Tournament blocks: Difference between revisions

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== 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 ==

Tournament blocks could be used to impose a fair distribution of winds, balancing out variables such as how many times has a player started as East/South/West/North, when players have a degree of control regarding such assignments. However, they have also been used to segregate players, guaranteeing a zero-opportunity between people for a variety of reasons (not wanting to play a spouse, a club member, or someone from the same country), effectively manipulating the random chance everyone has to a more skewed probablility of meeting certain choice players even before drawing in a manner more restrictive than simply knowing who drawed beforehand.

The other mathematically unobservable fact resides in the truth that mahjong tiles can develop scratches and other identifying marks. Simply rotating players by [x,x,x,x]:->[x,x+1,x+2,x+3] will leave a quarter of the table with a potential opportunity to learn and exploit the tile markings. Luckily, the easiest way to minimize that impact is to change the transformation posited above to [x,x,x,x]:->[x+1,x+2,x+3,x+4].

EMA tournaments have tried implementing in some of their events a "final round" where players are then re-ranked 1,2,3,4; 5,6,7,8; et cetera, enabling them to have 8 hanchan in an event where 24 or 48 people show up, as it is mathematically difficult or impossible to satisfy for more than 7. Otherwise, most of their tournament seating arrangements occur though Dutch cycles (transparent but with risks), or through software which may or may not be easily verifiable or reproducible (trusting the black box, security through obscurity). However, they have not statuated on a method for running tournaments. The two main software solutions have benefits but have significant drawbacks. Criticism for these kinds of points in participative tournaments is not generally a thing in Japan, as most tournaments there are knock-out events or small groups that accept duplication of pairings.

== Block repository ==

=== Tournament blocks ===

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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 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.

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 10, it would lead to a collision when h = 10, making the 11th hanchan place naively two players who met before across all 30 tables. It is also divisible by 5, but for larger events, a collision cannot occur if 1*factor, 2*factor or 3*factor does not equal or surpass the number of tables.

~~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". -- 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 =====

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| 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] ]

|- style="background:#ccf"

| 40 (10*4) || ~~Unkn.~~ || 5~~/6ND~~ ||

| 40 || all of || 5/6ND || Dutch cycles break after 5.

|-

| 40 || 1 || 9 || [ [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 || 2 || 9 || [ [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, 20, 24, 37], [26, 30, 34, 38], [27, 31, 35, 39], [28, 32, 36, 40] ]

|-

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

|-

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

|-

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

|-

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

|-

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

|-

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

|-

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

|-

| 44 (11*4) || Dutch || 11 ||

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| 68 (17*4) || Dutch || 17 ||

|-

| 76 (19*4) || Dutch || 19 ||

| (prime*4) || Dutch || prime ||

|}

=== 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~~.

Considering that league blocks contain 5 players, the counting mechanism has to be recalculated almost from scratch. These numbers are to satisfy 6-session events or seasons. For all Dutch cycles, as well as some non-Dutch SGP blocks can drop the last fifth in order to make groups of 4, expanding the solved ranges of players from as low as 80% to 100% of the maximal solutions. Considering all our solutions for 25+ are good for 6+ sessions, solutions for larger numbers can concatenate smaller groups with the minimum amount of sessions needed to make a larger group that satisfies that lower bound. The solution for 25 players in 6 sessions can drop one player, the solution for 40 present below can drop 5, although it may be possible that a solution allowing to drop 8 exists.

{| class="wikitable"

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==== 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.

== External links ==

* [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".

* [http://web.archive.org/web/20050407074608/http://www.icparc.ic.ac.uk/~wh/golf/solutions.html Warwick Harvey (web.archive.org)] had also published many solutions for groups with 10 or fewer groups and players of sizes in a simple to understand manner.

