
30 5th of 6 [ [1, 10, 15, 25, 30], [2, 9, 17, 24, 27], [3, 7, 12, 20, 21], [4, 6, 13, 18, 29], [5, 14, 16, 22, 28], [8, 11, 19, 23, 26] ]
30 6th of 6 [ [1, 9, 12, 19, 29], [5, 6, 20, 24, 30], [4, 7, 11, 22, 27], [8, 15, 17, 21, 28], [3, 10, 13, 16, 23], [2, 14, 18, 25, 26] ]
35 all of 7 Dutch cycles. {28P-35P}
40 1st of 6 [ [1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25], [26, 27, 28, 29, 30], [31, 32, 33, 34, 35], [36, 37, 38, 39, 40] ]
40 2nd of 6 [ [1, 6, 11, 16, 21], [2, 7, 12, 26, 31], [3, 8, 13, 17, 36], [4, 9, 14, 27, 32], [5, 10, 15, 22, 37], [18, 23, 28, 33, 38], [19, 24, 29, 34, 39], [20, 25, 30, 35, 40] ]
40 3rd of 6 [ [1, 7, 14, 17, 22], [2, 6, 13, 28, 34], [3, 10, 11, 27, 40], [4, 8, 12, 20, 33], [5, 9, 16, 25, 39], [15, 21, 29, 31, 38], [18, 24, 26, 35, 36], [19, 23, 30, 32, 37] ]
40 4th of 6 [ [1, 8, 15, 19, 35], [2, 16, 22, 27, 33], [3, 9, 12, 24, 38], [4, 6, 30, 31, 36], [5, 14, 23, 34, 40], [7, 11, 20, 28, 39], [10, 17, 21, 26, 32], [13, 18, 25, 29, 37] ]
40 5th of 6 [ [1, 9, 13, 31, 40], [2, 15, 20, 24, 32], [3, 14, 21, 33, 39], [4, 16, 26, 34, 37], [5, 7, 27, 35, 38], [6, 12, 17, 23, 29], [8, 11, 18, 22, 30], [10, 19, 25, 28, 36] ]
40 6th of 6 [ [1, 10, 12, 18, 39], [2, 9, 11, 23, 35], [3, 6, 15, 25, 26], [4, 13, 19, 22, 38], [5, 17, 24, 30, 33], [7, 16, 29, 32, 40], [8, 14, 28, 31, 37], [20, 21, 27, 34, 36] ]
45 1st of 6 [ [20, 24, 32, 39, 43], [6, 29, 31, 33, 42], [7, 11, 17, 34, 45], [12, 14, 15, 18, 25], [2, 5, 8, 9, 19], [1, 4, 16, 27, 28], [3, 13, 23, 30, 35], [10, 22, 26, 38, 40], [21, 36, 37, 41, 44] ]
45 2nd of 6 [ [5, 17, 28, 40, 41], [11, 13, 22, 24, 27], [8, 25, 32, 33, 37], [9, 10, 20, 30, 45], [2, 12, 16, 29, 44], [3, 15, 36, 42, 43], [4, 6, 18, 23, 26], [1, 14, 19, 21, 34], [7, 31, 35, 38, 39] ]
45 3rd of 6 [ [4, 21, 25, 31, 40], [13, 16, 41, 43, 45], [2, 20, 22, 33, 34], [9, 14, 23, 27, 38], [17, 18, 19, 29, 32], [7, 8, 12, 30, 36], [1, 15, 26, 37, 39], [3, 5, 6, 24, 44], [10, 11, 28, 35, 42] ]
45 4th of 6 [ [19, 27, 40, 42, 45], [4, 12, 17, 22, 39], [8, 13, 20, 26, 44], [18, 24, 30, 37, 38], [7, 23, 28, 29, 43], [6, 11, 15, 21, 32], [3, 9, 16, 25, 34], [1, 5, 33, 35, 36], [2, 10, 14, 31, 41] ]
45 5th of 6 [ [4, 7, 9, 15, 41], [11, 25, 26, 29, 36], [8, 16, 18, 39, 40], [27, 30, 31, 34, 44], [14, 22, 35, 37, 43], [5, 12, 13, 21, 42], [6, 19, 20, 28, 38], [1, 2, 3, 32, 45], [10, 17, 23, 24, 33] ]
45 6th of 6 [ [4, 8, 24, 29, 45], [6, 13, 14, 36, 40], [15, 16, 19, 22, 23], [5, 7, 10, 18, 27], [3, 17, 20, 31, 37], [1, 11, 38, 43, 44], [9, 21, 26, 28, 33], [12, 32, 34, 35, 41], [2, 25, 30, 39, 42] ]
50 all of 5 Dutch cycles work for 5 sessions only.
50 1st of 7 [ [5, 16, 18, 25, 29], [1, 7, 27, 42, 46], [4, 6, 15, 21, 41], [3, 11, 26, 30, 32], [9, 19, 24, 37, 38], [8, 13, 33, 43, 45], [35, 40, 47, 48, 49], [17, 23, 31, 34, 50], [2, 12, 14, 20, 36], [10, 22, 28, 39, 44] ]
50 1st of 7 [ [8, 20, 40, 42, 44], [15, 16, 17, 22, 47], [1, 10, 23, 29, 38], [3, 6, 28, 36, 48], [2, 24, 27, 33, 39], [7, 11, 13, 21, 31], [4, 32, 35, 45, 46], [9, 14, 18, 34, 41], [25, 30, 37, 43, 50], [5, 12, 19, 26, 49] ]
50 1st of 7 [ [19, 21, 29, 32, 47], [8, 24, 25, 28, 34], [3, 18, 31, 44, 46], [4, 14, 39, 43, 49], [2, 10, 15, 42, 45], [9, 11, 16, 20, 50], [5, 7, 23, 37, 40], [1, 13, 26, 41, 48], [22, 33, 35, 36, 38], [6, 12, 17, 27, 30] ]
50 1st of 7 [ [10, 14, 16, 21, 26], [23, 25, 36, 39, 42], [9, 13, 32, 44, 49], [3, 5, 8, 22, 27], [4, 11, 17, 38, 40], [12, 24, 43, 46, 48], [2, 7, 19, 41, 50], [1, 18, 28, 30, 33], [6, 34, 37, 45, 47], [15, 20, 29, 31, 35] ]
50 1st of 7 [ [13, 20, 22, 30, 34], [26, 27, 38, 45, 50], [1, 11, 36, 43, 44], [16, 32, 33, 37, 41], [4, 9, 12, 31, 47], [14, 28, 29, 40, 46], [5, 6, 24, 35, 42], [7, 15, 18, 39, 48], [2, 8, 21, 23, 49], [3, 10, 17, 19, 25] ]
50 1st of 7 [ [6, 26, 31, 39, 40], [11, 14, 22, 37, 42], [2, 5, 9, 17, 48], [4, 24, 29, 44, 50], [15, 19, 34, 36, 46], [3, 7, 20, 33, 49], [10, 18, 27, 32, 43], [13, 16, 23, 28, 35], [1, 12, 21, 25, 45], [8, 30, 38, 41, 47] ]
50 1st of 7 [ [11, 18, 23, 24, 47], [22, 31, 41, 45, 49], [3, 9, 21, 40, 43], [2, 6, 13, 38, 46], [14, 25, 27, 35, 44], [12, 15, 28, 32, 50], [4, 5, 10, 33, 34], [7, 8, 26, 29, 36], [1, 17, 20, 37, 39], [16, 19, 30, 42, 48] ]
55 all of 11 Dutch cycles. {44P-55P}
60 all of 6 G(25+35) or G(30+30).
65 all of 13 Dutch cycles. {52P-65P}
70 all of 7 G(35+35).
75 all of 6 G(35+40), G(30+45) {both 60P-75P}; G(25+25+25) {72P-75P}.
{5*prime} all of {prime} Dutch cycles. {{4*prime}P-{5*prime}P}; prime>=5. (25 is the special case, where it gets the sixth extra session by virtue of being 25 = 5^2.)

