Rule symmetries

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This thread is for discussing the symmetries that rules obey. This has been discussed to some extent on Discord, I am partially crossposting and partially adding some new ideas. Most prior research has only considered the MAP rulespace (the 2^512 range-1 Moore-neighborhood 2-state rules), so we will mostly be considering that. However, all of these ideas are still extendable to all CAs.

This is fairly big, so it will take up several posts. I think this is an acceptable use of multiposting (precedent).

Ultimately a CA (or anything, really) having a 'symmetry' just means that it is invariant under some sort of transformation. We can represent this as commutation between some action A on the state of the universe, and some rule R. If R(A(s)) = A(R(s)) for some grid state s, then we've found a symmetry. For instance, if our action A is translation by 1 cell, then A(Life(s)) = Life(A(s)), because Life is translation invariant, and thus has translation symmetry. Or, if our A is XORing with a checkerboard, then if A(R(s)) = R(A(s)), the rule has checkerboard dual symmetry.

This is the basic definition of rule symmetries.

For many rule symmetries, you can derive a subspace of the main rulespace that they work in. For instance, the MAP rules that are symmetric under D8 symmetry are the INT rules, which there are of course 2^102 of them. But this is not just a simple minmax like it is for patterns, it is more complicated, because some transitions are "bound" together, for instance, the 4 MAP transitions that make up the INT "B1c" transition are bound together, and so on for the other 101 groups.

I've also developed a program for exploring the rule symmetries in the MAP rulespace. It can calculate the subspace of a given rule symmetry and find the rule symmetries corresponding to a given rule. I'll explain more about how to use this below.

Edit: I also apologize for not crossposting this before, it should have been done a long time ago.

[speedydelete]

These are the rotations and reflections of patterns, see the wiki for a deeper explanation. For square-grid CA, they are: C1, C2, C4, D2|, D2-, D2\, D2/, D4+, D4x, and D8. Because the transition function is always centered on a cell, there is no need to distinguish between even and odd, only the odd ones are possible.

Here are examples of patterns with these symmetries:

Code: Select all

x = 32, y = 166, rule = LifeHistory
19.13F$19.F11.F$19.F2.4A2.A2.F$4F2.F12.F5.A5.F$F5.F12.F11.F$F5.F12.F4.
A3.2A.F$F5.F12.F4.A4.A.F$F5.F12.F2.2A5.A.F$F5.F12.F.2A4.A.A.F$4F2.F12.
FA6.3A.F$19.FA7.A2.F$19.FA10.F$19.13F5$19.13F$19.F10.AF$19.F10.AF$4F3.
2F10.F10.AF$F5.F2.F9.F8.2A.F$F8.F9.F7.2A2.F$F7.F10.F4.3A4.F$F6.F11.F2.
2A7.F$F5.F12.F.2A8.F$4F2.4F9.FA10.F$19.FA10.F$19.FA10.F$19.13F5$19.13F
$19.F3A7.AF$19.F3.A6.AF$4F2.F2.F9.F3.2A5.AF$F5.F2.F9.F4.A3.2A.F$F5.F2.
F9.F5.A.2A2.F$F5.4F9.F4.3A4.F$F8.F9.F2.2A.A5.F$F8.F9.F.2A3.A4.F$4F5.F
9.FA5.2A3.F$19.FA6.A3.F$19.FA7.3AF$19.13F5$19.13F$19.F5.A5.F$19.F5.A5.
F$3F4.2F3.F6.F5.A5.F$F2.F2.F2.F2.F6.F5.A5.F$F2.F5.F2.F6.F2.A2.A2.A2.F
$F2.F4.F3.F6.F3.A.A.A3.F$F2.F3.F4.F6.F.3A.A.3A.F$F2.F2.F5.F6.F5.A5.F$
3F3.4F2.F6.F5.A5.F$19.F5.A5.F$19.F5.A5.F$19.13F5$19.13F$19.F11.F$19.F
5.A5.F$3F4.2F10.F6.A4.F$F2.F2.F2.F9.F4.3A4.F$F2.F5.F9.F11.F$F2.F4.F3.
4F3.F11AF$F2.F3.F11.F11.F$F2.F2.F12.F4.3A4.F$3F3.4F9.F6.A4.F$19.F5.A5.
F$19.F11.F$19.13F5$19.13F$19.FA10.F$19.F.A6.A2.F$3F4.2F3.F6.F2.A6.A.F
$F2.F2.F2.F2.F6.F3.A3.3A.F$F2.F5.F3.F5.F4.A6.F$F2.F4.F4.2F4.F5.A5.F$F
2.F3.F6.F4.F6.A4.F$F2.F2.F8.F3.F3.A3.A3.F$3F3.4F5.F3.F.A.A4.A2.F$19.F
2.2A5.A.F$19.F10.AF$19.13F5$19.13F$19.F10.AF$19.F2.2A5.A.F$3F4.2F6.F3.
F.A.A4.A2.F$F2.F2.F2.F5.F3.F3.A3.A3.F$F2.F5.F4.F4.F6.A4.F$F2.F4.F4.2F
4.F5.A5.F$F2.F3.F5.F5.F4.A6.F$F2.F2.F5.F6.F3.A3.3A.F$3F3.4F2.F6.F2.A6.
A.F$19.F.A6.A2.F$19.FA10.F$19.13F5$19.13F$19.F5.A5.F$19.F2.A2.A2.A2.F
$3F3.F2.F4.F4.F3.A.A.A3.F$F2.F2.F2.F4.F4.F.3A.A.3A.F$F2.F2.F2.F4.F4.F
5.A5.F$F2.F2.4F2.5F2.F11AF$F2.F5.F4.F4.F5.A5.F$F2.F5.F4.F4.F.3A.A.3A.
F$3F6.F4.F4.F3.A.A.A3.F$19.F2.A2.A2.A2.F$19.F5.A5.F$19.13F5$19.13F$19.
FA2.A6.AF$19.F.A.A5.A.F$3F3.F2.F2.F3.F2.F2.A5.A2.F$F2.F2.F2.F2.F3.F2.
F2A.A3.A3.F$F2.F2.F2.F3.F.F3.F4.A.A4.F$F2.F2.4F4.F4.F5.A5.F$F2.F5.F3.
F.F3.F4.A.A4.F$F2.F5.F2.F3.F2.F3.A3.A.2AF$3F6.F2.F3.F2.F2.A5.A2.F$19.
F.A5.A.A.F$19.FA6.A2.AF$19.13F5$19.13F$19.FA4.A4.AF$19.F.A3.A3.A.F$3F
4.2F10.F2.A2.A2.A2.F$F2.F2.F2.F9.F3.A.A.A3.F$F2.F2.F2.F9.F4.3A4.F$F2.
F3.2F10.F11AF$F2.F2.F2.F9.F4.3A4.F$F2.F2.F2.F9.F3.A.A.A3.F$3F4.2F10.F
2.A2.A2.A2.F$19.F.A3.A3.A.F$19.FA4.A4.AF$19.13F!

For MAP, here are the total counts of rules with each kind of symmetry:

Code: Select all

C1 = 2^512
C2 = 2^272
C4 = 2^140
D2- = 2^288
D2| = 2^288
D2/ = 2^288
D2\ = 2^288
D4+ = 2^168
D4x = 2^168
D8 = 2^102

Or, for the rules that only have that symmetry and no higher symmetry (these may be incorrect, I checked them but I am not fully sure):

Code: Select all

C1 = 2^512 - 2^290 - 2^272 + 2^169 + 2^103
C2 = 2^272 - 2^140
C4 = 2^140 - 2^102
D2* = 2^288 - 2^168
D4* = 2^168 - 2^102
D8 = 2^102

These symmetries are not that interesting by themselves, but they are a good example of a rule symmetry.

