B35/S23

[confocaloid]

Several glider+sparker interactions selected from manual exploration; maybe somehow someone somewhere could combine them to possibly create more gliders:

Code: Select all

x = 376, y = 101, rule = B35/S23
19b3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o$19bo
2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo
2bo2bo$20bo3bo45bo3bo45bo3bo45bo3bo45bo3bo45bo3bo45bo3bo45bo3bo$21b3o
47b3o47b3o47b3o47b3o47b3o47b3o47b3o12$3o7b3o87b3o7b3o89bo9bo38b3o7b3o
38bo9bo$3o7b3o37b3o7b3o37b4o5b4o38b3o7b3o37bobo7bobo37b2o9b2o37bobo7bo
bo$o2bo5bo2bo37b3o7b3o37b2o2bo3bo2b2o38b3o7b3o36bo3bo5bo3bo35b3o2bo3bo
2b3o86bo11bo$2b3o3b3o39bo2bo5bo2bo36b2o11b2o37bo2bo5bo2bo37b3obo3bob3o
36b3o2b2ob2o2b3o38bob2ob2obo38b2o11b2o$52b3o3b3o37bo4bo5bo4bo38b3o3b3o
41b3o3b3o40bo2b2ob2o2bo37bo2b2o5b2o2bo35b2o11b2o$101bo9bo92bo5bo41b4o
3b4o38bob2o5b2obo40b2o3b2o$97bo6b2ob2o6bo136bo2bo3bo2bo38b2o9b2o40bo5b
o$97bo6b2ob2o6bo136bo2bo3bo2bo38b6ob6o$2b3o3b3o90bo9bo92bo5bo41b4o3b4o
38b6ob6o$o2bo5bo2bo39b3o3b3o37bo4bo5bo4bo38b3o3b3o41b3o3b3o40bo2b2ob2o
2bo38b2o9b2o40bo5bo$3o7b3o37bo2bo5bo2bo36b2o11b2o37bo2bo5bo2bo37b3obo
3bob3o36b3o2b2ob2o2b3o36bob2o5b2obo40b2o3b2o$3o7b3o37b3o7b3o37b2o2bo3b
o2b2o38b3o7b3o36bo3bo5bo3bo35b3o2bo3bo2b3o35bo2b2o5b2o2bo35b2o11b2o$
50b3o7b3o37b4o5b4o38b3o7b3o37bobo7bobo37b2o9b2o39bob2ob2obo38b2o11b2o$
100b3o7b3o89bo9bo38b3o7b3o87bo11bo$301bobo7bobo$302bo9bo40$19b3ob3o43b
3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o43b3ob3o$19bo2bo2bo43bo2b
o2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo43bo2bo2bo$20bo
3bo45bo3bo45bo3bo45bo3bo45bo3bo45bo3bo45bo3bo45bo3bo$21b3o47b3o47b3o
47b3o47b3o47b3o47b3o47b3o7$355b2o$254b2o51bobo47bo$254b2o51b2o46b2o$
255bo52b2o51bo$355b8o$254bo4b3o43bo48bob5o2bo$253bob4o3bo42b2o4b2o46b
2o$163b2o47b2o39bo2bo4bo2bo47bo43bo2b5obo$12b2o98bo50b2o47b3o40bo3b4ob
o92b8o$12b2o46bobo43b5o2b2o48bo47b4o41b3o4bo44b2o48bo$12bo45b2ob2o42bo
4bo4bo91b2o3b3o94b2o52b2o$58bo4b2o45b2o3bo41b3o4bo41bob2o52bo45bobo51b
o$6b3o4bo44bo5bo44b2o4bo40bo3b4obo42bo4b2o47b2o99b2o$5bo3b4obo42b3o4bo
38b3o6b3o39bo2bo4bo2bo40bobo2bobo48b2o$3bo2bo4bo2bo41bobo2bobo38bo4b2o
45bob4o3bo41b2o4bo$3bob4o3bo42bo4b3o39bo3b2o47bo4b3o47b2obo$4bo4b3o43b
o5bo40bo4bo4bo92b3o3b2o$55b2o4bo41b2o2b5o44bo48b4o$5bo51b2ob2o43bo49b
2o48b3o$4b2o51bobo95b2o49b2o$4b2o!

I did not do any extensive searches (either manual or automated). Maybe the p24, p82, p142 oscillators could work too. Then there might be a transparent block.

[Haycat2009]

38P142 (Someone please name it) has R sparks, which will work nicely. 46P6 will work as a middleweight sparkler. 36P82 is a T-nose, so its utility is doubtful.

[Haycat2009]

Would a gun or stable cap/turn for beehive/block fuse be more useful in omniperiodicity?

[confocaloid]

>> omniperiodicity?

