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Combinatorial Symmetry Folds

Topic closed. 2 replies. Last post 10 years ago by JADELottery.

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JADELottery's avatar - MeAtWork 03.PNG
The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
Posted: November 10, 2006, 5:43 pm - IP Logged

Combinatorial Symmetry Folds

    Reference - Combinatorial Symmetry

Combinatorial symmetry folding is a process that removes the non-self symmetric combinations. Using the combinatorial symmetry function, here's a example of a C(9,6) folded combinatorial set. Table 1 shows the combination set and the symmetry set side by side. The blue highlighted combinations are the primary set and the red highlighted combinations are the secondary set; the green highlighted are the self symmetric set. There are two types of folds, the primary fold and the secondary fold, shown in Table 2.

                    Table 1
  Combination Set        Symmetry Set
      (Primary)            (Secondary)

1

2

3

4

5

6

 

4

5

6

7

8

9

1

2

3

4

5

7

 

3

5

6

7

8

9

1

2

3

4

5

8

 

2

5

6

7

8

9

1

2

3

4

5

9

 

1

5

6

7

8

9

1

2

3

4

6

7

 

3

4

6

7

8

9

1

2

3

4

6

8

 

2

4

6

7

8

9

1

2

3

4

6

9

 

1

4

6

7

8

9

1

2

3

4

7

8

 

2

3

6

7

8

9

1

2

3

4

7

9

 

1

3

6

7

8

9

1

2

3

4

8

9

 

1

2

6

7

8

9

1

2

3

5

6

7

 

3

4

5

7

8

9

1

2

3

5

6

8

 

2

4

5

7

8

9

1

2

3

5

6

9

 

1

4

5

7

8

9

1

2

3

5

7

8

 

2

3

5

7

8

9

1

2

3

5

7

9

 

1

3

5

7

8

9

1

2

3

5

8

9

 

1

2

5

7

8

9

1

2

3

6

7

8

 

2

3

4

7

8

9

1

2

3

6

7

9

 

1

3

4

7

8

9

1

2

3

6

8

9

 

1

2

4

7

8

9

1

2

3

7

8

9

 

1

2

3

7

8

9

1

2

4

5

6

7

 

3

4

5

6

8

9

1

2

4

5

6

8

 

2

4

5

6

8

9

1

2

4

5

6

9

 

1

4

5

6

8

9

1

2

4

5

7

8

 

2

3

5

6

8

9

1

2

4

5

7

9

 

1

3

5

6

8

9

1

2

4

5

8

9

 

1

2

5

6

8

9

1

2

4

6

7

8

 

2

3

4

6

8

9

1

2

4

6

7

9

 

1

3

4

6

8

9

1

2

4

6

8

9

 

1

2

4

6

8

9

1

2

4

7

8

9

 

1

2

3

6

8

9

1

2

5

6

7

8

 

2

3

4

5

8

9

1

2

5

6

7

9

 

1

3

4

5

8

9

1

2

5

6

8

9

 

1

2

4

5

8

9

1

2

5

7

8

9

 

1

2

3

5

8

9

1

2

6

7

8

9

 

1

2

3

4

8

9

1

3

4

5

6

7

 

3

4

5

6

7

9

1

3

4

5

6

8

 

2

4

5

6

7

9

1

3

4

5

6

9

 

1

4

5

6

7

9

1

3

4

5

7

8

 

2

3

5

6

7

9

1

3

4

5

7

9

 

1

3

5

6

7

9

1

3

4

5

8

9

 

1

2

5

6

7

9

1

3

4

6

7

8

 

2

3

4

6

7

9

1

3

4

6

7

9

 

1

3

4

6

7

9

1

3

4

6

8

9

 

1

2

4

6

7

9

1

3

4

7

8

9

 

1

2

3

6

7

9

1

3

5

6

7

8

 

2

3

4

5

7

9

1

3

5

6

7

9

 

1

3

4

5

7

9

1

3

5

6

8

9

 

1

2

4

5

7

9

1

3

5

7

8

9

 

1

2

3

5

7

9

1

3

6

7

8

9

 

1

2

3

4

7

9

1

4

5

6

7

8

 

2

3

4

5

6

9

1

4

5

6

7

9

 