Now, let's explain how to use these in the rule symmetry program.

You can code them manually, the given example here is for D2| symmetry:

Screenshot 2026-09-07 133323.png (29.83 KiB) Viewed 279 times

Or, you can use the preprogrammed "D2h" for D2|. The "h" stands for horizontal, this matches the "flip horizontal" functions in LifeViewer and the "reflect_horizontal" symmetry in @TABLE. Yes, I know, it's a vertical line, so shouldn't it be D2v? I've tried it flipped that way, and it's more confusing.

And then, D2v is D2-, D2b is D2\, and D2s is D2/ (these two stand for "backslash" and "slash", respectively).

How this works is the [stuff] describes a permutation. You can even put boolean expressions in there, so for instance [nw=(n & e)] is valid, however, I haven't seen any uses of this feature yet. Multiple symmetries can be combined using commas, like "D2|, D2-" (which is equivalent to "D4+").

[speedydelete]

Now this is the interesting part!

Code: Select all

x = 80, y = 25, rule = B1e2c3eq4aikt5eq6c7e/S01c2-c3-eq4-aikt5-eq6-c7c8:T80,25
20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo
2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b
12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$6b3o
11bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$5b3o12bobobobobobo5b12ob2ob2o
5bo2bo2bo3bo4bo$5b3o12bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$6b3o11bo
bobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2b
o3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob
2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobob
obobobo5b12ob2ob2o5bo2bo2bo3bo4bo$6b6o8bobobobobobo5b12ob2ob2o5bo2bo2b
o3bo4bo$5b3o3b2o7bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$5b2o2b2ob3o5b
obobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$6bob4o2bo5bobobobobobo5b12ob2ob
2o5bo2bo2bo3bo4bo$7b2o3b3o5bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$8b5o
7bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2b
o2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12o
b2ob2o5bo2bo2bo3bo4bo$20bobobobobobo5b12ob2ob2o5bo2bo2bo3bo4bo$20bobo
bobobobo5b12ob2ob2o5bo2bo2bo3bo4bo!

Edit: Context soon

This was found with a very early version of the rule symmetry program. The conceptual understanding of how it works was very incomplete when this was found, but again I am sorry for not crossposting all this earlier. (A lot of the work here was done by me and AforAmpere).

So, a "duality" is a rule symmetry under XOR with an agar, that is, the symmetry is that if you XOR any pattern with the agar, run the rule, and then XOR it back, it will be the same.

To find what rules satisfy the agar, find out the transitions the agar uses, and then put that into the rule symmetry program.

We are using the confocaloid sense of transition here, where the state of the center cell in the next generation matters too, so there are 1024 MAP transitions instead of 512. So, these are all distinct transitions called A4c, B4c, D4c, and S4c (A = abstain [from birth], D = die):

Code: Select all

x = 47, y = 56, rule = LifeHistory
21.11F$21.F3A3.3AF$2.2F4.F2.F4.2F3.F3A3.3AF4.F$.F2.F3.F2.F3.F5.F3A3.3A
F5.F4.5F$.F2.F3.F2.F2.F6.F9.F6.F3.F3.F$.4F3.4F2.F6.F9.F2.5F3.F3.F$F4.
F5.F2.F6.F9.F6.F3.F3.F$F4.F5.F3.F5.F3A3.3AF5.F4.5F$F4.F5.F4.2F3.F3A3.
3AF4.F$21.F3A3.3AF$21.11F5$21.11F$21.F3A3.3AF$2.3F3.F2.F4.2F3.F3A3.3A
F4.F$2.F2.F2.F2.F3.F5.F3A3.3AF5.F4.5F$2.F2.F2.F2.F2.F6.F9.F6.F3.F3AF$
2.4F2.4F2.F6.F9.F2.5F3.F3AF$2.F2.F5.F2.F6.F9.F6.F3.F3AF$2.F2.F5.F3.F5.
F3A3.3AF5.F4.5F$2.3F6.F4.2F3.F3A3.3AF4.F$21.F3A3.3AF$21.11F5$21.11F$21.
F3A3.3AF$2.3F3.F2.F4.2F3.F3A3.3AF4.F$2.F2.F2.F2.F3.F5.F3A3.3AF5.F4.5F
$2.F2.F2.F2.F2.F6.F3.3A3.F6.F3.F3.F$2.F2.F2.4F2.F6.F3.3A3.F2.5F3.F3.F
$2.F2.F5.F2.F6.F3.3A3.F6.F3.F3.F$2.F2.F5.F3.F5.F3A3.3AF5.F4.5F$2.3F6.
F4.2F3.F3A3.3AF4.F$21.F3A3.3AF$21.11F5$21.11F$21.F3A3.3AF$3.3F2.F2.F4.
2F3.F3A3.3AF4.F$2.F5.F2.F3.F5.F3A3.3AF5.F4.5F$2.F5.F2.F2.F6.F3.3A3.F6.
F3.F3AF$3.2F3.4F2.F6.F3.3A3.F2.5F3.F3AF$5.F5.F2.F6.F3.3A3.F6.F3.F3AF$
5.F5.F3.F5.F3A3.3AF5.F4.5F$2.3F6.F4.2F3.F3A3.3AF4.F$21.F3A3.3AF$21.11F
!

Here's some examples to show you how to find what transitions an agar uses:

The checkerboard agar:

Code: Select all

x = 12, y = 12, rule = B/S4c:T12,12
obobobobobo$bobobobobobo$obobobobobo$bobobobobobo$obobobobobo$bobobob
obobo$obobobobobo$bobobobobobo$obobobobobo$bobobobobobo$obobobobobo$b
obobobobobo!

Uses A4e and S4c, because the dead cells cannot become alive.

However, the period-2 checkerboard agar:

Code: Select all

x = 12, y = 12, rule = B4e/S:T12,12
obobobobobo$bobobobobobo$obobobobobo$bobobobobobo$obobobobobo$bobobob
obobo$obobobobobo$bobobobobobo$obobobobobo$bobobobobobo$obobobobobo$b
obobobobobo!

Uses B4e and D4c, because the alive cells must die via D4c. You can think of adding S4c to the rule as removing D4c from it, and vice versa. The same applies with B4e and A4e, and indeed the 100 other such pairs of INT transitions.

These are coded in the rule symmetry program like this:

Screenshot 2026-09-07 135600.png (29.12 KiB) Viewed 268 times

Or, you can combine them, like "INT, ^A4e/S4c".