If you find a glider gun, it will allow to solve infinitely many oscillator periods (multiples of the period of the gun). Although probably not all of them.
If you find a stable 90-degree glider reflector, that might help to make guns of all sufficiently high periods, as soon as you find in addition a free-standing overunity reaction or an overunity reaction supported by still lives.
If you find a stable glider duplicator with one output glider at 90 degrees to the input glider (allowing to connect four duplicators arranged in a square-shaped formation) and the other output glider either 90 degrees in the other direction or 0 degrees, then the game is mostly over for all but finitely many periods.

Of course any of this is a tall order for a cellular automaton like this, where the most common glider (glider 3736 in this case) is fairly hard to obtain naturally out of random reactions. Likely possible, just not particularly easy, compared to cellular automata with very common gliders.

>> 46P6 will work as a middleweight sparkler

I think the correct term is 'sparker'.

[confocaloid]

Apgcodes of n-bit strict still lives for n < 25:

b35s23-still-4to24-apgcodes.txt.zip

apgcodes of n-bit strict still lives for n < 25

(2.03 MiB) Downloaded 94 times

Since the birth conditions of this CA form a superset of birth conditions of Life and the survival conditions of this CA form a subset of survival conditions of Life, every still life in this CA remains a still life in Life (but not vice versa). I filtered the data from CGoL-still-life-enumeration to leave only objects that remain still lives in b35s23.

Code: Select all

population 4 : 2/2
population 5 : 0/1
population 6 : 4/5
population 7 : 1/4
population 8 : 7/9
population 9 : 4/10
population 10 : 12/25
population 11 : 17/46
population 12 : 52/121
population 13 : 64/240
population 14 : 178/619
population 15 : 281/1353
population 16 : 680/3286
population 17 : 1193/7773
population 18 : 2877/19044
population 19 : 5241/45759
population 20 : 12154/112243
population 21 : 23276/273188
population 22 : 52696/672172
population 23 : 103780/1646147
population 24 : 232374/4051732

[Haycat2009]

What are the last 2 13-cell still lives to not have appeared naturally?

[confocaloid]

Update (2024-03-25): all 13-bit still lives are in b35s23/C1. The following 14-bit still lives are not yet in C1:

Code: Select all

< xs14_032qkzc93
< xs14_0358gz2lp1
< xs14_0358gz8ka221
< xs14_08kai3z6221
< xs14_0g4q23zbc1
< xs14_0ggo8b5z56
< xs14_0j5oz3kp
< xs14_0j9c826z121
< xs14_0o4871zbd
< xs14_0o4871zc452
< xs14_1784ozx34a4
< xs14_178k46z311
< xs14_178ka248c
< xs14_25icz69k8
< xs14_31248gzy21248go
< xs14_3146o8b5
< xs14_32qkzw254c
< xs14_358gzw116io
< xs14_358gzw11mq
< xs14_39cggmazx1
< xs14_3ia4zw11da
< xs14_3lozx3lo
< xs14_4ai3zwo9a4
< xs14_5b8og8426
< xs14_6421eozx65
< xs14_64k871z65
< xs14_8e1248gzy31246
< xs14_dbgzw1qm
< xs14_dj8gzx3426
< xs14_g8ka23z1ac
< xs14_ggc4871z56
< xs14_o4a5123z23
< xs14_wg853zgb6z11
< xs14_wg853zmdz11
< xs14_wo8b5zbd
< xs14_xog853zc93

png.png (103.08 KiB) Viewed 6777 times

[Haycat2009]

What search program should I use to find stable endings for the reburnable fuse? (bump)

[silversmith]

What search program should I use to find stable endings for the reburnable fuse? (bump)

Depending on how big the fuse is, you could use LLS.

[confocaloid]

What search program should I use to find stable endings for the reburnable fuse? (bump)

Depending on how big the fuse is, you could use LLS.

Here is the reburnable fuse, quoted from an earlier post. (Note: it burns the same way in several other Life-like CA; the neighbour-counting conditions B6, B7, B8, S0, S8 are optional.)

What the...?

Code: Select all

x = 54, y = 4, rule = B35/S23
52bo$2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2obobo$2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2obobo$52bo!

With enough effort, maybe an universal toolkit could be found for stationary periodic circuitry (with some base period). Such a toolkit would likely include several glider reflectors (one may be insufficient to account for all combinations of space/time parities), glider duplicators and/or over-unity reactions, "tee"-type glider collisions that could be used to form salvos, etc.

Once that becomes available, one could then use the following reaction to send one signal. (Which is carried by three consecutive pulses. A single pulse changes locations of the blocks; therefore three pulses must be sent, to be able to actually make this thing into a wire and use it repeatedly for communication.)

Code: Select all

x = 210, y = 48, rule = B35/S23
20b3ob3o$20bo2bo2bo$21bo3bo$22b3o7$10b3ob3o$10bo2bo2bo$11bo3bo$12b3o7$
3ob3o$o2bo2bo$bo3bo$2b3o13$207b2o$207b2o4$4b2ob2ob2ob2ob2ob2ob2ob2ob2o
b2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2o$4b2ob2ob2ob2ob2ob2ob2ob2ob2o
b2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2o4$207b2o$207b2o!

