ConwayLife.com

Posted: **January 17th, 2024, 3:59 pm**

1_1 wrote: January 17th, 2024, 12:42 pm
Sokwe wrote: January 9th, 2024, 6:17 am Nice work! Unfortunately, it seems to me that no finite NxN box has weights that would give a strict 1/2 density bound.
What if there was an algorithm which could, for any real number larger than 1/2, produce a set of weights in a finite box which gives a density bound less than that real number? Or would such an algorithm not be guaranteed to exist?

Welcome to the forums!!

I believe amling was using an SAT solver to find the most recent update to the number -- in that case there already is an algorithm.

Posted: **January 17th, 2024, 5:14 pm**

wirehead wrote: January 17th, 2024, 3:59 pm Welcome to the forums!!

I believe amling was using an SAT solver to find the most recent update to the number -- in that case there already is an algorithm.

Hi! I'm happy to be welcomed!

From what I gather, that could only find the configurations of weights assuming they already exist, which we haven't proved yet. I was more asking if it would be possible to prove that configurations of weights which give upper bounds arbitrarily close to 1/2 must exist.

Not that using a SAT solver to find weights isn't really cool on it's own, of course.

I also had another thought: finite weights can't enforce an upper bound of 1/2, but what about an infinite set of weights? As long as it was well defined, and added up to a finite number, I would believe such a set of weights could still prove an upper bound. Since Sokwe's proof requires looking at a weight in the top left corner, it wouldn't apply, so an infinite set of weights might be able to give us 1/2 as an upper bound.

Posted: **March 12th, 2024, 2:44 pm**

carsoncheng wrote: September 5th, 2023, 4:55 am

Macbi wrote: January 14th, 2022, 11:44 am The smallest you can get it is population 306. (Confirmed with LLS.)

Code: Select all

x = 34, y = 28, rule = LifeHistory
12.2C4.2C$12.C5.C$7.2C5.C5.C7.2C$6.C.A4D2A4D2A4D2AD.C$2.2C.A2DAD2A2DA
D2A2DAD2A2DADA$2.C.2ADA2D2ADA2D2ADA2D2ADA2D2A$3.4D2A4D2A4D2A4D2A3DC$
3.2ADA2D2ADA2D2ADA2D2ADA2D2AD.C$3.AD2A2DAD2A2DAD2A2DAD2A2DAD2AC$3.4D
2A4D2A4D2A4D2A4D$3.D2A2DAD2A2DAD2A2DAD2A2DAD2AD$3.D2ADA2D2ADA2D2ADA2D
2ADA2D2AD$3.4D2A4D2A4D2A4D2A4D.2C$2.C2ADA2D2ADA2D2ADA2D2ADA2D2ADA2.C$
C2.AD2A2DAD2A2DAD2A2DAD2A2DAD2AC$2C.4D2A4D2A4D2A4D2A4D$3.D2A2DAD2A2DA
D2A2DAD2A2DAD2AD$3.D2ADA2D2ADA2D2ADA2D2ADA2D2AD$3.4D2A4D2A4D2A4D2A4D$
2.C2ADA2D2ADA2D2ADA2D2ADA2D2ADA$2.C.D2A2DAD2A2DAD2A2DAD2A2DAD2A$3.C3D
2A4D2A4D2A4D2A4D$4.2A2DAD2A2DAD2A2DAD2A2DAD2A.C$5.ADA2D2ADA2D2ADA2D2A
DA2DA.2C$4.C.D2A4D2A4D2A4DA.C$4.2C7.C5.C5.2C$15.C5.C$14.2C4.2C!

