Unproven conjectures

[d/dx]

A phoenix is a pattern where every living cell dies in the next generation, i.e. lasts exactly one generation.
Do there exist oscillators where every living cell lasts exactly 2 generations? 3? 4? Any number?

[wwei47]

Do there exist oscillators where every living cell lasts exactly 2 generations? 3? 4? Any number?

I know of this for P3:

Code: Select all

x = 7, y = 4, rule = B35/S136
3o$bo2bo$2bo2bo$4b3o!

[confocaloid]

A phoenix is a pattern where every living cell dies in the next generation, i.e. lasts exactly one generation.
Do there exist oscillators where every living cell lasts exactly 2 generations? 3? 4? Any number?

For periods above 3, there are at least two different puzzles here:
(a) period-n oscillators where every living cell is alive for exactly k consecutive generations
(b) period-n oscillators where every living cell is alive for exactly k out of n generations (not necessarily consecutive)

(n > 3, k > 1, n - k > 1)

[wwei47]

Did we ever figure out what was going on here?

Code: Select all

x = 10, y = 25, rule = B246/S024
2bo5bo$2o7bo$2bo2$9bo3$6bo$6o$6bo6$3bo$3o3bo$3bo5$5bo$5o$5bo!

Code: Select all

x = 20, y = 50, rule = B02468/S0246
4b2o10b2o$4b2o10b2o$4o14b2o$4o14b2o$4b2o$4b2o3$18b2o$18b2o5$12b2o$12b
2o$12o$12o$12b2o$12b2o11$6b2o$6b2o$6o6b2o$6o6b2o$6b2o$6b2o9$10b2o$10b
2o$10o$10o$10b2o$10b2o!

Edit: Block emulation CAN break down in some cases here, but it's not the usual Margolus stuff and the fact that it seems to be emulating another OT rule (or close to it) is interesting.

[Azerty]

Conjecture: The growth rate of the function f(n) is exponential or even faster.
f(n) return the biggest non-infinite final population obtainable from a starting pattern that fit in a nxn bounding box.

[d/dx]

A phoenix is a pattern where every living cell dies in the next generation, i.e. lasts exactly one generation.
Do there exist oscillators where every living cell lasts exactly 2 generations? 3? 4? Any number?

The four center cells of gourmet are a demiphoenix 16/32:

Code: Select all

x = 20, y = 20, rule = LifeHistory
10.2A$10.A$4.2A.2A.A4.2A$2.A2.A.A.A5.A$2.2A3.BA8.A$6.4B6.2A$6.6B.2B$
7.9B2A$A5.4B3A2BA.A$3A3.3BDCBA2B.A$3.A.4BDCBAB3.3A$2.A.A9B5.A$2.2A9B$
5.2B.6B$2.2A6.4B$2.A8.AB3.2A$4.A5.A.A.A2.A$3.2A4.A.2A.2A$9.A$8.2A!

(let us define "demiphoenix a/b" as "demiphoenix of period b where every cell is alive for a generations")

[Supersuthiastic]

The 5 known still lives in b34s23 are: block, boat, ship, beehive, and long ship. I have conjectured that they are the only ones in this rule. If you proved that they are the only ones, remember to attach a proof to it. I am looking forward for this problem to be solved!

Code: Select all

x = 10, y = 10, rule = B34/S23
bo6b2o$obo4bobo$obo5bo$bo$4b2o$4b2o$8b2o$b2o4bobo$obo3bobo$2o4b2o!

EDIT by dvgrn:See this post by Tropylium from a decade ago.

[confocaloid]

I have a note to go after statorless p3 since the human definition and the program definition are the same and I think it's conceivable I could get to the obvious target of 140 for the known oscillator.

Alas, nope. Pop 125 took ~118GB (no results) and pop 126 hit 120GB limit and died.

A period-3 phoenix (i.e. strictly volatile period-3 oscillator where every cell is only one for one period) cannot exist, but can a strictly-volatile period-3 oscillator where every cell is on for two generations exist?