The above three-pulse reaction has the total effect of shortening the transmitting end of the block chain by 10 blocks. Then those ten blocks would have to be rebuilt again, to ensure reusability. The following reaction shows that three gliders suffice to rebuild one block:

Code: Select all

x = 96, y = 84, rule = B35/S23
5b3ob3o$5bo2bo2bo$6bo3bo$7b3o6$19b2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2o$19b2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2o12$11bo$10bobo$9b
o3bo$9bo3bo$9bobobo$11bo$10bobo49$2bo$bobo$o3bo$o3bo$obobo$2bo$bobo!

With the above reactions, 33 gliders would suffice to send one signal.
The above handwaving relies on the assumption that there's some way to create those 33 gliders in a repeatable way from just a single incoming glider. (And no idea how hard it would be to receive that signal in a reusable way at the other end of the block chain. A completed demo pattern is likely to end up being fairly large, both in terms of the bounding box area and in terms of the population.)

[WhiteHawk]

This seems to be the most common pulsar predecessor in this rule - appearing even in the pulsar soup already in the forums. Contrary to the Wiki, it is a "new" predecessor in the sense it fails in life and relies on B5 to produce a pulsar.

Code: Select all

x = 7, y = 4, rule = B35/S23
bo$o3b3o$o2b2o$2o2b3o!

Code: Select all

x = 7, y = 4, rule = B3/S23
bo$o3b3o$o2b2o$2o2b3o!

Is there another pulsar predecessor in this rule?

EDIT: Yup. Requires the blinker/block at the top or dies completely. Still seems the top predecessor is far more common.

Code: Select all

x = 54, y = 16, rule = B35/S23
6ob2o4bo24bo2bo3b2obobo$2o2b3o2bo2bo25bob3o3b2o2b2o$bo2b2ob2o2b2obo23b
3ob3o2bo2b2obo$bob2obobo3bobo24bobobo3bo4b2o$2o2bobob2o2bob2o22bo4b2o
bobobob2o$2bob3obo2b2ob2o23bob2ob6o$2b2o4bo2bobo28b2obob2o2b3o$b2obo3b
o3b4o22bo2b9o2b2o$obo2b2o7bo23bobo3b5o3bo$2ob3o2b3ob2obo22b4ob2obobob
ob2o$bobob2ob8o22bobobobo2bo2bobo$3obo5b2obo25b2o2bo2b4ob2o$b2obobob2o
b3o24bo2bo2bo2b2obob2o$bobo2b3o2b2o30b3obobo2bo$bo3b6obo2bo24b4o2b2ob
5o$o2b2ob5o2b3o22b2obo4b2o4b2o!

EDIT 2: these three predecessors may seem different, but they both converge to the same sequence as the original example given above

Code: Select all

x = 55, y = 54, rule = B35/S23
bo2b4obobobo29b10obo$bob2ob2o2b3o2bo23b4o2bobo4b2o$bobo5b4ob2o27bo4b3o
2b2o$b2obo5b3obo24b2obobo2bo3bob2o$3b3o4bob2o26b5o2b4o3bo$bo2b3o2bobo
3bo23bo2bobo2b2ob2ob2o$3ob2obo3bob3o23bobo2bo4bo3bo$4ob3o2bobobo24b3o
b4ob2o2b3o$4bo4bob2obo25b6o4bo2b2o$bobob3ob2obob2o27b3o2b3o$4o2bobo6b
o23bobobo7b2obo$b2o2b5o2b2obo24b3obob5o2bo$o4bo2b2o4b2o23b2ob8o4bo$2b
2obob2ob6o23b6ob2o3b4o$2obobo2bob2o2bo24b3ob3o5bo$o2bo2b2obobo27b3o2b
2ob2ob5o23$2obob2o3bobo2bo$bo3bo3bo2bo$4bob4o3bobo$bo4b5o2b3o$2o2b2ob
4ob3o$2bo4b2o3bob2o$o4b2ob2obobo$obob2ob2o2b2o$bo2bobobob3o$3o7bo2bo$
2b3o2bo4bobo$obo2b4obobo2bo$b2obobob2o$b2o3bob4o3bo$bo3bo2b2obob3o$4o
3bo2bo2bo!

Also, the last phase of the predecessor which triggers B5 (t=59)

Code: Select all

x = 9, y = 21, rule = B35/S23
4bo$4bo$3b3o$4bo$4bo2$b2o3b2o$o2bobo2bo$b2o3b2o4$b2o3b2o$o2bobo2bo$b2o
3b2o2$4bo$4bo$3b3o$4bo$4bo!

[Sylvani]

I call these the happy dog space ships. They resemble a dog wagging its tail while prancing.