1

3

4

5

6

9

1

4

5

6

8

9

 

1

2

4

5

6

9

1

4

5

7

8

9

 

1

2

3

5

6

9

1

4

6

7

8

9

 

1

2

3

4

6

9

1

5

6

7

8

9

 

1

2

3

4

5

9

2

3

4

5

6

7

 

3

4

5

6

7

8

2

3

4

5

6

8

 

2

4

5

6

7

8

2

3

4

5

6

9

 

1

4

5

6

7

8

2

3

4

5

7

8

 

2

3

5

6

7

8

2

3

4

5

7

9

 

1

3

5

6

7

8

2

3

4

5

8

9

 

1

2

5

6

7

8

2

3

4

6

7

8

 

2

3

4

6

7

8

2

3

4

6

7

9

 

1

3

4

6

7

8

2

3

4

6

8

9

 

1

2

4

6

7

8

2

3

4

7

8

9

 

1

2

3

6

7

8

2

3

5

6

7

8

 

2

3

4

5

7

8

2

3

5

6

7

9

 

1

3

4

5

7

8

2

3

5

6

8

9

 

1

2

4

5

7

8

2

3

5

7

8

9

 

1

2

3

5

7

8

2

3

6

7

8

9

 

1

2

3

4

7

8

2

4

5

6

7

8

 

2

3

4

5

6

8

2

4

5

6

7

9

 

1

3

4

5

6

8

2

4

5

6

8

9

 

1

2

4

5

6

8

2

4

5

7

8

9

 

1

2

3

5

6

8

2

4

6

7

8

9

 

1

2

3

4

6

8

2

5

6

7

8

9

 

1

2

3

4

5

8

3

4

5

6

7

8

 

2

3

4

5

6

7

3

4

5

6

7

9

 

1

3

4

5

6

7

3

4

5

6

8

9

 

1

2

4

5

6

7

3

4

5

7

8

9

 

1

2

3

5

6

7

3

4

6

7

8

9

 

1

2

3

4

6

7

3

5

6

7

8

9

 

1

2

3

4

5

7

4

5

6

7

8

9

 

1

2

3

4

5

6

 

                  Table 2
    Primary Fold        Secondary Fold

1

2

3

4

5

6

 

1

2

3

7

8

9

1

2

3

4

5

7

 

1

2

4

6

8

9

1

2

3

4

5

8

 

1

2

4

7

8

9

1

2

3

4

5

9

 

1

2

5

6

8

9

1

2

3

4

6

7

 

1

2

5

7

8

9

1

2

3

4

6

8

 

1

2

6

7

8

9

1

2

3

4

6

9

 

1

3

4

5

8

9

1

2

3

4

7

8

 

1

3

4

6

7

9

1

2

3

4

7

9

 

1

3

4

6

8

9

1

2

3

4

8

9

 

1

3

4

7

8

9

1

2

3

5

6

7

 

1

3

5

6

7

9

1

2

3

5

6

8

 

1

3

5

6

8

9

1

2

3

5

6

9

 

1

3

5

7

8

9

1

2

3

5

7

8

 

1

3

6

7

8

9

1

2

3

5

7

9

 

1

4

5

6

7

9

1

2

3

5

8

9

 

1

4

5

6

8

9

1

2

3

6

7

8

 

1

4

5

7

8

9

1

2

3

6

7

9

 

1

4

6

7

8

9

1

2

3

6

8

9

 

1

5

6

7

8

9

1

2

3

7

8

9

 

2

3

4

5

6

9

1

2

4

5

6

7

 

2

3

4

5

7

9

1

2

4

5

6

8

 

2

3

4

5

8

9

1

2

4

5

6

9

 

2

3

4

6

7

8

1

2

4

5

7

8

 

2

3

4

6

7

9

1

2

4

5

7

9

 

2

3

4

6

8

9

1

2

4

5

8

9

 

2

3

4

7

8

9

1

2

4

6

7

8

 

2

3

5

6

7

8

1

2

4

6

7

9

 

2

3

5

6

7

9

1

2

4

6

8

9

 

2

3

5

6

8

9

1

2

5

6

7

8

 

2

3

5

7

8

9

1

2

5

6

7

9

 