Now, it's time to explain how exactly the above "line transparency" works. The transitions required for it are A3i, A6i, S2i, S5i, and S8, but notice that all of these are implied by ^A3i, ^S2i, because for instance XORing A3i and S2i together gives you S5i. A3i is the most important one, it lets patterns run the same on the side of lines. You can try this yourself in this pattern, set the paste mode to XOR and copy the "A3i" region into the "S2i" region:

Code: Select all

x = 47, y = 41, rule = LifeHistory
21.11F$21.F3A6.F$2.2F5.2F3.3F4.F3A6.F4.F$.F2.F6.F3.F5.F3A6.F5.F4.5F$.
F2.F6.F3.F5.F3A6.F6.F3.F3.F$.4F4.2F4.F5.F3A6.F2.5F3.F3.F$F4.F5.F3.F5.
F3A6.F6.F3.F3.F$F4.F5.F3.F5.F3A6.F5.F4.5F$F4.F3.2F3.3F4.F3A6.F4.F$21.
F3A6.F$21.11F5$21.11F$21.F3.3A3.F$3.3F3.2F3.3F4.F3.3A3.F4.F$2.F5.F2.F
3.F5.F3.3A3.F5.F4.5F$2.F8.F3.F5.F3.3A3.F6.F3.F3AF$3.2F5.F4.F5.F3.3A3.
F2.5F3.F3AF$5.F3.F5.F5.F3.3A3.F6.F3.F3AF$5.F2.F6.F5.F3.3A3.F5.F4.5F$2.
3F3.4F2.3F4.F3.3A3.F4.F$21.F3.3A3.F$21.11F5$21.11F$21.F6A3.F$3.3F2.4F
2.3F4.F6A3.F4.F$2.F5.F6.F5.F6A3.F5.F4.5F$2.F5.F6.F5.F6A3.F6.F3.F3AF$3.
2F3.4F3.F5.F6A3.F2.5F3.F3AF$5.F5.F3.F5.F6A3.F6.F3.F3AF$5.F5.F3.F5.F6A
3.F5.F4.5F$2.3F3.4F2.3F4.F6A3.F4.F$21.F6A3.F$21.11F!

And finally, I've computed all 474 "fully resolved" (all XORs are expanded) groups of XOR transitions that are attainable in INT, here they are (I removed all the A0's from the list, every rule satisfies ^A0 after all):