Code: Select all

x = 27, y = 26, rule = B35/S23
2b3o7b3o7b3o$bo3bo5bo3bo5bo3bo$o5bo3bo5bo3bo5bo$o5bo3bo5bo3bo5bo$2b3o
7b3o7b3o$2o3b2o3b2o3b2o3b2o3b2o2$2obob2o3b2obob2o3b2obob2o$o5bo3bo5bo
3bo5bo$3bo9bo9bo$obobobo3bobobobo3bobobobo$ob3obo3bob3obo3bob3obo$bobo
bo5bobobo5bobobo$2o3b2o3b2o3b2o3b2o3b2o$b2ob2o5b2ob2o5b2ob2o$bob3o5bob
3o5bob3o$4b2o8b2o8b2o$obo2bo4bobo2bo4bobo2bo$b2o8b2o8b2o$2bo2bo6bo2bo
6bo2bo$2bo9bo9bo$2bo9bo9bo$2bo9bo3bo5bo3bo$3bobo6bo9bo3bo$13bobo6bo$
23bobo!

Their tails can be extended in the same way as small, medium, and heavy weight spaceships.

[WhiteHawk]

Just noticed a "new" seminatural p26 oscillator (the third in this rule) from D4+2 symmetry in Catagolue.

Code: Select all

x = 16, y = 15, rule = B35/S23
3o10b3o$b3o8b3o5$3bo2bo2bo2bo$3bo2bo2bo2bo$3bo2bo2bo2bo5$b3o8b3o$3o10b
3o!

EDIT: Some low-period semi-natural oscillators which are not in the stamp collection on the wiki (Why is washing machine in the stamp collection, it works in life)

Code: Select all

x = 152, y = 62, rule = B35/S23History
10.2A$.2D3.2A.A2.A$D2.D2.2A.A2.A$3.D6.2A$.2D3.2A$3.D.A2.A.2A$D2.D.A2.
A.2A103.2A.2A$.2D3.2A108.A.A$95.B2.B4.B2.B9.A.A$33.A43.4A3.4A6.B4AB2.
B4AB7.2A.2A26.A$3.D29.A42.A.3A3.3A.A5.A.3A4.3A.A8.A.A27.2A$2.2D29.2A12.
A.2A.2A.A21.4A3.4A6.B4A4.4AB7.2A.2A26.A$.D.D4.A.A6.2A15.2A10.A.A.A.A.
A.A5.3A.A.3A8.2A3.2A11.2A4.2A9.A.A.A.A5.3A4.2A14.3A$D2.D2.4A.A.A4.A.A
2.2A4.3A3.A9.A4.A.A4.A4.2A2.A2.2A7.3A3.3A9.3A4.3A8.A5.A8.2A2.A10.A5.A
$5D.A.A.4A4.A.A2.A.A5.2A4.2A6.A4.A.A4.A4.2A2.A2.2A7.3A3.3A9.3A4.3A8.A
5.A6.A.3A.3A.A5.3A$3.D5.A.A7.2A2.A.A6.2A2.3A7.A.A.A.A.A.A5.3A.A.3A8.2A
3.2A10.3A4.3A8.A.A.A.A10.A2.2A12.A$3.D21.2A8.A2.A8.A.2A.2A.A21.4A3.4A
9.2A4.2A10.2A.2A10.2A4.3A8.2A$34.2A40.A.3A3.3A.A5.B4A4.4AB8.A.A29.A$34.
3A40.4A3.4A6.A.3A4.3A.A7.2A.2A$94.B4AB2.B4AB8.A.A$7.A87.B2.B4.B2.B9.A
.A$6.A.A2.2A7.3A.3A88.2A.2A$4D3.A.A2.A8.2A.2A$D7.A8.3A7.3A$D16.A4.A.A
4.A$.2D19.A.A$3.D4.A13.A.A$D2.D3.A.A2.A4.A4.A.A4.A$.2D3.A.A2.2A4.3A7.
3A$7.A13.2A.2A$20.3A.3A3$.2D9.3A$D2.D8.3A$D13.3A$3D14.A$D2.D13.3A$D2.
D10.A$.2D3.2A5.2A$6.2A4.A$6.3A2.2A$8.A$8.A$9.2A$10.A$10.A3$23.A.A$.2D
4.A3.A10.A2.A$D2.D3.A3.A9.A2.A$D2.D2.2A3.2A11.A17.3A$.2D4.2A.2A8.A3.A
19.3A$D2.D4.A.A6.A2.A3.A6.A4.2A2.A5.A2.2A$D2.D13.A2.A3.A6.A4.2A2.A5.A
2.2A$.2D4.A3.A5.A2.A3.A6.A8.3A$7.A3.A8.A3.A17.3A$7.A3.A12.A$21.A2.A$22.
A2.A$23.A.A!

