Row2 Possibles - Grouped
A study of the possible Row2 combinations. The carat symbol is a wildcard so that 7 ^ indicates that Rule 7 is invoked but the state (active/passive) is undetermined.
Start with Row1:
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| |
|
|
1 ^ |
2 ^ |
4 ^ |
|
|
|
Organized based on the number of wildcards in Row2
No wildcards:
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
10 |
20 |
40 |
00 |
00 |
00 |
| 00 |
00 |
00 |
00 |
00 |
00 |
00 |
00 |
00 |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
11 |
20 |
40 |
00 |
00 |
00 |
| 00 |
00 |
11 |
20 |
40 |
00 |
00 |
00 |
00 |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
10 |
21 |
40 |
00 |
00 |
00 |
| 00 |
00 |
00 |
10 |
21 |
40 |
00 |
00 |
00 |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
10 |
20 |
41 |
00 |
00 |
00 |
| 00 |
00 |
00 |
00 |
10 |
20 |
41 |
00 |
00 |
Notes:
- These four show Result Group 00, 04, 02, and 01.
- Rule 0 = 0, i.e. Rule 0 invokes a passive state.
- At most, only one of the Row 1 cells is active.
- All are reducible. The state of any cell can be calculated without running the automaton. In order:
- All cells are passive
- In any row, count to the left of the center column (below the seed cell) the number of rows and that cell is active. All others are passive. For instance, in the tenth row, the cell ten positions to the left of center is active.
- All cells in the center column are active. All others are passive.
- In any row, count to the right of center the number of rows and that cell is active. All others are passive.
- In multi-seed conditions, the rules above may not apply for all rulesets in a Result Group. The contention is that formulas can be developed for each ruleset even so.
One wildcard:
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
11 |
20 |
41 |
00 |
00 |
00 |
| 00 |
00 |
11 |
20 |
5 ^ |
20 |
41 |
00 |
00 |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
10 |
20 |
40 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
7 ^ |
40 |
01 |
10 |
7 ^ |
7 ^ |
7 ^ |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
11 |
21 |
41 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
7 ^ |
7 ^ |
7 ^ |
7 ^ |
7 ^ |
7 ^ |
7 ^ |
Two wildcards:
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
11 |
21 |
40 |
00 |
00 |
00 |
| 00 |
00 |
11 |
3 ^ |
6 ^ |
40 |
00 |
00 |
00 |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
10 |
21 |
41 |
00 |
00 |
00 |
| 00 |
00 |
00 |
10 |
3 ^ |
6 ^ |
41 |
00 |
00 |
Three wildcards:
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 00 |
00 |
00 |
11 |
21 |
41 |
00 |
00 |
00 |
| 00 |
00 |
11 |
3 ^ |
7 ^ |
6 ^ |
41 |
00 |
00 |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
11 |
20 |
40 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
7 ^ |
6 ^ |
40 |
11 |
3 ^ |
7 ^ |
7 ^ |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
10 |
20 |
41 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
6 ^ |
41 |
10 |
3 ^ |
7 ^ |
7 ^ |
7 ^ |
Four wildcards:
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
10 |
21 |
40 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
6 ^ |
5 ^ |
21 |
5 ^ |
3 ^ |
7 ^ |
7 ^ |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
11 |
21 |
40 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
7 ^ |
7 ^ |
6 ^ |
5 ^ |
3 ^ |
7 ^ |
7 ^ |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
11 |
20 |
41 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
7 ^ |
6 ^ |
5 ^ |
3 ^ |
7 ^ |
7 ^ |
7 ^ |
| 90 |
90 |
90 |
90 |
91 |
90 |
90 |
90 |
90 |
| 01 |
01 |
01 |
10 |
21 |
41 |
01 |
01 |
01 |
| 7 ^ |
7 ^ |
6 ^ |
5 ^ |
3 ^ |
7 ^ |
7 ^ |
7 ^ |
7 ^ |
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