One can try running various p3 oscillators in this 8-state cellular automaton. Barring bugs, it should be equivalent to Conway's Life (light = on, dark = off), with an additional property that the p3 cells that are on in 2/3 phases are highlighted as green, and all other p3 cells are highlighted as red.
The shown oscillators are the statorless p3 and the pulsar.

Now the challenge can be stated in the form "Find a statorless p3 oscillator where every rotor cell is green, under these rules":

Code: Select all

x = 51, y = 28, rule = demo_p3
9.G.G5.G3.G5.G.G$8.G3.G4.G3.G4.G3.G$9.G7.G3.G7.G$11.2G.2G.G3.G.2G.2G$
17.G3.G$10.G3.G9.G3.G$8.G.G17.G.G2$8.G19.G11.3G3.3G$7.2G18.2G$7.G19.G
10.G4.G.G4.G$5.G2.G16.G2.G9.G4.G.G4.G$G4.G14.G4.G12.G4.G.G4.G$4G.2G.
3G9.4G.2G.3G9.3G3.3G$3G.2G.4G9.3G.2G.4G$5.G4.G14.G4.G9.3G3.3G$2.G2.G
16.G2.G12.G4.G.G4.G$3.G19.G14.G4.G.G4.G$2.2G18.2G14.G4.G.G4.G$2.G19.G
$40.3G3.3G$G.G17.G.G$2.G3.G9.G3.G$9.G3.G$3.2G.2G.G3.G.2G.2G$.G7.G3.G
7.G$G3.G4.G3.G4.G3.G$.G.G5.G3.G5.G.G!

@RULE demo_p3

0 = 000 permanently off; dark
1 = 001 bad rotor; on; light red
2 = 010 bad rotor; off; dark red
3 = 011 good rotor; on; light green
4 = 100 bad rotor; off; dark red
5 = 101 good rotor; on; light green
6 = 110 good rotor; off; dark green
7 = 111 permanently on; light
lsb = current generation

@COLORS

0 40 40 40
1 240 40 40
2 120 40 40
3 40 240 40
4 120 40 40
5 40 240 40
6 40 120 40
7 240 240 240

@TABLE
n_states:8
neighborhood:Moore
symmetries:permute

var d4={0,2,4,6}
var d5=d4
var d6=d4
var d7=d4
var d8=d4
var a1={1,3,5,7}
var a2=a1
var a3=a1
var x1={0,1,2,3,4,5,6,7}
var x2=x1
var x3=x1
var x4=x1
var x5=x1
var x6=x1
var x7=x1
var x8=x1

0, a1,a2,a3,d4,d5,d6,d7,d8, 1
0, x1,x2,x3,x4,x5,x6,x7,x8, 0
1, a1,a2,x3,d4,d5,d6,d7,d8, 3
1, x1,x2,x3,x4,x5,x6,x7,x8, 2
2, a1,a2,a3,d4,d5,d6,d7,d8, 5
2, x1,x2,x3,x4,x5,x6,x7,x8, 4
3, a1,a2,x3,d4,d5,d6,d7,d8, 7
3, x1,x2,x3,x4,x5,x6,x7,x8, 6
4, a1,a2,a3,d4,d5,d6,d7,d8, 1
4, x1,x2,x3,x4,x5,x6,x7,x8, 0
5, a1,a2,x3,d4,d5,d6,d7,d8, 3
5, x1,x2,x3,x4,x5,x6,x7,x8, 2
6, a1,a2,a3,d4,d5,d6,d7,d8, 5
6, x1,x2,x3,x4,x5,x6,x7,x8, 4
7, a1,a2,x3,d4,d5,d6,d7,d8, 7
7, x1,x2,x3,x4,x5,x6,x7,x8, 6

edit: reworded

[muzik]

Now the challenge can be stated in the form "Find a statorless p3 oscillator where every rotor cell is green, under these rules":

Probably even simpler would be "find a period-3 oscillator where every non-dead cell in its frequency map is the same colour". The only possibilities that would satisfy this condition would be a still life, a p3 phoenix, or this type of oscillator; we can rule out the first due to its "real" period not being 3, and the second has been disproven. Sadly there doesn't seem to be any program support for this type of oscillator map so far.