The bound was *(provisionally) improved to 278 cells on Discord by "400spartans", using a different self-forcing patch:

Code: Select all

x = 31, y = 28, rule = B3/S23
16bo4b2o$15bobo2bo2bo$7b2o6bo2bo2bo2bo$6bobo4b2ob2ob2ob2obo$2b2obo2bob2o2bo2b
o2bo2bobo$2bob2obo2b2obo2bo2bo2bo2b2o$7b2o4b2ob2ob2ob2o3bo$3b2obo2b2obo2bo2bo
2bo2b2o2bo$3bob2o2bob2o2bo2bo2bo2bob3o$7b2o4b2ob2ob2ob2o$4b2o2bob2o2bo2bo2bo
2bob2o$4b2obo2b2obo2bo2bo2bo2b2o$7b2o4b2ob2ob2ob2o5b2o$ob3obo2b2obo2bo2bo2bo
2b2obo2bo$2obob2o2bob2o2bo2bo2bo2bob3o$7b2o4b2ob2ob2ob2o$4b2o2bob2o2bo2bo2bo
2bob2o$4b2obo2b2obo2bo2bo2bo2b2o$7b2o4b2ob2ob2ob2o$2b3obo2b2obo2bo2bo2bo2b2ob
o$2bo2b2o2bob2o2bo2bo2bo2bob2o$3bo3b2o4b2ob2ob2ob2o$4b2o2bob2o2bo2bo2bo2bob3o
$5bobo2b2obo2bo2bo2bo2bo2bo$4bo2b2o4b3o4b2o3bobo$4b2o3b4o13bo$9bo3bo$12b2o!

* the result has not been verified independently yet; can anyone verify this?

Here's a still life which gives a new bound of 236 cells:

Code: Select all

x = 28, y = 26, rule = B3/S23
2bo8b2o$2b3o5bobo$5bo4bo2b2o2b2o$4bo2b2ob2obo2bo2bobo$4b2ob2o2bo2bob2o
b2obo$5bo4bo2b2o2bo4bo$2obo2bob2ob2o2bo2bob2ob2o$ob3ob2obo2bo2b2ob2obo
2bo$5bo4bo2b2o2bo4bo$4bo2b2ob2obo2bo2b2ob2o$4b2ob2o2bo2bob2ob2o2bo$5bo
4bo2b2o2bo4bo$2obo2bob2ob2o2bo2bob2ob3obo$ob3ob2obo2bo2b2ob2obo2bob2o$
5bo4bo2b2o2bo4bo$4bo2b2ob2obo2bo2b2ob2o$4b2ob2o2bo2bob2ob2o2bo$5bo4bo
2b2o2bo4bo$3bo2bob2ob2o2bo2bob2ob3obo$3b2ob2obo2bo2b2ob2obo2bob2o$5bo
4bo2b2o2bo4bo$5bob2ob2obo2bo2b2ob2o$6bobo2bo2bob2ob2o2bo$9b2o2b2o2bo4b
o$15bobo5b3o$15b2o8bo!

Posted: **March 12th, 2024, 4:13 pm**

400spartans wrote: March 12th, 2024, 2:44 pm Here's a still life which gives a new bound of 236 cells:

Code: Select all

x = 28, y = 26, rule = B3/S23
2bo8b2o$2b3o5bobo$5bo4bo2b2o2b2o$4bo2b2ob2obo2bo2bobo$4b2ob2o2bo2bob2o
b2obo$5bo4bo2b2o2bo4bo$2obo2bob2ob2o2bo2bob2ob2o$ob3ob2obo2bo2b2ob2obo
2bo$5bo4bo2b2o2bo4bo$4bo2b2ob2obo2bo2b2ob2o$4b2ob2o2bo2bob2ob2o2bo$5bo
4bo2b2o2bo4bo$2obo2bob2ob2o2bo2bob2ob3obo$ob3ob2obo2bo2b2ob2obo2bob2o$
5bo4bo2b2o2bo4bo$4bo2b2ob2obo2bo2b2ob2o$4b2ob2o2bo2bob2ob2o2bo$5bo4bo
2b2o2bo4bo$3bo2bob2ob2o2bo2bob2ob3obo$3b2ob2obo2bo2b2ob2obo2bob2o$5bo
4bo2b2o2bo4bo$5bob2ob2obo2bo2b2ob2o$6bobo2bo2bob2ob2o2bo$9b2o2b2o2bo4b
o$15bobo5b3o$15b2o8bo!

I checked this with JLS. The pattern below shows a highlighted self-forcing patch. I don't know if it's a minimal patch (Edit: it's not).

