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 | |