[d/dx]

edit 2:

the superstring makes debris faster than the hive fuse can destabilise it

Code: Select all

x = 32, y = 9, rule = B3/S23:T32,0
32o$3ob4obob2ob3o3bobob2ob3o$2b2o2b4o4bo2bo6bo2bob2o$b4obo2bo4b2o4bo2b
o3bo2b2o$ob3o2b2ob2ob3ob2ob8o3bo$2bo2bobobo2b2o4bob2ob2o2bobobo$2o2b2o
2bo2bob2o2bo4b4obobo$2o5b3o3b3obo2bob3obob2obo$b2ob2obo2bo4b2o2bo2bob
4o3bo!

conjecture: there exist finite puffers with this property

[dvgrn]

the superstring makes debris faster than the hive fuse can destabilise it

conjecture: there exist finite puffers with this property

See Line puffer for some of what's known about finite puffers that look vaguely like these superstrings, but travel at c/2 instead of c.

I think this conjecture is disprovable, if I understand the meaning of "this property" correctly -- it's not very clearly stated.

The hive fuse travels at 4c/5, and finite puffers can only travel at c/2, so if the conjecture is about something finite outrunning that particular hive fuse then it's definitely false. If something more general is meant -- a puffer puffing something that a slower fuse eats -- then see growing spaceship for one example -- there are a lot more known, some a lot more "natural-looking" than others. Maybe see also growing/shrinking spaceship, depending on what you're interested in.

[d/dx]

this is kind of hard to explain

growing spaceship, except instead of eating the debris, the slower fuse perturbs it in some way

[dvgrn]

this is kind of hard to explain

growing spaceship, except instead of eating the debris, the slower fuse perturbs it in some way

Sure -- that's reasonably well explained, it's just not an unproven conjecture. You'll find a lot of hits on "growing puffer" for OCA rules, here on the forums, but they don't get much attention in B3/S23 just because they're a lot easier to invent than growing spaceships are -- and growing spaceships themselves don't get a lot of attention because they're too easy to invent.

Some types of growing spaceships can easily be converted into growing puffers just by damaging them slightly. Here's a nice example based on an improbable growing spaceship by Sokwe from back in 2020:

Code: Select all

x = 106, y = 84, rule = B3/S23
60bo$59b2o$59bobo2$62b2o$62b2o2$9bob2o44b2ob2o$7b3ob2obo42bo3bo$7bo3b
o2bo41bo2b3o$bo5bo4bob2o41b3o$2o6bo4bobo40bobo$obo3b2o6bobo16b2o23b2o
$4bo13b3o5bobo4bobo19b2o2bo$3bo4bo9bo2bo2b2obobo3bo23bob2o$3bo2bo11bo
3b2obo4bo3b2o8bo14bobo3b2o$15b2o4b2obo9bo2bo4b5o12bobo2b2o$14b2o11bob
o4bo6b2o4bo12b4ob2o$16bo19bobobo3b2obo13b2obo$27b4o33bo$26b4o3bo4bo5b
3o14bo26b3o$26bo5b3o4b5o4bo12bob2o23bo5b3o$34bo13bo4bo9bo25bo2b2ob2o$
32bobo10bo6bob2o6b2o20b3o4bo$32bobo3b2o5bo2b2obo5bo4b2o20bo7b2o$32bob
o3bobo4b2ob2o2b2o3bo27bo2b3o$29b2o2b2o3bo7b3o9bo3bo2bo3bo8b2o7bo2bo$29b
o4bo13bo6b2o4bo2bo3b2o8bobo9bo$29bo3bo27bo2bo3bobo7bo$26b2obo2b2o21b2o
bo5bo10b2obo10bo$26bo3bo7b2ob2o12bo2bo3b2o4b2o5bo4b2ob6o$26bob3obo5b2o
b2o12bobo10bo6bo4b2obo$32bo6bobo30b2ob2o6b3o$30bo9bo20b2obo4b2ob3o2bo
bo5b2o$30bobo28bo3bo2bo10bo6bobo$29b2o30bo4bo2bo7bo8b2o$22b3o38b2o$22b
o5bo3b2o31bobo4bo$23bo4bo4bo39b3o$25bo2bo4bo14bo18b2o$19b3o2b3o2bobo15b
obo18bo2bo$19bo27bobo16b2o$20bo27bo16bobo4bo$22bo3bo38bobo16b2o$21b2o
bo39bobobo14bo2bo$21bo42b2o2bo2bobo10b2o$20b2o43bo2b3o2bobo$20bo35bo8b
o3b2o2bobo$21bo33bobo8bo5b3o$55bobo13bo2bo$56bo$74bo17b2o$72bo18bo2bo
$71b2o19b2o2$64bo6bobo4b2o$63bobo7b2o3bobo$63bobo9b2o$64bo8bob4o$74b2o
2bo19bo$96b3o2$93bo6b2o$72bo10bobo6b2o6b2o$71bobo8bo9bo5bobo$71bobo9b
o2bo6bo3b2o$72bo12b3o10b2o4$104bo$80bo22bobo$79bobo22bo$79bobo$80bo$91b
o$90bobo$91bo4$97b2o$96bo2bo$97b2o!

[erictom333]

Can a quintuple (or higher) psuedo oscillator exist? It would require two noninteracting oscillators that cross each other, which can (boringly) be achieved with pairs of glider guns pointed at each other.

[confocaloid]

Can a quintuple (or higher) psuedo oscillator exist? It would require two noninteracting oscillators that cross each other, which can (boringly) be achieved with pairs of glider guns pointed at each other.

Crossposting from "Thread for Discord crossposts" (unfortunately so far there was no actual discord crosspost, as far as I can tell):

I was told (offline) that Discord has a proof of "pseudo-oscillators" being minimally-separable into more than 4 pieces (thus unlike pseudo still lives) that works by introducing intersection component (so connection graph is no longer planar)

I don't think it's in News Archive on wiki, nor on any pages (tho that I might have missed)

[erictom333]

Here's a potential start, with two intersecting but noninteracting oscillators:

Code: Select all

x = 39, y = 39, rule = LifeSuper
14.A$14.3A$17.A$16.2A$16.4B$18.3B$18.3B$17.5B$17.5B$16.7B$16.7B$16.7B
$16.7B$16.2A3B2A$16.2B3A2B14.2I$17.A3BA15.I$9.JI4J2.BABAB2.4JIJ4.JI.I
$7.JIJI6J.BAB.6JIJIJ2.J2I$5.2JIJI9JB9JIJI3J$4.2JI2JI9J.9JI2JI2J$4.3JI
JI9JB9JIJI2J$2.2IJ2.JIJI6J.BAB.6JIJIJ$.I.IJ4.JI4J2.BABAB2.4JIJ$.I15.A
3BA$2I14.2B3A2B$16.2A3B2A$16.7B$16.7B$16.7B$16.7B$17.5B$17.5B$18.3B$
18.3B$19.4B$21.2A$21.A$22.3A$24.A!

[Bullet51]

Conjecture: The growth rate of the function f(n) is exponential or even faster.
f(n) return the biggest non-infinite final population obtainable from a starting pattern that fit in a nxn bounding box.

The function f(n) grows no slower than a busy beaver function, as we can simulate a Turing machine in Life, and let the Turing machine spit a glider out if it doesn't halt, and self-destruct if it halts.

[WhiteHawk]

Conjecture: 24p24 (Caterer on Figure Eight) is the only SPOP(n) (Smallest Possible Oscillator for each Period, akin to SKOP, which is Smallest Known), discovered and undiscovered, in life which has a minipop value equal to it's period.