2

3

6

7

8

9

1

3

4

5

6

7

 

2

4

5

6

7

8

1

3

4

5

6

8

 

2

4

5

6

7

9

1

3

4

5

6

9

 

2

4

5

6

8

9

1

3

4

5

7

8

 

2

4

5

7

8

9

1

3

4

5

7

9

 

2

4

6

7

8

9

1

3

4

6

7

8

 

2

5

6

7

8

9

1

3

4

6

7

9

 

3

4

5

6

7

8

1

3

5

6

7

8

 

3

4

5

6

7

9

1

4

5

6

7

8

 

3

4

5

6

8

9

2

3

4

5

6

7

 

3

4

5

7

8

9

2

3

4

5

6

8

 

3

4

6

7

8

9

2

3

4

5

7

8

 

3

5

6

7

8

9

2

3

4

6

7

8

 

4

5

6

7

8

9

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.
Use at your own risk.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

    Hyperdimension's avatar - latest trace_171.gif

    United States
    Member #9059
    November 26, 2004
    128 Posts
    Offline
    Posted: November 25, 2006, 10:40 pm - IP Logged

    Hi JADELottery,

      In your software Jade Lottery system generator it built symmetric designs or apply the symmetry to non-symmetric designs, are there any advantage using this method of wheels creation over traditional wheels?

      Any answer is appreciated, thank's..

    El pensamiento ordena el caos..

    http://1x2quinielas.blogspot.com

      JADELottery's avatar - MeAtWork 03.PNG
      The Quantum Master
      West Concord, MN
      United States
      Member #21
      December 7, 2001
      3675 Posts
      Offline
      Posted: November 26, 2006, 5:25 pm - IP Logged

      Hi JADELottery,

        In your software Jade Lottery system generator it built symmetric designs or apply the symmetry to non-symmetric designs, are there any advantage using this method of wheels creation over traditional wheels?

        Any answer is appreciated, thank's..

      Hyperdimension,

      The Incorporate Symmetric Combinations under the Edit menu of JADE Lottery System Generator allows someone to balance a wheel through combinatorial symmetry. Wheels created by JADE LSG or by pasting into JADE LSG are analyzed for symmetry. If there are any symmetric combinations, JADE LSG gives the option of adding those combinations to the wheel and make it more balanced through combinatorial symmetry.

      The advantage of having a symmetrically balanced wheel has to do with number coverage. I use Cover Master, excellent wheel software by the way, to create wheels and paste them into JADE LSG for analysis and lottery betting. Cover Master works very well at creating a wheel to cover the combinations that produce a desired conditional win, however, there are combinations that have an equal symmetrical validity for covering the possibilities. JADE LSG allows a user to incorporate those combinations into the wheel and make the wheel more balanced symmetrically. Here's an example of a wheel created by Cover Master v0.55.2.0, a free wheel creator that anyone can generate using the software:

      Cover Master - Pool = 18, Pick = 6, Match = 3, Hits = 6
      01 02 03 04 05 06 
      01 02 03 04 05 17 
      01 02 03 04 11 12 
      01 02 07 08 13 14 
      03 04 09 10 15 16 
      05 06 07 08 09 10 
      05 06 11 12 17 18 
      05 06 13 14 15 16 
      07 08 09 10 11 12 
      13 14 15 16 17 18 

      Looking at the 2nd combination in the list, {01, 02, 03, 04, 05, 17}, there is an equally valid symmetric combination, {02, 14, 15, 16, 17, 18}, that is not present in the list of combinations.

      Below is the distribution of numbers by column and total.

      Cover Master Distribution - Pool = 18, Pick = 6, Match = 3, Hits = 6

       

       

      1

      2

      3

      4

      5

      6

      Total

      1

      4

       

       

       

       

       

      4

      2

       

      4

       

       

       

       

      4

      3

      1

       

      3

       

       

       

      4

      4

       

      1

       

      3

       

       

      4

      5

      3

       

       

       

      2

       

      5

      6

       

      3

       

       

       

      1

      4

      7

      1

       

      2

       

       

       

      3

      8

       

      1

       

      2

       

       

      3

      9

       

       

      2

       

      1

       

      3

      10

       

       

       

      2

       