Code: Select all

0: 2^62: ^A4c
1: 2^62: ^A4e
2: 2^54: ^A8
3: 2^51: ^D0
4: 2^51: ^D4c
5: 2^51: ^D4e
6: 2^51: ^D8
7: 2^51: ^S0
8: 2^51: ^S4c
9: 2^51: ^S4e
10: 2^51: ^S8
11: 2^38: ^A2i4e
12: 2^38: ^A2n4c
13: 2^38: ^A4c6n
14: 2^38: ^A4ce8
15: 2^38: ^A4e6i
16: 2^31: ^A4c/D04c
17: 2^31: ^A4c/D2n
18: 2^31: ^A4c/D4e8
19: 2^31: ^A4c/D6n
20: 2^31: ^A4e/D04e
21: 2^31: ^A4e/D2i
22: 2^31: ^A4e/D4c8
23: 2^31: ^A4e/D6i
24: 2^31: ^A4c/S04c
25: 2^31: ^A4e/S04e
26: 2^31: ^A4e/S2i
27: 2^31: ^A4c/S2n
28: 2^31: ^A4e/S4c8
29: 2^31: ^A4c/S4e8
30: 2^31: ^A4e/S6i
31: 2^31: ^A4c/S6n
32: 2^27: ^A8/D08
33: 2^27: ^A8/D4ce
34: 2^27: ^A8/S08
35: 2^27: ^A8/S4ce
36: 2^26: ^A2i4ce6i8
37: 2^26: ^A2n4ce6n8
38: 2^24: ^A2cn4c
39: 2^24: ^A2ei4e
40: 2^24: ^A4ci6n
41: 2^24: ^A4ew6i
42: 2^22: ^A4cez8
43: 2^20: ^A2i4e6e
44: 2^20: ^A2n4c6c
45: 2^20: ^A4ct6n
46: 2^20: ^A4eq6i
47: 2^19: ^A2i4e/D02i4e
48: 2^19: ^A2i4e/D2e
49: 2^19: ^A2i4e/D4c6i8
50: 2^19: ^A2i4e/D6e
51: 2^19: ^A2n4c/D02n4c
52: 2^19: ^A2n4c/D2c
53: 2^19: ^A2n4c/D4e6n8
54: 2^19: ^A2n4c/D6c
55: 2^19: ^A4c6n/D04c6n
56: 2^19: ^A4c6n/D2n4e8
57: 2^19: ^A4c6n/D4i
58: 2^19: ^A4c6n/D4t
59: 2^19: ^A4ce8/D04ce8
60: 2^19: ^A4ce8/D2i6i
61: 2^19: ^A4ce8/D2n6n
62: 2^19: ^A4ce8/D4z
63: 2^19: ^A4e6i/D04e6i
64: 2^19: ^A4e6i/D2i4c8
65: 2^19: ^A4e6i/D4q
66: 2^19: ^A4e6i/D4w
67: 2^19: ^A2i4e/S02i4e
68: 2^19: ^A2n4c/S02n4c
69: 2^19: ^A4c6n/S04c6n
70: 2^19: ^A4ce8/S04ce8
71: 2^19: ^A4e6i/S04e6i
72: 2^19: ^A2n4c/S2c
73: 2^19: ^A2i4e/S2e
74: 2^19: ^A4e6i/S2i4c8
75: 2^19: ^A4ce8/S2i6i
76: 2^19: ^A4c6n/S2n4e8
77: 2^19: ^A4ce8/S2n6n
78: 2^19: ^A2i4e/S4c6i8
79: 2^19: ^A2n4c/S4e6n8
80: 2^19: ^A4c6n/S4i
81: 2^19: ^A4e6i/S4q
82: 2^19: ^A4c6n/S4t
83: 2^19: ^A4e6i/S4w
84: 2^19: ^A4ce8/S4z
85: 2^19: ^A2n4c/S6c
86: 2^19: ^A2i4e/S6e
87: 2^18: ^A2in4cez6in8
88: 2^16: ^A2cn4ce6cn8
89: 2^16: ^A2ei4ce6ei8
90: 2^16: ^A2i4ceqw6i8
91: 2^16: ^A2n4ceit6n8
92: 2^13: ^A2i4ce6i8/D02i4ce6i8
93: 2^13: ^A2i4ce6i8/D2e6e
94: 2^13: ^A2i4ce6i8/D2n4z6n
95: 2^13: ^A2i4ce6i8/D4qw
96: 2^13: ^A2n4ce6n8/D02n4ce6n8
97: 2^13: ^A2n4ce6n8/D2c6c
98: 2^13: ^A2n4ce6n8/D2i4z6i
99: 2^13: ^A2n4ce6n8/D4it
100: 2^13: ^A2i4ce6i8/S02i4ce6i8
101: 2^13: ^A2n4ce6n8/S02n4ce6n8
102: 2^13: ^A2n4ce6n8/S2c6c
103: 2^13: ^A2i4ce6i8/S2e6e
104: 2^13: ^A2n4ce6n8/S2i4z6i
105: 2^13: ^A2i4ce6i8/S2n4z6n
106: 2^13: ^A2n4ce6n8/S4it
107: 2^13: ^A2i4ce6i8/S4qw
108: 2^12: ^B1c3c/A2cn4c
109: 2^12: ^B1e3e/A2ei4e
110: 2^12: ^B3a5k/A4ci6n
111: 2^12: ^B3i5y/A4ew6i
112: 2^12: ^B3k5A4ci6n
113: 2^12: ^B3y5i/A4ew6i
114: 2^12: ^B5c7c/A2cn4c
115: 2^12: ^B5e7e/A2ei4e
116: 2^12: ^A1c2cn3c4c
117: 2^12: ^A1e2ei3e4e
118: 2^12: ^A2cin4ceitz6cin8
119: 2^12: ^A2cn4c/D02cn4c
120: 2^12: ^A2cn4c/D1c3c
121: 2^12: ^A2cn4c/D4e6cn8
122: 2^12: ^A2cn4c/D5c7c
123: 2^12: ^A2cn4c5c7c
124: 2^12: ^A2ei4e/D02ei4e
125: 2^12: ^A2ei4e/D1e3e
126: 2^12: ^A2ei4e/D4c6ei8
127: 2^12: ^A2ei4e/D5e7e
128: 2^12: ^A2ei4e5e7e
129: 2^12: ^A2ein4ceqwz6ein8
130: 2^12: ^A3a4ci5k6n
131: 2^12: ^A3i4ew5y6i
132: 2^12: ^A3k4ci5a6n
133: 2^12: ^A3y4ew5i6i
134: 2^12: ^A4ci6n/D04ci6n
135: 2^12: ^A4ci6n/D2n4et8
136: 2^12: ^A4ci6n/D3a5k
137: 2^12: ^A4ci6n/D3k5a
138: 2^12: ^A4ew6i/D04ew6i
139: 2^12: ^A4ew6i/D2i4cq8
140: 2^12: ^A4ew6i/D3i5y
141: 2^12: ^A4ew6i/D3y5i
142: 2^12: ^A2cn4c/S02cn4c
143: 2^12: ^A2ei4e/S02ei4e
144: 2^12: ^A4ci6n/S04ci6n
145: 2^12: ^A4ew6i/S04ew6i
146: 2^12: ^A2cn4c/S1c3c
147: 2^12: ^A2ei4e/S1e3e
148: 2^12: ^A4ew6i/S2i4cq8
149: 2^12: ^A4ci6n/S2n4et8
150: 2^12: ^A4ci6n/S3a5k
151: 2^12: ^A4ew6i/S3i5y
152: 2^12: ^A4ci6n/S3k5a
153: 2^12: ^A4ew6i/S3y5i
154: 2^12: ^A2ei4e/S4c6ei8
155: 2^12: ^A2cn4c/S4e6cn8
156: 2^12: ^A2cn4c/S5c7c
157: 2^12: ^A2ei4e/S5e7e
158: 2^11: ^A4cez8/D04cez8
159: 2^11: ^A4cez8/D2in6in
160: 2^11: ^A4cez8/S04cez8
161: 2^11: ^A4cez8/S2in6in
162: 2^10: ^A2i4e6e/D02i4e6e
163: 2^10: ^A2i4e6e/D2e4c6i8
164: 2^10: ^A2in4acekz6in8
165: 2^10: ^A2n4c6c/D02n4c6c
166: 2^10: ^A2n4c6c/D2c4e6n8
167: 2^10: ^A4ct6n/D04ct6n
168: 2^10: ^A4ct6n/D2n4ei8
169: 2^10: ^A4eq6i/D04eq6i
170: 2^10: ^A4eq6i/D2i4cw8
171: 2^10: ^A2i4e6e/S02i4e6e
172: 2^10: ^A2n4c6c/S02n4c6c
173: 2^10: ^A4ct6n/S04ct6n
174: 2^10: ^A4eq6i/S04eq6i
175: 2^10: ^A2n4c6c/S2c4e6n8
176: 2^10: ^A2i4e6e/S2e4c6i8
177: 2^10: ^A4eq6i/S2i4cw8
178: 2^10: ^A4ct6n/S2n4ei8
179: 2^9: ^A2in4cez6in8/D02in4cez6in8
180: 2^9: ^A2in4cez6in8/D2c4it6c
181: 2^9: ^A2in4cez6in8/D2e4qw6e
182: 2^9: ^A2in4cez6in8/D4ak
183: 2^9: ^A2in4cez6in8/S02in4cez6in8
184: 2^9: ^A2in4cez6in8/S2c4it6c