Code: Select all

x = 18, y = 6, rule = B3/S23
4b2o6bo$2bob2o4bo3b4o$bo8bo3bo$4bo5bo$2obo9bo$2o9b2o!

For reference, here are all oscillator periods in life which have a SPOP(n) with minipop higher than their period (This list is finite since eventually rectifier loops achieve smaller populations): 1-14, 16-23, 25-29, 31-35, 37-39, 41-44, 47, 50-51, 53, 57, 59, 61-63, 67, 69, 71, 74, 79, 81, 89, 95, 97, 101, 109, 121, and 131. (EDIT: 125 is removed, EDIT in Feb 2026: 103 is removed)

Due to the existence of a (EDIT 2026: List of oscillators with 3 to 16 cells is not a complete list, more accurately LifeWiki:Object counts for specific periods), Blinker, Caterer, Mazing/Mold, and Pseudo-barberpole are the only SKOPs which have been proven to (1) also be the true SPOP(n) of their respective period, and (2) prove that their periods do not violate this conjecture.

Of course, there are several SKOP(n)s which come very close, e.g. 45p44 (Mold on Rattlesnake), 64p63 (32P21 phase-shifting p5), and 79p80 (p79 pi-heptomino hassler). Additionally, there is an infinite subset of the periods with SKOP(n) minipop LOWER than their period with oscillators which have oscillators matching minipops and periods, but they are not SKOP(n)s. For example, the Queen Bee shuttle supported by integrals on both sides below has minipop 30, same as it's period, but it is not SKOP 30

Code: Select all

x = 64, y = 46, rule = B3/S23
3$bo5bo4bobobo3bobobo8bo5bo3bo3bobobo3bobobo$31bo3bo24bo$bobo3bo3bo5b
o4bo16bo2bo3bo5bo2bo$31bo28bo$bo3bobo3bo5bo4bo10bo5bobo4bo5bo2bobobo$
35bo$bo5bo3bo5bo4bo16bo2bo3bo5bo2bo$31bo3bo$bo5bo4bobobo5bo10bo5bo3bo
3bobobo3bo8$28bo$26bobo$25bobo$24bo2bo$25bobo$20b2o4bobo9b2o$19bobo6b
o9bobo$19bo20bo$17bobo20bobo$17b2o22b2o7$13bo6bobo4bobobo6bo6bobo$11b
o3bo3bo3bo8bo3bo3bo3bo3bo$27bo$15bo2bo5bo7bo7bo2bo5bo$13bo13bobobo6bo
$15bo2bo5bo15bo2bo5bo$27bo$11bo3bo3bo3bo12bo3bo3bo3bo$13bo6bobo4bo10b
o6bobo!

Obviously the conjecture could theoretically be disproven by the discovery of a smaller p24 or smaller sparking p3 or p98

Finally, a few more SKOP/SPOP-based conjectures which are yet unproven:

All SPOP(n)s in life have a minimum population less than 200 (Only SKOP 67 has a higher minipop as it's a snark loop)
All SPOP(n)s in life have a minimum population less than 150 (In addition to SKOP 67, periods with SKOP 41, 89, 97, 109, and 121 (EDIT: 125 and 103 are removed) have SKOPs with higher minipops)
Cribbage is not SPOP 19 (TBH if this one is wrong, I will accept that)

[confocaloid]

[...] SKOP (Smallest Known Oscillator for each Period), discovered and undiscovered, [...]

There is an internal contradiction here. The abbreviation SKOP means "smallest known oscillator of such-and-such period". However, the part "discovered and undiscovered" is apparently supposed to mean "smallest possible oscillator of such-and-such period".

The smallest possible oscillator of period n may or may not coincide with the current SKOP(n).

The smallest possible oscillator of a specific period may fail to be unique (there may be two or more oscillators with the same minimum population).
Likewise, SKOP(n) may fail to be unique.