Trying to group a league with a variable amount of entries poses its own challenges. In order to start a league asymmetrically, it is possible to start seeding groups from 17 members and every 5 onward. However, this alone will guarantee that groups can be made but not that no rematches will occur. For an ideal solution like 35, in theory, 7 groups of 4 can use the same solution, however, with 3 groups pre-formed at 27 members, the minimum requirement would be to end up with 31 players [31, 35]. Likewise, for the 55 solution, overloading groups of 5 may render the ability to make 11 groups but with more groups of 4. If [48, 50] players are present, two separate blocks of 24/25 are usable for 6-session seasons.

Min Groups of 5 Groups of 4 Spare Max Comment
27 3 0 12 35 <=30 : 6 groups, >=31 : 7 groups.
32 4 0 12 40 <=35 : 7 groups, >=36 : 8 groups.
37 5 0 12 45 <=40 : 8 groups, >=41 : 9 groups.
42 5 0 17 50 <=45 : 9 groups, >=46 : 10 groups.
48 7 0 12 50 <=50 : 10 groups in 25s, >=51 : 11 groups in 55.
52 7 0 17 60 <=55 : 11 groups in 55, >=56 : 12 groups in 25 and 35.
57 8 0 17 60 <=60 : 12 groups in 25 and 35, >=61 : 14 groups in 35s (5*8 + 4*4 + 5*1).
62 10 0 12 60 <=65 : 13 groups, >=66 : 14 groups.
67 11 0 12 60 <=70 : 14 groups, >=71 : 15 groups.
72 12 0 12 60 <=75 : 15 groups, >=76 : 16 groups.


X-player season

Min. Players Max Players Mechanism Comment
10 10 5P randomize Repeats will forcibly occur.
11 12 Not possible With thirds promotion, 10 in top group, 18 in next, top group gets 13+.
13 15 Mix and randomize Repeats will forcibly occur.
16 16 4P perfect + Siberian or Random (6th) 5 sessions covered, 6th pairing could be done randomly at the start or drawn later (at a pre-announced time) relative to current ranking.
17 23 Mix and randomize Repeats will most likely occur.
24 25 5P perfect For 24 players, ignore 25th player in grid, those sessions are 4P
26 27 Mix and randomize {TBD.}
28 28 Use 4P
29 30 Use 5P For 29 players, ignore 30th player in grid, those sessions are 4P
31 35 Use 5P grid for 35
36 40 Use 5P grid for 40 Grid can ignore last 5 players.


WAML-relevant summary

If we can get player blocks of 25♥, 35, 40, 55 (even 65) or sums of their multiples (60♥ = 25♥ + 35; 65 = 25♥ + 40; 70 = 35 + 35; 75yes; 80yes; 85♥♥yes; 90yes; 95yes; 100♥yes; ...) or a number significantly close, 80% + 4 per heart, then we can cover the whole span from 48+ if we stop forming asymmetrical groups after 4 are made. A fifth group can be drawn at 57 with no dificulty.

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