185: 2^9: ^A2in4cez6in8/S2e4qw6e
186: 2^9: ^A2in4cez6in8/S4ak
187: 2^8: ^B1c3c5c7c/A2cn4ce6cn8
188: 2^8: ^B1e3e5e7e/A2ei4ce6ei8
189: 2^8: ^B3ak5ak/A2n4ceit6n8
190: 2^8: ^B3iy5iy/A2i4ceqw6i8
191: 2^8: ^B3j5j/A2n4ceit6n8
192: 2^8: ^B3n5n/A2i4ceqw6i8
193: 2^8: ^B3q5q/A2ei4ce6ei8
194: 2^8: ^B3r5r/A2cn4ce6cn8
195: 2^8: ^A1c2cn3c4ce5c6cn7c8
196: 2^8: ^A1e2ei3e4ce5e6ei7e8
197: 2^8: ^A2-ak4-jnry6-ak8
198: 2^8: ^A2cn3r4ce5r6cn8
199: 2^8: ^A2cn4ce6cn8/D02cn4ce6cn8
200: 2^8: ^A2cn4ce6cn8/D1c3c5c7c
201: 2^8: ^A2cn4ce6cn8/D2i4itz6i
202: 2^8: ^A2cn4ce6cn8/D3r5r
203: 2^8: ^A2ei3q4ce5q6ei8
204: 2^8: ^A2ei4ce6ei8/D02ei4ce6ei8
205: 2^8: ^A2ei4ce6ei8/D1e3e5e7e
206: 2^8: ^A2ei4ce6ei8/D2n4qwz6n
207: 2^8: ^A2ei4ce6ei8/D3q5q
208: 2^8: ^A2i3iy4ceqw5iy6i8
209: 2^8: ^A2i3n4ceqw5n6i8
210: 2^8: ^A2i4ceqw6i8/D02i4ceqw6i8
211: 2^8: ^A2i4ceqw6i8/D2en4z6en
212: 2^8: ^A2i4ceqw6i8/D3iy5iy
213: 2^8: ^A2i4ceqw6i8/D3n5n
214: 2^8: ^A2n3ak4ceit5ak6n8
215: 2^8: ^A2n3j4ceit5j6n8
216: 2^8: ^A2n4ceit6n8/D02n4ceit6n8
217: 2^8: ^A2n4ceit6n8/D2ci4z6ci
218: 2^8: ^A2n4ceit6n8/D3ak5ak
219: 2^8: ^A2n4ceit6n8/D3j5j
220: 2^8: ^A2cn4ce6cn8/S02cn4ce6cn8
221: 2^8: ^A2ei4ce6ei8/S02ei4ce6ei8
222: 2^8: ^A2i4ceqw6i8/S02i4ceqw6i8
223: 2^8: ^A2n4ceit6n8/S02n4ceit6n8
224: 2^8: ^A2cn4ce6cn8/S1c3c5c7c
225: 2^8: ^A2ei4ce6ei8/S1e3e5e7e
226: 2^8: ^A2n4ceit6n8/S2ci4z6ci
227: 2^8: ^A2i4ceqw6i8/S2en4z6en
228: 2^8: ^A2cn4ce6cn8/S2i4itz6i
229: 2^8: ^A2ei4ce6ei8/S2n4qwz6n
230: 2^8: ^A2n4ceit6n8/S3ak5ak
231: 2^8: ^A2i4ceqw6i8/S3iy5iy
232: 2^8: ^A2n4ceit6n8/S3j5j
233: 2^8: ^A2i4ceqw6i8/S3n5n
234: 2^8: ^A2ei4ce6ei8/S3q5q
235: 2^8: ^A2cn4ce6cn8/S3r5r
236: 2^6: ^B1c3c/A2cn4c/S02cn4c/D1c3c
237: 2^6: ^B1c3c/A2cn4c/S1c3c/D02cn4c
238: 2^6: ^B1c3c/A2cn4c/S4e6cn8/D5c7c
239: 2^6: ^B1c3c/A2cn4c/S5c7c/D4e6cn8
240: 2^6: ^B1c3cr5cr7c/A2cin4ceitz6cin8
241: 2^6: ^B1e3e/A2ei4e/S02ei4e/D1e3e
242: 2^6: ^B1e3e/A2ei4e/S1e3e/D02ei4e
243: 2^6: ^B1e3e/A2ei4e/S4c6ei8/D5e7e
244: 2^6: ^B1e3e/A2ei4e/S5e7e/D4c6ei8
245: 2^6: ^B1e3eq5eq7e/A2ein4ceqwz6ein8
246: 2^6: ^B3a5k/A4ci6n/S04ci6n/D3a5k
247: 2^6: ^B3a5k/A4ci6n/S2n4et8/D3k5a
248: 2^6: ^B3a5k/A4ci6n/S3a5k/D04ci6n
249: 2^6: ^B3a5k/A4ci6n/S3k5D2n4et8
250: 2^6: ^B3ajk5ajk/A2cin4ceitz6cin8
251: 2^6: ^B3i5y/A4ew6i/S04ew6i/D3i5y
252: 2^6: ^B3i5y/A4ew6i/S2i4cq8/D3y5i
253: 2^6: ^B3i5y/A4ew6i/S3i5y/D04ew6i
254: 2^6: ^B3i5y/A4ew6i/S3y5i/D2i4cq8
255: 2^6: ^B3iny5iny/A2ein4ceqwz6ein8
256: 2^6: ^B3k5A4ci6n/S04ci6n/D3k5a
257: 2^6: ^B3k5A4ci6n/S2n4et8/D3a5k
258: 2^6: ^B3k5A4ci6n/S3a5k/D2n4et8
259: 2^6: ^B3k5A4ci6n/S3k5D04ci6n
260: 2^6: ^B3y5i/A4ew6i/S04ew6i/D3y5i
261: 2^6: ^B3y5i/A4ew6i/S2i4cq8/D3i5y
262: 2^6: ^B3y5i/A4ew6i/S3i5y/D2i4cq8
263: 2^6: ^B3y5i/A4ew6i/S3y5i/D04ew6i
264: 2^6: ^B5c7c/A2cn4c/S02cn4c/D5c7c
265: 2^6: ^B5c7c/A2cn4c/S1c3c/D4e6cn8
266: 2^6: ^B5c7c/A2cn4c/S4e6cn8/D1c3c
267: 2^6: ^B5c7c/A2cn4c/S5c7c/D02cn4c
268: 2^6: ^B5e7e/A2ei4e/S02ei4e/D5e7e
269: 2^6: ^B5e7e/A2ei4e/S1e3e/D4c6ei8
270: 2^6: ^B5e7e/A2ei4e/S4c6ei8/D1e3e
271: 2^6: ^B5e7e/A2ei4e/S5e7e/D02ei4e
272: 2^6: ^A1c2cin3cr4ceitz5cr6cin7c8
273: 2^6: ^A1c2cn3c4c/D01c2cn3c4c
274: 2^6: ^A1c2cn3c4c/D4e5c6cn7c8
275: 2^6: ^A1e2ei3e4e/D01e2ei3e4e
276: 2^6: ^A1e2ei3e4e/D4c5e6ei7e8
277: 2^6: ^A1e2ein3eq4ceqwz5eq6ein7e8
278: 2^6: ^A2cin3ajk4ceitz5ajk6cin8
279: 2^6: ^A2cin4ceitz6cin8/D02cin4ceitz6cin8
280: 2^6: ^A2cin4ceitz6cin8/D1c3cr5cr7c
281: 2^6: ^A2cin4ceitz6cin8/D2e4akqw6e
282: 2^6: ^A2cin4ceitz6cin8/D3ajk5ajk
283: 2^6: ^A2cn4c5c7c/D02cn4c5c7c
284: 2^6: ^A2cn4c5c7c/D1c3c4e6cn8
285: 2^6: ^A2ei4e5e7e/D02ei4e5e7e
286: 2^6: ^A2ei4e5e7e/D1e3e4c6ei8
287: 2^6: ^A2ein3iny4ceqwz5iny6ein8
288: 2^6: ^A2ein4ceqwz6ein8/D02ein4ceqwz6ein8
289: 2^6: ^A2ein4ceqwz6ein8/D1e3eq5eq7e
290: 2^6: ^A2ein4ceqwz6ein8/D2c4aikt6c
291: 2^6: ^A2ein4ceqwz6ein8/D3iny5iny
292: 2^6: ^A3a4ci5k6n/D03a4ci5k6n
293: 2^6: ^A3a4ci5k6n/D2n3k4et5a8
294: 2^6: ^A3i4ew5y6i/D03i4ew5y6i
295: 2^6: ^A3i4ew5y6i/D2i3y4cq5i8
296: 2^6: ^A3k4ci5a6n/D03k4ci5a6n
297: 2^6: ^A3k4ci5a6n/D2n3a4et5k8
298: 2^6: ^A3y4ew5i6i/D03y4ew5i6i
299: 2^6: ^A3y4ew5i6i/D2i3i4cq5y8
300: 2^6: ^A1c2cn3c4c/S01c2cn3c4c
301: 2^6: ^A1e2ei3e4e/S01e2ei3e4e
302: 2^6: ^A2cin4ceitz6cin8/S02cin4ceitz6cin8
303: 2^6: ^A2cn4c5c7c/S02cn4c5c7c
304: 2^6: ^A2ei4e5e7e/S02ei4e5e7e
305: 2^6: ^A2ein4ceqwz6ein8/S02ein4ceqwz6ein8
306: 2^6: ^A3a4ci5k6n/S03a4ci5k6n
307: 2^6: ^A3i4ew5y6i/S03i4ew5y6i
308: 2^6: ^A3k4ci5a6n/S03k4ci5a6n
309: 2^6: ^A3y4ew5i6i/S03y4ew5i6i
310: 2^6: ^A2cn4c5c7c/S1c3c4e6cn8