[confocaloid]

I think these are all highest-population still-life patterns in square-shaped regions up to 11-by-11:

Code: Select all

x = 74, y = 110, rule = B35/S23
2o$2o6$2o$obo$b2o6$b2o$o2bo$o2bo$b2o6$2ob2o$2ob2o2$2ob2o$2ob2o6$2ob2o
5b2ob2o$2obobo4b2ob2o$4b2o$10b2ob2o$2o2b2o4b2obobo$2o2b2o8b2o6$o2bo10b
o$6o6b5o$6bo4bo5bo$7o4b7o$o$b6o4b7o$3bo2bo4bo2bo2bo6$2ob2ob2o$2ob2ob2o
2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o6$2ob2ob2o5b2ob2ob2o12b2o4b2ob2o
2b2o4b2ob2o2b2o4b2o2b2ob2o$2ob2ob2o5b2ob2obobo4b2ob2obobo4b2ob2o2b2o4b
2ob2o3bo4b2o2b2ob2o$20b2o4b2ob2ob2o25bo$2ob2ob2o31b2ob2o2b2o4b2ob2o2b
2o8b2ob2o$2ob2obobo4b2ob2o2b2o4b2ob2ob2o5b2obobo2bo4b2obobobo5b2obobob
2o$7b2o4b2ob2o2b2o4b2ob2ob2o9b2obo9b2o2bo4b2ob2o$46b2o10bobo$2ob2o2b2o
4b2ob2o2b2o4b2ob2ob2o5bob3obo6bob3o2bo5b2ob2o2b2o$2ob2o2b2o4b2ob2o2b2o
4b2ob2ob2o5b2obob2o6b2obob2o6b2ob2o2b2o6$2ob2o2b2o5b2o2b2o2b2o5b2ob2o
8b2ob2o2b2o$2obo2bo2bo4bo2bo2bo2bo4bobobobob2o5bobo3b2o$4bob4o5bob4obo
5bobobobob2o4bo3bo$2ob2o9b2o6b2o5b2ob2o8b2ob2o2b2o$bobo2b4o4bo2b4o2bo
22bobo2bo$o3bobo2bo5bobo2bobo5bob2ob3obo4b2obo2b4o$2ob2o2b2o5b2o2b2o2b
2o4b2ob2obob2o4b2ob2o$4bo10bo6bo23bob4o$2obo3b2o5bo3b2o3bo4b2ob2obob2o
4b2obo2bo2bo$2ob2o2b2o5b2o2b2o2b2o4bob2ob2ob2o4b2ob2o2b2o6$2ob2ob2ob2o
$2ob2ob2ob2o2$2ob2ob2ob2o$2ob2ob2ob2o2$2ob2ob2ob2o$2ob2ob2ob2o2$2ob2ob
2ob2o$2ob2ob2ob2o!

Here are lower bounds for 12x12 and 13x13 (possibly suboptimal):

Code: Select all

x = 13, y = 30, rule = B35/S23
o2bobo2bob2o$4ob4ob2o2$4ob4ob2o$o2bobo2bob2o$6b2o$4b2o$2obo2bobo2bo$2o
b4ob4o2$2ob4ob4o$2obo2bobo2bo6$o2bo2bo2bo2bo$13o2$13o$o11bo$b11o2$b11o
$o11bo$13o2$13o$o2bo2bo2bo2bo!

I used Logic Life Search with kissat to find all of these.

[Naszvadi]

I think these are all highest-population still-life patterns in square-shaped regions up to 11-by-11:

Code: Select all

x = 74, y = 110, rule = B35/S23
2o$2o6$2o$obo$b2o6$b2o$o2bo$o2bo$b2o6$2ob2o$2ob2o2$2ob2o$2ob2o6$2ob2o
5b2ob2o$2obobo4b2ob2o$4b2o$10b2ob2o$2o2b2o4b2obobo$2o2b2o8b2o6$o2bo10b
o$6o6b5o$6bo4bo5bo$7o4b7o$o$b6o4b7o$3bo2bo4bo2bo2bo6$2ob2ob2o$2ob2ob2o
2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o6$2ob2ob2o5b2ob2ob2o12b2o4b2ob2o
2b2o4b2ob2o2b2o4b2o2b2ob2o$2ob2ob2o5b2ob2obobo4b2ob2obobo4b2ob2o2b2o4b
2ob2o3bo4b2o2b2ob2o$20b2o4b2ob2ob2o25bo$2ob2ob2o31b2ob2o2b2o4b2ob2o2b
2o8b2ob2o$2ob2obobo4b2ob2o2b2o4b2ob2ob2o5b2obobo2bo4b2obobobo5b2obobob
2o$7b2o4b2ob2o2b2o4b2ob2ob2o9b2obo9b2o2bo4b2ob2o$46b2o10bobo$2ob2o2b2o
4b2ob2o2b2o4b2ob2ob2o5bob3obo6bob3o2bo5b2ob2o2b2o$2ob2o2b2o4b2ob2o2b2o
4b2ob2ob2o5b2obob2o6b2obob2o6b2ob2o2b2o6$2ob2o2b2o5b2o2b2o2b2o5b2ob2o
8b2ob2o2b2o$2obo2bo2bo4bo2bo2bo2bo4bobobobob2o5bobo3b2o$4bob4o5bob4obo
5bobobobob2o4bo3bo$2ob2o9b2o6b2o5b2ob2o8b2ob2o2b2o$bobo2b4o4bo2b4o2bo
22bobo2bo$o3bobo2bo5bobo2bobo5bob2ob3obo4b2obo2b4o$2ob2o2b2o5b2o2b2o2b
2o4b2ob2obob2o4b2ob2o$4bo10bo6bo23bob4o$2obo3b2o5bo3b2o3bo4b2ob2obob2o
4b2obo2bo2bo$2ob2o2b2o5b2o2b2o2b2o4bob2ob2ob2o4b2ob2o2b2o6$2ob2ob2ob2o
$2ob2ob2ob2o2$2ob2ob2ob2o$2ob2ob2ob2o2$2ob2ob2ob2o$2ob2ob2ob2o2$2ob2ob
2ob2o$2ob2ob2ob2o!