Code: Select all

x = 28, y = 26, rule = LifeHistory
2.A8.2A$2.3A5.A.A$3.2DC4DC2DCA2.2A$3.DC2D2CD2CDC2DC2DC.A$3.D2CD2C2DC
2DCD2CD2CDA$3.2DC4DC2D2C2DC4DC2D$2A.C2DCD2CD2C2DC2DCD2CD2C$A.A2CD2CDC
2DC2D2CD2CDC2DC$3.2DC4DC2D2C2DC4DC2D$3.DC2D2CD2CDC2DC2D2CD2CD$3.D2CD
2C2DC2DCD2CD2C2DCD$3.2DC4DC2D2C2DC4DC2D$2A.C2DCD2CD2C2DC2DCD2CD2CA.A$
A.A2CD2CDC2DC2D2CD2CDC2DC.2A$3.2DC4DC2D2C2DC4DC2D$3.DC2D2CD2CDC2DC2D
2CD2CD$3.D2CD2C2DC2DCD2CD2C2DCD$3.2DC4DC2D2C2DC4DC2D$3.C2DCD2CD2C2DC
2DCD2CD2CA.A$3.2CD2CDC2DC2D2CD2CDC2DC.2A$3.2DC4DC2D2C2DC4DC2D$5.AD2CD
2CDC2DC2D2CD2CD$6.A.C2DC2DCD2CD2C2DCD$9.2A2.AC2DC4DC2D$15.A.A5.3A$15.
2A8.A!

Edit: here is a smaller self-forcing patch in the same still life. In this case, removing any "corner" cell from the self-forcing patch results in a non-self-forcing patch. I don't know if this is a minimal patch.

Code: Select all

x = 28, y = 26, rule = LifeHistory
2.A8.2A$2.3A5.A.A$5.A4.A2.2A2.2A$4.C2D2CD2CDC2DCD.A.A$4.2CD2C2DC2DCD
2CD2C.A$4.DC4DC2D2C2DC3D.A$2A.C2DCD2CD2C2DC2DCD2CDCA$A.A2CD2CDC2DC2D
2CD2CDC2DA$3.2DC4DC2D2C2DC4DCD$3.DC2D2CD2CDC2DC2D2CD2C$3.D2CD2C2DC2DC
D2CD2C2DC$3.2DC4DC2D2C2DC4DCD$2A.C2DCD2CD2C2DC2DCD2CDC2A.A$A.2ACD2CDC
2DC2D2CD2CDC2DC.2A$4.DC4DC2D2C2DC4DC2D$4.C2D2CD2CDC2DC2D2CD2CD$4.2CD
2C2DC2DCD2CD2C2DCD$4.DC4DC2D2C2DC4DC2D$3.A2DCD2CD2C2DC2DCD2CD2CA.A$3.
ACD2CDC2DC2D2CD2CDC2DA.2A$5.A.3DC2D2C2DC4DCD$5.A.2CD2CDC2DC2D2CD2C$6.
A.A.DC2DCD2CD2C2DC$9.2A2.2A2.A4.A$15.A.A5.3A$15.2A8.A!

Notice that while the still life is symmetric, the shape of the self-forcing patch is not.

Edit 2: can you give any details about how you found it?

Posted: **March 12th, 2024, 7:06 pm**

Sokwe wrote: March 12th, 2024, 4:13 pm Edit 2: can you give any details about how you found it?

I'm just using Torma and Salo's program (https://github.com/ilkka-torma/gol-agars) with two minor tweaks to speed things up. First, I exclude patterns which are just a repetition of some smaller tiling. And second, this came up in a search which only looks at p1 agars in which every live cell is adjacent to 3 live cells and every dead cell is adjacent to 4 live cells. There are plenty of infinite self-forcing p1 agars which do not have this property, but the only known finite self-forcing p1 agar patterns all seem to have this property.

Posted: **March 12th, 2024, 9:58 pm**

Conjecture: 29 is the lowest population for which an infinite number of distinct, non-pseudo oscillators exist.

Code: Select all

x = 22, y = 48, rule = B3/S23
3o$obo$3o$3o$3o$3o$obo$3o4$o$b2o$2o27$19b3o$19bobo$19b3o$19b3o$19b3o$19b3o$
19bobo$19b3o!