311: 2^6: ^A2cin4ceitz6cin8/S1c3cr5cr7c
312: 2^6: ^A2ei4e5e7e/S1e3e4c6ei8
313: 2^6: ^A2ein4ceqwz6ein8/S1e3eq5eq7e
314: 2^6: ^A2ein4ceqwz6ein8/S2c4aikt6c
315: 2^6: ^A2cin4ceitz6cin8/S2e4akqw6e
316: 2^6: ^A3y4ew5i6i/S2i3i4cq5y8
317: 2^6: ^A3i4ew5y6i/S2i3y4cq5i8
318: 2^6: ^A3k4ci5a6n/S2n3a4et5k8
319: 2^6: ^A3a4ci5k6n/S2n3k4et5a8
320: 2^6: ^A2cin4ceitz6cin8/S3ajk5ajk
321: 2^6: ^A2ein4ceqwz6ein8/S3iny5iny
322: 2^6: ^A1e2ei3e4e/S4c5e6ei7e8
323: 2^6: ^A1c2cn3c4c/S4e5c6cn7c8
324: 2^5: ^A2in4acekz6in8/D02in4acekz6in8
325: 2^5: ^A2in4acekz6in8/D2ce4iqtw6ce
326: 2^5: ^A2in4acekz6in8/S02in4acekz6in8
327: 2^5: ^A2in4acekz6in8/S2ce4iqtw6ce
328: 2^4: ^B1c3acjkr5acjkr7c/A2-ak4-jnry6-ak8
329: 2^4: ^B1c3c5c7c/A2cn4ce6cn8/S02cn4ce6cn8/D1c3c5c7c
330: 2^4: ^B1c3c5c7c/A2cn4ce6cn8/S1c3c5c7c/D02cn4ce6cn8
331: 2^4: ^B1c3c5c7c/A2cn4ce6cn8/S2i4itz6i/D3r5r
332: 2^4: ^B1c3c5c7c/A2cn4ce6cn8/S3r5r/D2i4itz6i
333: 2^4: ^B1e3e5e7e/A2ei4ce6ei8/S02ei4ce6ei8/D1e3e5e7e
334: 2^4: ^B1e3e5e7e/A2ei4ce6ei8/S1e3e5e7e/D02ei4ce6ei8
335: 2^4: ^B1e3e5e7e/A2ei4ce6ei8/S2n4qwz6n/D3q5q
336: 2^4: ^B1e3e5e7e/A2ei4ce6ei8/S3q5q/D2n4qwz6n
337: 2^4: ^B1e3einqy5einqy7e/A2-ak4-jnry6-ak8
338: 2^4: ^B2ak4jnry6ak/A2-ak4-jnry6-ak8
339: 2^4: ^B3ak5ak/A2n4ceit6n8/S02n4ceit6n8/D3ak5ak
340: 2^4: ^B3ak5ak/A2n4ceit6n8/S2ci4z6ci/D3j5j
341: 2^4: ^B3ak5ak/A2n4ceit6n8/S3ak5ak/D02n4ceit6n8
342: 2^4: ^B3ak5ak/A2n4ceit6n8/S3j5j/D2ci4z6ci
343: 2^4: ^B3iy5iy/A2i4ceqw6i8/S02i4ceqw6i8/D3iy5iy
344: 2^4: ^B3iy5iy/A2i4ceqw6i8/S2en4z6en/D3n5n
345: 2^4: ^B3iy5iy/A2i4ceqw6i8/S3iy5iy/D02i4ceqw6i8
346: 2^4: ^B3iy5iy/A2i4ceqw6i8/S3n5n/D2en4z6en
347: 2^4: ^B3j5j/A2n4ceit6n8/S02n4ceit6n8/D3j5j
348: 2^4: ^B3j5j/A2n4ceit6n8/S2ci4z6ci/D3ak5ak
349: 2^4: ^B3j5j/A2n4ceit6n8/S3ak5ak/D2ci4z6ci
350: 2^4: ^B3j5j/A2n4ceit6n8/S3j5j/D02n4ceit6n8
351: 2^4: ^B3n5n/A2i4ceqw6i8/S02i4ceqw6i8/D3n5n
352: 2^4: ^B3n5n/A2i4ceqw6i8/S2en4z6en/D3iy5iy
353: 2^4: ^B3n5n/A2i4ceqw6i8/S3iy5iy/D2en4z6en
354: 2^4: ^B3n5n/A2i4ceqw6i8/S3n5n/D02i4ceqw6i8
355: 2^4: ^B3q5q/A2ei4ce6ei8/S02ei4ce6ei8/D3q5q
356: 2^4: ^B3q5q/A2ei4ce6ei8/S1e3e5e7e/D2n4qwz6n
357: 2^4: ^B3q5q/A2ei4ce6ei8/S2n4qwz6n/D1e3e5e7e
358: 2^4: ^B3q5q/A2ei4ce6ei8/S3q5q/D02ei4ce6ei8
359: 2^4: ^B3r5r/A2cn4ce6cn8/S02cn4ce6cn8/D3r5r
360: 2^4: ^B3r5r/A2cn4ce6cn8/S1c3c5c7c/D2i4itz6i
361: 2^4: ^B3r5r/A2cn4ce6cn8/S2i4itz6i/D1c3c5c7c
362: 2^4: ^B3r5r/A2cn4ce6cn8/S3r5r/D02cn4ce6cn8
363: 2^4: ^A1c2-ak3acjkr4-jnry5acjkr6-ak7c8
364: 2^4: ^A1c2cn3c4ce5c6cn7c8/D01c2cn3c4ce5c6cn7c8
365: 2^4: ^A1c2cn3c4ce5c6cn7c8/D2i3r4itz5r6i
366: 2^4: ^A1e2-ak3einqy4-jnry5einqy6-ak7e8
367: 2^4: ^A1e2ei3e4ce5e6ei7e8/D01e2ei3e4ce5e6ei7e8
368: 2^4: ^A1e2ei3e4ce5e6ei7e8/D2n3q4qwz5q6n
369: 2^4: ^A2-ak4-jnry6-ak8/D02-ak4-jnry6-ak8
370: 2^4: ^A2-ak4-jnry6-ak8/D1c3acjkr5acjkr7c
371: 2^4: ^A2-ak4-jnry6-ak8/D1e3einqy5einqy7e
372: 2^4: ^A2-ak4-jnry6-ak8/D2ak4jnry6ak
373: 2^4: ^A2468
374: 2^4: ^A2cn3r4ce5r6cn8/D02cn3r4ce5r6cn8
375: 2^4: ^A2cn3r4ce5r6cn8/D1c2i3c4itz5c6i7c
376: 2^4: ^A2ei3q4ce5q6ei8/D02ei3q4ce5q6ei8
377: 2^4: ^A2ei3q4ce5q6ei8/D1e2n3e4qwz5e6n7e
378: 2^4: ^A2i3iy4ceqw5iy6i8/D02i3iy4ceqw5iy6i8
379: 2^4: ^A2i3iy4ceqw5iy6i8/D2en3n4z5n6en
380: 2^4: ^A2i3n4ceqw5n6i8/D02i3n4ceqw5n6i8
381: 2^4: ^A2i3n4ceqw5n6i8/D2en3iy4z5iy6en
382: 2^4: ^A2n3ak4ceit5ak6n8/D02n3ak4ceit5ak6n8
383: 2^4: ^A2n3ak4ceit5ak6n8/D2ci3j4z5j6ci
384: 2^4: ^A2n3j4ceit5j6n8/D02n3j4ceit5j6n8
385: 2^4: ^A2n3j4ceit5j6n8/D2ci3ak4z5ak6ci
386: 2^4: ^A1c2cn3c4ce5c6cn7c8/S01c2cn3c4ce5c6cn7c8
387: 2^4: ^A1e2ei3e4ce5e6ei7e8/S01e2ei3e4ce5e6ei7e8
388: 2^4: ^A2-ak4-jnry6-ak8/S02-ak4-jnry6-ak8
389: 2^4: ^A2cn3r4ce5r6cn8/S02cn3r4ce5r6cn8
390: 2^4: ^A2ei3q4ce5q6ei8/S02ei3q4ce5q6ei8
391: 2^4: ^A2i3iy4ceqw5iy6i8/S02i3iy4ceqw5iy6i8
392: 2^4: ^A2i3n4ceqw5n6i8/S02i3n4ceqw5n6i8
393: 2^4: ^A2n3ak4ceit5ak6n8/S02n3ak4ceit5ak6n8
394: 2^4: ^A2n3j4ceit5j6n8/S02n3j4ceit5j6n8
395: 2^4: ^A2cn3r4ce5r6cn8/S1c2i3c4itz5c6i7c
396: 2^4: ^A2-ak4-jnry6-ak8/S1c3acjkr5acjkr7c
397: 2^4: ^A2ei3q4ce5q6ei8/S1e2n3e4qwz5e6n7e
398: 2^4: ^A2-ak4-jnry6-ak8/S1e3einqy5einqy7e
399: 2^4: ^A2-ak4-jnry6-ak8/S2ak4jnry6ak
400: 2^4: ^A2n3j4ceit5j6n8/S2ci3ak4z5ak6ci
401: 2^4: ^A2n3ak4ceit5ak6n8/S2ci3j4z5j6ci
402: 2^4: ^A2i3n4ceqw5n6i8/S2en3iy4z5iy6en
403: 2^4: ^A2i3iy4ceqw5iy6i8/S2en3n4z5n6en
404: 2^4: ^A1c2cn3c4ce5c6cn7c8/S2i3r4itz5r6i
405: 2^4: ^A1e2ei3e4ce5e6ei7e8/S2n3q4qwz5q6n
406: 2^3: ^B1c3cr5cr7c/A2cin4ceitz6cin8/S02cin4ceitz6cin8/D1c3cr5cr7c
407: 2^3: ^B1c3cr5cr7c/A2cin4ceitz6cin8/S1c3cr5cr7c/D02cin4ceitz6cin8