Here are lower bounds for 12x12 and 13x13 (possibly suboptimal):

Code: Select all

x = 13, y = 30, rule = B35/S23
o2bobo2bob2o$4ob4ob2o2$4ob4ob2o$o2bobo2bob2o$6b2o$4b2o$2obo2bobo2bo$2o
b4ob4o2$2ob4ob4o$2obo2bobo2bo6$o2bo2bo2bo2bo$13o2$13o$o11bo$b11o2$b11o
$o11bo$13o2$13o$o2bo2bo2bo2bo!

I used Logic Life Search with kissat to find all of these.

I wrote two GNU mathprog models, both looks for a maxpop still life in a given square.
I can confirm these small values for B35/S23:

Code: Select all

#C FOUND A 2 by 2 SOLUTION WITH POPULATION 4
OO
OO
Model has been successfully processed
#C FOUND A 3 by 3 SOLUTION WITH POPULATION 6
OO.
O.O
.OO
Model has been successfully processed
#C FOUND A 4 by 4 SOLUTION WITH POPULATION 8
.OO.
O..O
O..O
.OO.
Model has been successfully processed
#C FOUND A 5 by 5 SOLUTION WITH POPULATION 16
OO.OO
OO.OO
.....
OO.OO
OO.OO
Model has been successfully processed
#C FOUND A 6 by 6 SOLUTION WITH POPULATION 18
OO..OO
OO..OO
......
....OO
OO.O.O
OO.OO.
Model has been successfully processed

[confocaloid]

I think these are all highest-population still-life patterns in square-shaped regions up to 11-by-11:

Code: Select all

x = 74, y = 110, rule = B35/S23

Here are lower bounds for 12x12 and 13x13 (possibly suboptimal):

Code: Select all

x = 13, y = 30, rule = B35/S23
o2bobo2bob2o$4ob4ob2o2$4ob4ob2o$o2bobo2bob2o$6b2o$4b2o$2obo2bobo2bo$2o
b4ob4o2$2ob4ob4o$2obo2bobo2bo6$o2bo2bo2bo2bo$13o2$13o$o11bo$b11o2$b11o
$o11bo$13o2$13o$o2bo2bo2bo2bo!

I used Logic Life Search with kissat to find all of these.

I ran LLS further to confirm that 68 bits is the maximum possible for 12-by-12, but I don't yet have a full list of all solutions.

Code: Select all

68
Getting search pattern...
Done

Preprocessing...
Done

Width: 14
Height: 14
Duration: 2

Number of undetermined cells: 144
Number of variables: 1177
Number of clauses: 41774

x = 14, y = 14, rule = B35/S23
bbbbbbbbbbbbbb$
bboobbbboobbob$
bobbobobboooob$
booooboobbbbbb$
bbbbbbboboooob$
boooobobbobbob$
bobboboobbbbbb$
bbbbbbbobobbob$
bobbobobboooob$
booooboobbbbbb$
bbbbbbboboooob$
boooobobbobbob$
bobboboobboobb$
bbbbbbbbbbbbbb!


Total solver time: 108.49493455886841
69
Getting search pattern...
Done

Preprocessing...
Done

Width: 14
Height: 14
Duration: 2

Number of undetermined cells: 144
Number of variables: 1179
Number of clauses: 41794

Unsatisfiable

Total solver time: 29856.561425209045

[eRroR_6o6]

so I was trying to complete this p12 partial when I accidentally discovered a p13

Code: Select all

x = 40, y = 38, rule = B35/S23
2$22b2o$21bobo$21b2o2$19bo$19bo$19bo$19bo$19bo$19bo$16bo5bo$15bobo3bob
o$3b2o9bobo5bobo$3bobo7bobo7bobo$4b2o8bo9bo2$7b6o13b6o2$14bo9bo8b2o$
13bobo7bobo7bobo$14bobo5bobo9b2o$15bobo3bobo$16bo5bo$19bo$19bo$19bo$
19bo$19bo$19bo2$16b2o$15bobo$15b2o!