Posted: **March 12th, 2024, 10:21 pm**

muzik wrote: March 12th, 2024, 9:58 pm Conjecture: 29 is the lowest population for which an infinite number of distinct, non-pseudo oscillators exist.

I believe it. I think the lower bound will always be three more than whatever cell count for which we've fully enumerated all infinitely growth patterns. Right now this is 9+3 = 12. There is still technically the possibility of a 10-cell puffer/wickstretcher/gun that travels/fires towards a far off blinker which, upon reaching the blinker causes a burning reaction along the puffer trail/wick/glider stream until the pattern collapses back into the original 10-cell infinite growth + blinker.

Posted: **March 12th, 2024, 11:39 pm**

[ DATA EXPUNGED ]

Posted: **March 12th, 2024, 11:48 pm**

Haycat2009 wrote: March 12th, 2024, 11:39 pm Help me prove: Every patch where the dead cells have 4 neighbours and the alive cells have 3 neighbours is self-forcing.

This 8-by-8 patch should count as a counterexample. (Because the 36-bit 3-by-3 block array is glider-constructible.)

Code: Select all

x = 8, y = 8, rule = B3/S23
2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o!

Posted: **March 13th, 2024, 12:53 am**

[ CENSORED ]

Posted: **March 13th, 2024, 1:17 am**

Haycat2009 wrote: March 13th, 2024, 12:53 amThe only condition in the search was what I listed above.

No, it wasn't.

They wrote:

400spartans wrote: March 12th, 2024, 7:06 pm
Sokwe wrote: March 12th, 2024, 4:13 pm Edit 2: can you give any details about how you found it?
I'm just using Torma and Salo's program (https://github.com/ilkka-torma/gol-agars) with two minor tweaks to speed things up. First, I exclude patterns which are just a repetition of some smaller tiling. And second, this came up in a search which only looks at p1 agars in which every live cell is adjacent to 3 live cells and every dead cell is adjacent to 4 live cells. There are plenty of infinite self-forcing p1 agars which do not have this property, but the only known finite self-forcing p1 agar patterns all seem to have this property.

You wrote:

Haycat2009 wrote: March 12th, 2024, 11:39 pmHelp me prove: Every patch where the dead cells have 4 neighbours and the alive cells have 3 neighbours is self-forcing.

Let the property P be "every specified alive cell has =3 alive neighbours, and every specified dead cell has =4 alive neighbours".

o They searched for some self-forcing patches with the property P, without assuming that every self-forcing patch has property P, and also without assuming that every patch with the property P is self-forcing.

o In contrast, you are claiming that every patch with the property P is self-forcing. That is disproven by the counterexample I posted above (the 3-by-3 block array viewed as a 8-by-8 patch).

Posted: **April 2nd, 2024, 1:53 am**

Is there a proof/disproof that there is an infinite amount of elementary oscillator periods?

Posted: **April 2nd, 2024, 2:47 am**

eRroR_6o6 wrote: April 2nd, 2024, 1:53 am Is there a proof/disproof that there is an infinite amount of elementary oscillator periods?

A proof would need to rely on some precise definition of elementary.
Almost certainly there are large chaotic-looking blobs without any visible structure somewhere in search space that are p2023 oscillators, and p142857 oscillators, and p1048577 oscillators, and so on for arbitrarily high periods. There are no simple "local" reasons preventing existence of such oscillators.

Posted: **April 27th, 2024, 9:29 am**

Conjecture: No glider eater can be constructed with a recovery time of three ticks or less.
(All known glider eaters take at least four ticks to recover to their original state after eating a glider.)

Well, the boat-bit has a stage in 3 ticks (boat appears).
How considerate this 2-states eater (edit: since recovering the original state depends on the stream period) ?

Code: Select all

x = 30, y = 25, rule = LifeHistory
3$26.A.A$26.2A$27.A$24.A$22.2A$23.2A2$19.A.A$19.2A$20.A$17.A$15.2A$
16.2A6$3.2A$3.A2.A$5.2A!

Posted: **April 27th, 2024, 9:57 am**

I think that was proven in 2021 viewtopic.php?p=125496#p125496

Posted: **May 2nd, 2024, 11:10 pm**

Tawal wrote: April 27th, 2024, 9:29 am Conjecture: No glider eater can be constructed with a recovery time of three ticks or less.
(All known glider eaters take at least four ticks to recover to their original state after eating a glider.)