408: 2^3: ^B1c3cr5cr7c/A2cin4ceitz6cin8/S2e4akqw6e/D3ajk5ajk
409: 2^3: ^B1c3cr5cr7c/A2cin4ceitz6cin8/S3ajk5ajk/D2e4akqw6e
410: 2^3: ^B1e3eq5eq7e/A2ein4ceqwz6ein8/S02ein4ceqwz6ein8/D1e3eq5eq7e
411: 2^3: ^B1e3eq5eq7e/A2ein4ceqwz6ein8/S1e3eq5eq7e/D02ein4ceqwz6ein8
412: 2^3: ^B1e3eq5eq7e/A2ein4ceqwz6ein8/S2c4aikt6c/D3iny5iny
413: 2^3: ^B1e3eq5eq7e/A2ein4ceqwz6ein8/S3iny5iny/D2c4aikt6c
414: 2^3: ^B3ajk5ajk/A2cin4ceitz6cin8/S02cin4ceitz6cin8/D3ajk5ajk
415: 2^3: ^B3ajk5ajk/A2cin4ceitz6cin8/S1c3cr5cr7c/D2e4akqw6e
416: 2^3: ^B3ajk5ajk/A2cin4ceitz6cin8/S2e4akqw6e/D1c3cr5cr7c
417: 2^3: ^B3ajk5ajk/A2cin4ceitz6cin8/S3ajk5ajk/D02cin4ceitz6cin8
418: 2^3: ^B3iny5iny/A2ein4ceqwz6ein8/S02ein4ceqwz6ein8/D3iny5iny
419: 2^3: ^B3iny5iny/A2ein4ceqwz6ein8/S1e3eq5eq7e/D2c4aikt6c
420: 2^3: ^B3iny5iny/A2ein4ceqwz6ein8/S2c4aikt6c/D1e3eq5eq7e
421: 2^3: ^B3iny5iny/A2ein4ceqwz6ein8/S3iny5iny/D02ein4ceqwz6ein8
422: 2^3: ^A1c2cin3cr4ceitz5cr6cin7c8/D01c2cin3cr4ceitz5cr6cin7c8
423: 2^3: ^A1c2cin3cr4ceitz5cr6cin7c8/D2e3ajk4akqw5ajk6e
424: 2^3: ^A1e2ein3eq4ceqwz5eq6ein7e8/D01e2ein3eq4ceqwz5eq6ein7e8
425: 2^3: ^A1e2ein3eq4ceqwz5eq6ein7e8/D2c3iny4aikt5iny6c
426: 2^3: ^A2cin3ajk4ceitz5ajk6cin8/D02cin3ajk4ceitz5ajk6cin8
427: 2^3: ^A2cin3ajk4ceitz5ajk6cin8/D1c2e3cr4akqw5cr6e7c
428: 2^3: ^A2ein3iny4ceqwz5iny6ein8/D02ein3iny4ceqwz5iny6ein8
429: 2^3: ^A2ein3iny4ceqwz5iny6ein8/D1e2c3eq4aikt5eq6c7e
430: 2^3: ^A1c2cin3cr4ceitz5cr6cin7c8/S01c2cin3cr4ceitz5cr6cin7c8
431: 2^3: ^A1e2ein3eq4ceqwz5eq6ein7e8/S01e2ein3eq4ceqwz5eq6ein7e8
432: 2^3: ^A2cin3ajk4ceitz5ajk6cin8/S02cin3ajk4ceitz5ajk6cin8
433: 2^3: ^A2ein3iny4ceqwz5iny6ein8/S02ein3iny4ceqwz5iny6ein8
434: 2^3: ^A2cin3ajk4ceitz5ajk6cin8/S1c2e3cr4akqw5cr6e7c
435: 2^3: ^A2ein3iny4ceqwz5iny6ein8/S1e2c3eq4aikt5eq6c7e
436: 2^3: ^A1e2ein3eq4ceqwz5eq6ein7e8/S2c3iny4aikt5iny6c
437: 2^3: ^A1c2cin3cr4ceitz5cr6cin7c8/S2e3ajk4akqw5ajk6e
438: 2^2: ^B1357/A2468
439: 2^2: ^B1c2ak3acjkr4jnry5acjkr6ak7c/A1e2-ak3einqy4-jnry5einqy6-ak7e8
440: 2^2: ^B1c3acjkr5acjkr7c/A2-ak4-jnry6-ak8/S02-ak4-jnry6-ak8/D1c3acjkr5acjkr7c
441: 2^2: ^B1c3acjkr5acjkr7c/A2-ak4-jnry6-ak8/S1c3acjkr5acjkr7c/D02-ak4-jnry6-ak8
442: 2^2: ^B1c3acjkr5acjkr7c/A2-ak4-jnry6-ak8/S1e3einqy5einqy7e/D2ak4jnry6ak
443: 2^2: ^B1c3acjkr5acjkr7c/A2-ak4-jnry6-ak8/S2ak4jnry6ak/D1e3einqy5einqy7e
444: 2^2: ^B1e2ak3einqy4jnry5einqy6ak7e/A1c2-ak3acjkr4-jnry5acjkr6-ak7c8
445: 2^2: ^B1e3einqy5einqy7e/A2-ak4-jnry6-ak8/S02-ak4-jnry6-ak8/D1e3einqy5einqy7e
446: 2^2: ^B1e3einqy5einqy7e/A2-ak4-jnry6-ak8/S1c3acjkr5acjkr7c/D2ak4jnry6ak
447: 2^2: ^B1e3einqy5einqy7e/A2-ak4-jnry6-ak8/S1e3einqy5einqy7e/D02-ak4-jnry6-ak8
448: 2^2: ^B1e3einqy5einqy7e/A2-ak4-jnry6-ak8/S2ak4jnry6ak/D1c3acjkr5acjkr7c
449: 2^2: ^B2ak4jnry6ak/A2-ak4-jnry6-ak8/S02-ak4-jnry6-ak8/D2ak4jnry6ak
450: 2^2: ^B2ak4jnry6ak/A2-ak4-jnry6-ak8/S1c3acjkr5acjkr7c/D1e3einqy5einqy7e
451: 2^2: ^B2ak4jnry6ak/A2-ak4-jnry6-ak8/S1e3einqy5einqy7e/D1c3acjkr5acjkr7c
452: 2^2: ^B2ak4jnry6ak/A2-ak4-jnry6-ak8/S2ak4jnry6ak/D02-ak4-jnry6-ak8
453: 2^2: ^A12345678
454: 2^2: ^A1c2-ak3acjkr4-jnry5acjkr6-ak7c8/D01c2-ak3acjkr4-jnry5acjkr6-ak7c8
455: 2^2: ^A1c2-ak3acjkr4-jnry5acjkr6-ak7c8/D1e2ak3einqy4jnry5einqy6ak7e
456: 2^2: ^A1e2-ak3einqy4-jnry5einqy6-ak7e8/D01e2-ak3einqy4-jnry5einqy6-ak7e8
457: 2^2: ^A1e2-ak3einqy4-jnry5einqy6-ak7e8/D1c2ak3acjkr4jnry5acjkr6ak7c
458: 2^2: ^A2468/D02468
459: 2^2: ^A2468/D1357
460: 2^2: ^A1c2-ak3acjkr4-jnry5acjkr6-ak7c8/S01c2-ak3acjkr4-jnry5acjkr6-ak7c8
461: 2^2: ^A1e2-ak3einqy4-jnry5einqy6-ak7e8/S01e2-ak3einqy4-jnry5einqy6-ak7e8
462: 2^2: ^A2468/S02468
463: 2^2: ^A2468/S1357
464: 2^2: ^A1e2-ak3einqy4-jnry5einqy6-ak7e8/S1c2ak3acjkr4jnry5acjkr6ak7c
465: 2^2: ^A1c2-ak3acjkr4-jnry5acjkr6-ak7c8/S1e2ak3einqy4jnry5einqy6ak7e
466: 2^1: ^B1357/A2468/S02468/D1357
467: 2^1: ^B1357/A2468/S1357/D02468
468: 2^1: ^B1c2ak3acjkr4jnry5acjkr6ak7c/A1e2-ak3einqy4-jnry5einqy6-ak7e8/S01e2-ak3einqy4-jnry5einqy6-ak7e8/D1c2ak3acjkr4jnry5acjkr6ak7c
469: 2^1: ^B1c2ak3acjkr4jnry5acjkr6ak7c/A1e2-ak3einqy4-jnry5einqy6-ak7e8/S1c2ak3acjkr4jnry5acjkr6ak7c/D01e2-ak3einqy4-jnry5einqy6-ak7e8
470: 2^1: ^B1e2ak3einqy4jnry5einqy6ak7e/A1c2-ak3acjkr4-jnry5acjkr6-ak7c8/S01c2-ak3acjkr4-jnry5acjkr6-ak7c8/D1e2ak3einqy4jnry5einqy6ak7e
471: 2^1: ^B1e2ak3einqy4jnry5einqy6ak7e/A1c2-ak3acjkr4-jnry5acjkr6-ak7c8/S1e2ak3einqy4jnry5einqy6ak7e/D01c2-ak3acjkr4-jnry5acjkr6-ak7c8
472: 2^1: ^A12345678/D012345678
473: 2^1: ^A12345678/S012345678