If you're wondering, the p12 partial in question:

Code: Select all

x = 39, y = 37, rule = B35/S23
$24bo$23bobo$22bobo$21bobo$22bo$19bo$19bo$19bo$19bo$19bo$19bo$3bo12bo
5bo$2bobo10bobo3bobo$3bobo8bobo5bobo$4bobo6bobo7bobo$5bo8bo9bo2$7b6o
13b6o2$14bo9bo8bo$13bobo7bobo6bobo$14bobo5bobo8bobo$15bobo3bobo10bobo$
16bo5bo12bo$19bo$19bo$19bo$19bo$19bo$19bo$16bo$15bobo$14bobo$13bobo$
14bo!

(something funny is that this would be a rephasing of the p11 and the p13 I found would be a rephasing of that, making this oscillator a rephasing of a rephaser)

[confocaloid]

I think these are all highest-population still-life patterns in square-shaped regions up to 11-by-11: [...]
Here are lower bounds for 12x12 and 13x13 (possibly suboptimal): [...]

I ran LLS further to confirm that 68 bits is the maximum possible for 12-by-12, but I don't yet have a full list of all solutions. [...]

I ran Logic Life Search further to find all 68-bit still-life patterns in the 12-by-12 square-shaped region. I believe these are all solutions:

Code: Select all

#C 29 unique apgcodes.
#C stamp collection:
#CLL state-numbering golly
x = 460, y = 12, rule = B35/S23
b2o3b2o2b2o5b2ob2ob2ob2o5b2o6b2o5bo2b2o4b2o5bo2bobo2bob2o5b2o6b2o
6b2ob2ob2ob2o5b2ob2ob2ob2o5b2o6b2o6b2ob2ob2ob2o4bo2bobob2ob2o4b2ob
2o2b2ob2o5b2ob2ob2ob2o4b2ob2ob2ob2o5b2ob2o2b2ob2o5b2o3b2o8b2ob2o2b
2ob2o4b2obo2bobob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob
2o2b2ob2o4b2ob2o2b2ob2o4b2ob2obobo2bo4b2ob2o2b2ob2o4b2ob2o2b2ob2o
4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o$o2bobo2bo2bo5b2obo3bobo
5bo2bob2obo2bo4b4o2bobo2bo4b4ob5obo4bo2bob2obo2bo4bobobo3bobo6b2ob
o3bob2o4bo2bob2obo2bo4bobobo3bobo5b4ob2obob2o4bo3bo2b2ob2o4bobobo
3bobo6bob2ob2ob2o6bobo2bobobo5bo2bobo2bob2o5bobo3b2ob2o4b2ob4ob2ob
o5bobo3b2ob2o5bob2o3bobo5b2ob2o2bo3bo5bob2o3bobo6bob2o3bobo5b2obob
2ob4o4bo3bo2b2obo6bobo2bobobo5b2obo2bobobo6bobo2bobobo5bo3bo2b2obo
$4ob4obo10bobo3bo4b4ob2ob4o9b2ob4o20b4ob2ob4o4bobobo3bobo10bobo8b
4ob2ob4o4bobobo3bobo22bobo12bobobo3bobo5bo15bo3bobobobo5b4ob4ob2o
5bobo29bobo12bo7bobo13bobo5bo7bobo5bo7bobo22bobo7bo4bo3bobobobo9bo
bobobo5bo3bobobobo6bobo7bo$10b2o5b2ob2ob2ob2o20b4obo10b4ob5obo21b
2ob2ob2ob2o5b2ob2ob2ob2o21b2ob2ob2ob2o4b4ob2obob2o4b2ob2o2b2ob2o5b
2ob2ob2ob2o4b2ob2ob2ob2o5b2ob2o2b2ob2o20b2ob2o2b2ob2o4b2ob4ob2obo
4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b
2ob2o4b2obob2ob4o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o
2b2ob2o4b2ob2o2b2ob2o$4ob4o2bo4bobobo3bobo5b4ob2ob4o4bo2bo2bob4o4b
o2bobo2bob2o4b4ob2ob4o12bobo5bobobo3bobo5b4ob2ob4o20bo2bobob3obo
10bobobo10bobo3bo5bobo3bobobo8bo6bo4b4ob4ob2o10bobobo5b2obo2bobob
2o10bobobo6bobobo11bobobo11bobobo11bobobo10bob3obobo2bo10bobobo9bo
6bo4bo3bo15bo17bobobo$o2bobo2bobo5bobobo3bobo5bo2bob2obo2bo9b2obo
2bo20bo2b2obobo2bo5b2ob2obo3bo4bobobo3bobo5bo2bobob2o2bo5b2ob2ob2o
b2o5b2o13b2ob2obobobo6b2obo3bobo6bobo3bobobo4b2obo3b2obo5bo2bobo2b
ob2o4b2ob2obobobo9b2o10b2ob2obobobo6bobobob2ob2o5bobobob2ob2o5bobo
bob2ob2o5bobobob2ob2o13b2o5b2ob2obobobo5b2obo3b2obo6bobo3b2ob2o4b
2obo3b2ob2o4b2ob2obobobo$10b2o5b2ob2ob2ob2o20bo2bobo10bo2bobo2bob
2o21b2ob2ob2ob2o5b2ob2ob2ob2o21b2ob2ob2ob2o13b2o6bobobob2ob2o5b2ob
2ob2ob2o4b2ob2ob2ob2o5b2ob2o3bob2o21bobobob2ob2o10b2o9bobobob2ob2o
4b2ob2obobobo5b2ob2obobobo5b2ob2obobobo5b2ob2obobobo6b2o14bobobob
2ob2o5bob2o3bob2o4b2ob2o3bobo5b2ob2o3bob2o5bobobob2ob2o$o2bobo2bo
2bo20bo2bob2obo2bo4b4o2bobo2bo4b4ob4ob2o4bo2b2obobo2bo36bo2bobob2o
2bo20bob3obobo2bo5bobobo49bo8bo2bobo2bob2o5bobobo10b2obobo2bob2o5b
obobo16bobobo11bobobo11bobobo11bobobo5bo2bobob3obo5bobobo10bo6bo
15bo3bo11bo9bobobo$4ob4obo6b2ob2ob2ob2o4b4ob2ob4o9b2ob4o20b4ob2ob
4o5b2ob2ob2ob2o5b2ob2ob2ob2o4b4ob2ob4o5b2ob2ob2ob2o4b2obob2ob4o4b
2ob2o2b2ob2o5b2ob2ob2ob2o4b2ob2ob2ob2o5b2ob2o2b2ob2o4b4ob4ob2o4b2o
b2o2b2ob2o4bob2ob4ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b
2ob2o2b2ob2o4b2ob2o2b2ob2o4b4ob2obob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o
4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o$10b2o4bobobo3bobo21b4ob
o10b4ob4ob2o20bobobo3bobo5bobobo3bobo21bobobo3bobo29bobo5bobobo3bo
bo6bobo3bobobo5bobobobo3bo28bobo21bo7bobo6bobo7bo5bobo13bobo13bobo
28bo7bobo6bobobobo3bo5bobobobo9bobobobo3bo4bo7bobo$4ob4o2bo4bobobo
3bobo5b4ob2ob4o4bo2bo2bob4o4bo2bobo2bob2o4b4ob2ob4o4bobobo3bobo5bo
bobo3bobo5b4ob2ob4o4bobobo3bobo5b2obob2ob4o4b2ob2o2bo3bo4bobobo3bo
bo6bobo3bobobo5bobobo2bobo5b4ob4ob2o4b2ob2o3bobo5bob2ob4ob2o5bob2o
2bo3bo5bobo3b2obo6bobo3b2ob2o5bobo3b2ob2o4bo3bo2b2ob2o4b4ob2obob2o
5bob2o3bobo6bobobo2bobo6bobobo2bob2o5bobobo2bobo6bob2o2bo3bo$o2bob
o2b2o7b2ob2ob2ob2o4bo2bobob2o2bo5b2o2b2obo2bo5b2o3b2o8bo2bobob2o2b
o5b2ob2ob2ob2o5b2ob2ob2ob2o4bo2bob2obo2bo5b2ob2ob2ob2o4bob2obob2o
2bo4b2ob2o2b2ob2o5b2ob2ob2ob2o4b2ob2ob2ob2o5b2ob2o2b2ob2o4bo2bobo
2bob2o4b2ob2o2b2ob2o4b2obobo2b2obo4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2o
b2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4bo2bob2ob2obo4b2ob2o2b2ob2o
4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o4b2ob2o2b2ob2o!