      1

      3

      11

       

       

      1

       

      2

       

      3

      12

       

       

       

      1

       

      2

      3

      13

      1

       

      1

       

      1

       

      3

      14

       

      1

       

      1

       

      1

      3

      15

       

       

      1

       

      2

       

      3

      16

       

       

       

      1

       

      2

      3

      17

       

       

       

       

      2

      1

      3

      18

       

       

       

       

       

      2

      2

      Total

      10

      10

      10

      10

      10

      10

       

       

       

      Looking at the number 1 in column 1 there are 4 occurrences. The symmetrical counter part of the number 1 is number 18 in column 6; it has an occurrence of only 2. To be symmetrical, number 18 should have the same occurrence count as number 1. JADE LSG creates a list of combinations that can be added to the wheel. The symmetric combinations are as follows:

      Cover Master Symmetric Combinations - Pool = 18, Pick = 6, Match = 3, Hits = 6
      02 14 15 16 17 18
      07 08 15 16 17 18
      09 10 11 12 13 14
      03 04 05 06 13 14

      When added, the wheel then becomes balanced through combinatorial symmetry. 

      Cover Master Symmetric Balanced - Pool = 18, Pick = 6, Match = 3, Hits = 6
      01 02 03 04 05 06 
      01 02 03 04 05 17 
      01 02 03 04 11 12 
      01 02 07 08 13 14 
      02 14 15 16 17 18 
      03 04 05 06 13 14 
      03 04 09 10 15 16 
      05 06 07 08 09 10 
      05 06 11 12 17 18 
      05 06 13 14 15 16 
      07 08 09 10 11 12 
      07 08 15 16 17 18 
      09 10 11 12 13 14 
      13 14 15 16 17 18

      Below are the side by side distributions of the symmetrically balanced and unbalanced wheels.

      Cover Master Distribution of Balanced and Unbalanced wheels

                          Balanced                                  Unbalanced

       

       

      1

      2

      3

      4

      5

      6

      Total

       

      1

      2

      3

      4

      5

      6

      Total

      1

      4

       

       

       

       

       

      4

       

      4

       

       

       

       

       

      4

      2

      1

      4

       

       

       

       

      5

       

       

      4

       

       

       

       

      4

      3

      2

       

      3

       

       

       

      5

       

      1

       

      3

       

       

       

      4

      4

       

      2

       

      3

       

       

      5

       

       

      1

       

      3

       

       

      4

      5

      3

       

      1

       

      2

       

      6

       

      3

       

       

       

      2

       

      5

      6

       

      3

       

      1

       

      1

      5

       

       

      3

       

       

       

      1

      4

      7

      2

       

      2

       

       

       

      4

       

      1

       

      2

       

       

       

      3

      8

       

      2

       

      2

       

       

      4

       

       

      1

       

      2

       

       

      3

      9

      1

       

      2

       

      1

       

      4

       

       

       

      2

       

      1

       

      3

      10

       

      1

       

      2

       

      1

      4

       

       

       

       

      2

       

      1

      3

      11

       

       

      2

       

      2

       

      4

       

       

       

      1

       

      2

       

      3

      12

       

       

       

      2

       

      2

      4

       

       

       

       

      1

       

      2

      3

      13

      1

       

      1

       

      3

       

      5

       

      1

       

      1

       

      1

       

      3

      14

       

      2

       

      1

       

      3

      6

       

       

      1

       

      1

       

      1

      3

      15

       

       

      3

       

      2

       

      5

       

       

       

      1

       

      2

       

      3

      16

       

       

       

      3

       

      2

      5

       

       

       

       

      1

       

      2

      3

      17

       

       

       

       

      4

      1

      5

       

       

       

       

       

      2

      1

      3

      18

       

       

       

       

       

      4

      4

       

       

       

       

       

       

      2

      2

      Total

      14

      14

      14

      14

      14

      14

       

       

      10

      10

      10

      10

      10

      10

       

      Presented 'AS IS' and for Entertainment Purposes Only.
      Any gain or loss is your responsibility.
      Use at your own risk.

      Order is a Subset of Chaos
      Knowledge is Beyond Belief
      Wisdom is Not Censored
      Douglas Paul Smallish
      Jehocifer