The lowest possible minimum population of a period-n oscillator in Conway's Life may or may not be currently known, but it is a positive integer number that's a function of n (now that it's known that Conway's Life is omniperiodic, so a nontrivial oscillator of every period exists).

[WhiteHawk]

[...] SKOP (Smallest Known Oscillator for each Period), discovered and undiscovered, [...]

There is an internal contradiction here. The abbreviation SKOP means "smallest known oscillator of such-and-such period". However, the part "discovered and undiscovered" is apparently supposed to mean "smallest possible oscillator of such-and-such period".

Fixed my post.

EDIT:

The smallest possible oscillator of a specific period may fail to be unique (there may be two or more oscillators with the same minimum population).
Likewise, SKOP(n) may fail to be unique.

Since I am looking for at least 1 counterexample, I don't think it's relevant whether an oscillator is unique. Also, by definition, if there were 2 SKOPs for a specific period, they must have the same minipop or else one wouldn't be a SKOP.

EDIT 2:

New conjecture: All SPOPs in life below 29 cells are known (not necessarily all oscillators) - (18 periods 2-8, 10, 14-16, 24, 30, 36, 40, 46, 60, and 120) 29 cells is the minipop of smallest infinite family of oscillators, as well as the minipop of 29p9.

[confocaloid]

Conjecture: there exists an elementary statorless flipping oscillator that's loopable (with an odd loopability >= 3), in some CA that can be defined using Hensel notation. (In other words: two cellstates, range-1 Moore neighbourhood, isotropic rules.)

In [R]History, the envelope of such an oscillator might look (roughly) like this. The set of alive cells would always remain relatively compact, going along the "lemniscate", and evolving into a mirror image of itself after (period/2) ticks.

Code: Select all

#C [[ VIEWONLY ]]
x = 181, y = 84, rule = LifeHistory
34.18B$30.27B71.19B$27.33B62.29B$24.38B56.36B$22.43B50.42B$20.47B45.
47B$19.50B41.51B$17.53B38.54B$16.56B35.57B$15.58B32.60B$13.62B29.62B$
12.64B26.66B$11.66B24.68B$10.69B21.70B$9.71B19.72B$9.72B17.74B$8.74B
15.75B$7.76B13.77B$6.78B11.79B$6.80B8.81B$5.32B13.37B6.82B$5.29B20.
34B4.35B17.32B$4.28B24.33B2.33B23.29B$4.26B28.63B28.28B$3.26B31.59B
32.26B$3.25B34.56B34.26B$3.24B36.53B37.25B$2.24B39.50B39.24B$2.23B41.
48B41.24B$2.22B43.46B43.23B$.22B45.44B45.22B$.22B46.42B47.22B$.21B48.
40B48.22B$.21B49.38B50.21B$.21B50.36B51.21B$.7B6C7B52.34B52.21B$5B3C
2B2CBC7B53.32B53.8B6D7B$8BC3B3C6B54.30B54.8BDB2D2B3D5B$5BCBCBCBC9B55.
29B54.7B3D3BD8B$5B2CBCB5C5B57.27B55.10BDBDBDBD5B$5B2CBCB3C7B57.27B55.
7B5DBDB2D5B$5B2C2B2C2BC6B56.29B54.9B3DBDB2D5B$5B3CB2CBC7B56.29B54.8BD
2B2D2B2D5B$20B55.31B53.9BDB2DB3D5B$20B54.33B52.22B$.20B52.35B51.22B$.
20B51.37B50.21B$.20B51.37B50.21B$.20B50.39B49.21B$.21B48.41B47.22B$.
21B47.43B46.22B$2.21B45.45B45.22B$2.21B44.47B43.22B$2.22B42.49B42.22B
$2.23B39.52B41.22B$3.23B37.55B38.23B$3.24B35.29B.27B36.23B$4.25B31.
30B3.27B34.24B$4.26B29.30B5.28B30.26B$5.27B25.31B7.29B27.26B$5.30B19.
34B8.30B23.28B$6.32B13.36B10.31B18.31B$6.80B12.33B12.33B$7.78B14.36B
6.35B$8.77B15.75B$8.76B17.74B$9.74B19.72B$10.72B21.70B$11.70B23.69B$
12.68B25.67B$13.66B27.65B$14.64B29.63B$15.62B31.61B$16.60B33.59B$17.
58B35.57B$19.54B39.53B$20.52B41.51B$22.48B45.47B$24.44B49.43B$26.40B
53.39B$28.36B58.34B$31.30B64.28B$35.21B74.20B$141.5B!