@@ Line 3: / Line 3: @@
 == 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 ==
+Tournament blocks could be used to impose a fair distribution of winds, balancing out variables such as how many times has a player started as East/South/West/North, when players have a degree of control regarding such assignments. However, they have also been used to segregate players, guaranteeing a zero-opportunity between people for a variety of reasons (not wanting to play a spouse, a club member, or someone from the same country), effectively manipulating the random chance everyone has to a more skewed probablility of meeting certain choice players even before drawing in a manner more restrictive than simply knowing who drawed beforehand.
+The other mathematically unobservable fact resides in the truth that mahjong tiles can develop scratches and other identifying marks. Simply rotating players by [x,x,x,x]:->[x,x+1,x+2,x+3] will leave a quarter of the table with a potential opportunity to learn and exploit the tile markings. Luckily, the easiest way to minimize that impact is to change the transformation posited above to [x,x,x,x]:->[x+1,x+2,x+3,x+4].
+EMA tournaments have tried implementing in some of their events a "final round" where players are then re-ranked 1,2,3,4; 5,6,7,8; et cetera, enabling them to have 8 hanchan in an event where 24 or 48 people show up, as it is mathematically difficult or impossible to satisfy for more than 7. Otherwise, most of their tournament seating arrangements occur though Dutch cycles (transparent but with risks), or through software which may or may not be easily verifiable or reproducible (trusting the black box, security through obscurity). However, they have not statuated on a method for running tournaments. The two main software solutions have benefits but have significant drawbacks. Criticism for these kinds of points in participative tournaments is not generally a thing in Japan, as most tournaments there are knock-out events or small groups that accept duplication of pairings.
 == Block repository ==
 === Tournament blocks ===
@@ Line 18: / Line 28: @@
 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.
+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 10, it would lead to a collision when h = 10, making the 11th hanchan place naively two players who met before across all 30 tables. It is also divisible by 5, but for larger events, a collision cannot occur if 1*factor, 2*factor or 3*factor does not equal or surpass the number of tables.
-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". -- 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 =====
@@ Line 116: / Line 123: @@
 | 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"
-| 40 (10*4) || Unkn. || 5/6ND ||
+| 40 || all of || 5/6ND || Dutch cycles break after 5.
+|-
+| 40 || 1 || 9 || [&#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;]
+|-
+| 40 || 2 || 9 || [&#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, 20, 24, 37], [26, 30, 34, 38], [27, 31, 35, 39], [28, 32, 36, 40]&#160;]
+|-
+| 40 || 3 || 9 || [&#160;[1, 6, 11, 25], [2, 7, 9, 29], [3, 5, 10, 15], [4, 13, 17, 24], [8, 14, 21, 34], [12, 16, 22, 39], [18, 23, 28, 37], [19, 26, 31, 36], [20, 32, 35, 38], [27, 30, 33, 40]&#160;]
+|-
+| 40 || 4 || 9 || [&#160;[1, 12, 15, 29], [2, 22, 28, 34], [3, 6, 30, 35], [4, 9, 14, 36], [5, 25, 33, 37], [7, 10, 24, 32], [8, 13, 19, 27], [11, 17, 23, 40], [16, 18, 31, 38], [20, 21, 26, 39]&#160;]
+|-
+| 40 || 5 || 9 || [&#160;[1, 8, 24, 28], [2, 5, 11, 39], [3, 19, 22, 25], [4, 15, 21, 38], [6, 14, 26, 32], [7, 18, 35, 40], [9, 20, 23, 27], [10, 30, 36, 37], [12, 13, 31, 34], [16, 17, 29, 33]&#160;]
+|-
+| 40 || 6 || 9 || [&#160;[1, 14, 17, 39], [2, 8, 35, 37], [3, 18, 27, 32], [4, 22, 31, 33], [5, 16, 23, 26], [6, 12, 21, 36], [7, 13, 28, 38], [9, 15, 24, 30], [10, 20, 25, 40], [11, 19, 29, 34]&#160;]
+|-
+| 40 || 7 || 9 || [&#160;[1, 22, 36, 38], [2, 23, 25, 32], [3, 26, 29, 37], [4, 5, 18, 34], [6, 13, 33, 39], [7, 15, 17, 27], [8, 11, 20, 30], [9, 21, 28, 31], [10, 16, 19, 35], [12, 14, 24, 40]&#160;]
+|-
+| 40 || 8 || 9 || [&#160;[1, 19, 21, 40], [2, 12, 18, 33], [3, 8, 17, 31], [4, 7, 16, 30], [5, 20, 28, 29], [6, 24, 27, 34], [9, 22, 26, 35], [10, 14, 23, 38], [11, 13, 32, 37], [15, 25, 36, 39]&#160;]
+|-
+| 40 || 9 || 9 || [&#160;[1, 7, 23, 31], [2, 13, 20, 36], [3, 16, 34, 40], [4, 10, 29, 39], [5, 12, 27, 38], [6, 15, 22, 37], [8, 9, 32, 33], [11, 18, 24, 26], [14, 19, 28, 30], [17, 21, 25, 35]&#160;]
 |-
 | 44 (11*4) || Dutch || 11 ||
@@ Line 124: / Line 149: @@
 | 68 (17*4) || Dutch || 17 ||
 |-
-| 76 (19*4) || Dutch || 19 ||
+| (prime*4) || Dutch || prime ||
 |}
 === 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.
+Considering that league blocks contain 5 players, the counting mechanism has to be recalculated almost from scratch. These numbers are to satisfy 6-session events or seasons. For all Dutch cycles, as well as some non-Dutch SGP blocks can drop the last fifth in order to make groups of 4, expanding the solved ranges of players from as low as 80% to 100% of the maximal solutions. Considering all our solutions for 25+ are good for 6+ sessions, solutions for larger numbers can concatenate smaller groups with the minimum amount of sessions needed to make a larger group that satisfies that lower bound. The solution for 25 players in 6 sessions can drop one player, the solution for 40 present below can drop 5, although it may be possible that a solution allowing to drop 8 exists.
 {| class="wikitable"
@@ Line 273: / Line 298: @@
 ==== 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.
+== External links ==
+* [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".
+* [http://web.archive.org/web/20050407074608/http://www.icparc.ic.ac.uk/~wh/golf/solutions.html Warwick Harvey (web.archive.org)] had also published many solutions for groups with 10 or fewer groups and players of sizes in a simple to understand manner.

Tournament blocks: Difference between revisions

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