Of course, not all of these have physical interpretations (can be matched to an agar). They should be sorted through to figure out what cool patterns we can find, as well as perhaps figuring out "canonical names" for each one. The value after the colon is the size of the subspace that the symmetry is attainable in, that's how the list is sorted.

Please don't hesitate to ask questions about this stuff, I know it is very confusing.

[speedydelete]

And finally a side note on replicators. Notice that for B1357/S1357 the rule is its own XOR symmetries, this is also true for the von Neumann replicator rules:

Screenshot 2026-09-07 141313.png (29.76 KiB) Viewed 268 times

Therefore, the rules are also maximally XOR symmetric, because in MAP a rule cannot have more than 512 XOR symmetries without a contradiction happening.

I conjecture that the set of rules which are maximally XOR symmetric is equal to the set of rules where every pattern is a replicator. For this, we have to expand the definition of "replicator" a bit, B/S is a rule where every pattern is a replicator that produces 0 copies of itself, and in B/S012345678 every pattern produces 1 copy of itself.

[erictom333]

And finally a side note on replicators. Notice that for B1357/S1357 the rule is its own XOR symmetries, this is also true for the von Neumann replicator rules:

Screenshot 2026-09-07 141313.png

Therefore, the rules are also maximally XOR symmetric, because in MAP a rule cannot have more than 512 XOR symmetries without a contradiction happening.

I conjecture that the set of rules which are maximally XOR symmetric is equal to the set of rules where every pattern is a replicator. For this, we have to expand the definition of "replicator" a bit, B/S is a rule where every pattern is a replicator that produces 0 copies of itself, and in B/S012345678 every pattern produces 1 copy of itself.

I call these rules linear rules, as any two patterns A and B satisfy the identity R(A) ⊕ R(B) = R(A ⊕ B) (where ⊕ denotes bitwise XOR). There are a total of 512 linear two-state range-1 rules, which can be expressed as the transition-wise XOR of 9 basis rules: the stasis rule (B/S012345678) and the translation rules in each of the 8 directions (the state of a cell depends only on the state of the cell in the opposite direction); XORing multiple of these directional rules yields replication in the specified directions.

[NNlk05]

I call these rules linear rules, *snip*

Parity rules

[erictom333]

I call these rules linear rules, *snip*

Parity rules

I use the term "linear rule" since the property R(a) + R(b) = R(a + b) is called linearity, and because it generalises to more than two states by replacing bitwise XOR with addition modulo n.

[speedydelete]

I call these rules linear rules, *snip*

Very nice work!

Does this idea generalize in more than one way (other than addition modulo state count) to multistate grids?

[erictom333]

I call these rules linear rules, *snip*

Very nice work!

Does this idea generalize in more than one way (other than addition modulo state count) to multistate grids?

Yes, actually: I believe any kind of addition that forms a group can be used. (I'm not sure if it has to be a field, or if being a group is sufficient.) For example, a 2^n state rule could have bitwise XOR of state numbers as its addition.

[erictom333]

A rule R has a generalised gutter symmetry if there is a set of cells G (the generalised gutter) and a symmetry S of the plane except for G such that if at t=0 G is empty and the remainder of the plane has the symmetry S, then at t=1 (and hence for all t) G is empty and the remainder of the plane has the symmetry S. Examples include:

• Gutter symmetry: A02ci4ci6i
• Skew gutter symmetry: A01c2kn3ny4yz5r6i
• Double skew gutter symmetry: A012aikn3cqe4cnyz5er6i
• Diagonal gutter symmetry: A02en4ew6n
• Skew diagonal gutter symmetry: A012ak3akq4q
• Double skew diagonal gutter symmetry: A012aikn3eqr4ejrz5cq6n S1c=S2c=S3c S2a=S3n=S4y S2k=S3y S3i=S4n S3j=S4k S4a=S5j
• 2-fold rotational gutter symmetry: A02in4cez6in8
• 4-fold rotational gutter symmetry: A04ce8
• Orthogonal boundedness: A012ac3i
• Diagonal boundedness: A012ae3a

The black-white reversals of rules with a gutter symmetry yields a corresponding antigutter symmetry, where the generalised gutter will stay completely on forever. An example is S4a5aijn6-in78, where the following pattern will never die regardless of what other cells are on:

Code: Select all

x = 0, y = 0, rule = B3/S4a5aijn6-in78
b2o$4o$4o$b2o!