[dvgrn]

Conjecture: there exists an elementary statorless flipping oscillator that's loopable (with an odd loopability >= 3), in some CA that can be defined using Hensel notation. (In other words: two cellstates, range-1 Moore neighbourhood, isotropic rules.)

It seems like an interesting challenge to figure out a search method that can find one of these. The methods that people have used so far to dig up RROs with record-breaking multiplicity, tend to produce simple small objects like B-heptominoes that go back to the same simple small object rotated 90 degrees.

It seems like there's no particular reason why even a standard B-heptomino might not, in some INT rule, turn once in one direction and twice in the other direction before producing a mirror-image copy of itself -- maybe just nobody has thought about setting up a search to notice those instead of the 90-degree-turn no-mirror-image type of RRO. (?)

[400spartans]

184-cell unsynthesizable still life:

Code: Select all

x = 23, y = 25, rule = B3/S23
11b2o$10bo2bo$8bo2bobo$8b2obo2b2o$5b2obo2b2obo2b2o$3b3obo2bo2bo2b2obo$
2bo5b2obo2b2o4bo$3bob2obo2b2obo2b2obo$4b2obo2bo2bo2b2obo$2o6b2obo2b2o$
o2bob2obo2b2obo2b2o$b3ob2o2bo2bo2bob2o$8b2obo2b2o$3b2ob2o2bob2o2bob3o$
2bob2obo2bo2bo2b2obobo$bo6b2obo2b2o6bo$2b3ob2o2bob2o2bob4o$4b2obo2bo2b
o2b2obo$8b2obo2b2o$3bob2obo2b2obo2b2o$2bob2obo2bo2bo2b2obo$2bo5b2obo2b
2o4bo$3bob2obo2b2obo2b3o$4b2obo2bo2bo2b2o$11b2o!

EDIT: bb reduction, same population:

Code: Select all

x = 22, y = 25, rule = B3/S23
10b2o$9bo2bo$7bo2bobo$b2o4b2obo2b2o$bo2b2obo2b2obo2b2o$2b3obo2bo2bo2b
2obo$7b2obo2b2o4bo$4b2obo2b2obo2b2obo$2b3obo2bo2bo2b2obo$bo5b2obo2b2o$
2bob2obo2b2obo2b2o$b2ob2o2bo2bo2bob2o$7b2obo2b2o$b3ob2o2bob2o2bob3o$o
2b2obo2bo2bo2b2obobo$o6b2obo2b2o6bo$b3ob2o2bob2o2bob4o$3b2obo2bo2bo2b
2obo$7b2obo2b2o$2bob2obo2b2obo2b2o$bob2obo2bo2bo2b2obo$bo5b2obo2b2o4bo
$2bob2obo2b2obo2b3o$3b2obo2bo2bo2b2o$10b2o!

[Chai]

There are no still lives other than the block where all live cells have 3 live neighbors.
There are infinitely many with all-but-4 (long^n barges), all-but-3 (long^n boats), and all-but-2 (long^n ships).
Is there a still life where all but 1 live cell has 3 live neighbors? Are there infinitely many?

[WhiteHawk]

There are no still lives other than the block where all live cells have 3 live neighbors.

Proven for all finite strict still-lifes, yes. An infinite barge is a counterexample, though, as is bi-block for pseudo still-lives.