[Sylvani]

How many spaceship speeds (or oscillator periods) have been proven to not exist in both B35/S23 and B3/S23? Ones I know off the top of my head are c/2, c/3, c/4, 2c/4, c/5, and 2c/5.

[confocaloid]

How many spaceship speeds (or oscillator periods) have been proven to not exist in both B35/S23 and B3/S23? Ones I know off the top of my head are c/2, c/3, c/4, 2c/4, c/5, and 2c/5.

I found a forum post that seems to state a "there are no shared spaceships at this speed+period combination" result for orthogonal c/2, c/3, 2c/4, and 2c/5.

However, I can't seem to find where the orthogonal c/5 and c/4 cases were handled, can you add references to sources? For those cases, all I see in this thread is mentions of incomplete searches (e.g. "up to 60G").

[...] When such a search terminates with no results it shows, barring bugs, that there are no (finite) ships at all. This was easily and quickly the case for c/2, c/3, 2c/4, and 2c/5. [...]

[amling]

I wrote some more tricky hacks for LLSSS to be able to search for ships that share only a single phase. Performance is perhaps not so great, but I was able to extend the results here. Barring bugs I believe I have shown there are no ships shared between B3/S23 and B35/S23 for c/2, c/3, or 2c/4, even allowing to share only a single phase.