Well, the boat-bit has a stage in 3 ticks (boat appears).
How considerate this 2-states eater (edit: since recovering the original state depends on the stream period) ?
Code: Select all
x = 30, y = 25, rule = LifeHistory
3$26.A.A$26.2A$27.A$24.A$22.2A$23.2A2$19.A.A$19.2A$20.A$17.A$15.2A$
16.2A6$3.2A$3.A2.A$5.2A!

Doesn't the original eater have a recovery time of 3 ticks?

Posted: **May 3rd, 2024, 3:52 am**

eRroR_6o6 wrote: May 2nd, 2024, 11:10 pm …
Doesn't the original eater have a recovery time of 3 ticks?

Yes, the Eater1 recovers in 3 ticks.
But does the last reaction's cell count ? In this case, the RT is 4 ticks.

Posted: **May 3rd, 2024, 7:09 am**

Tawal wrote: May 3rd, 2024, 3:52 am Yes, the Eater1 recovers in 3 ticks.
But does the last reaction's cell count ? In this case, the RT is 4 ticks.

Yup, Scorbie pointed this out several years ago.

Probably the intention of the Did-You-Know from which this item was copied was something like

"All known glider eaters take at least four ticks to remove the glider completely and recover to their original state after eating a glider."

Posted: **May 7th, 2024, 9:32 am**

Conjecture: There're infinite distinct 4-glider collisions even when excluding trivial collisions by adjusting the position of a glider that collides a glider produced from glider collisions.

Probable prove: You can synthesis a 3G GPSE then have it's gliders pointed toward a glider:

Code: Select all

x = 2982, y = 2841, rule = B3/S23
95bobo$96b2o$96bo$92b3o$94bo$93bo69$2o$b2o$o2762$2979b3o$2979bo$2980b
o!

I'm thinking of making "RCT4", which has the same concept as this.

Posted: **May 7th, 2024, 11:33 pm**

b-engine wrote: May 7th, 2024, 9:32 am

Relevant discussion:
viewtopic.php?p=21948#p21948
viewtopic.php?p=151662#p151662

It feels pretty safe to assume or expect that there is no "RCT4" (or a moral equivalent of that) in B3/S23 Life, and it feels pretty safe to assume or expect that there is no four-glider synthesis of Caterpillar. But actually proving that seems much harder.
And even if someone somehow manages to exclude the case "4 gliders", it feels pretty safe to expect that the question will probably still be open for 5 or 6 gliders.

Posted: **May 12th, 2024, 10:04 am**

I edited this post because it have been proven that fatherless still-life exist.

Conjecture: All glider-constructible still-life can appear from a single continuous row of cells of any lenght.

Examples:

Code: Select all

x = 22, y = 1, rule = B3/S23
22o!

Code: Select all

x = 22, y = 1, rule = B3/S23
34o!

If the conjecture is false, how many different still-life can you get?

Posted: **May 12th, 2024, 11:25 am**

Azerty wrote: May 12th, 2024, 10:04 amConjecture: All types of still-life can appear from a single row of cell. [...]

If taken literally, this is disproven by the existence of an unconstructible still life that cannot form in any way, unless present at generation 0.
If one modifies the conjecture to "All glider-constructible still lives can appear..." then it might be true or false.

Posted: **May 12th, 2024, 12:11 pm**

Does the "single row of still life" mean a continuous line of ON cells, as in the two examples given?

If any pattern of ON and OFF cells is allowed in the row, then I don't think this counts as "unproven", due to the outline given in the unidiemsional spaceship thread showing that any constructible still life (or any other constructible object, for that matter) can be constructed starting from a one-cell-thick pattern.

If only solid lines are allowed, that's the "all cells alive" restriction mentioned on the LifeWiki... Nathaniel Johnson did some work on this in 2009, as described in Sloane's A061342.

People have tried a lot of widths, and it looks like you most likely aren't going to get infinite novelty from all-cells-alive 1xN patterns -- but the LifeWiki only mentions things being "conjectured (and strongly believed)", not proven. So apparently nobody has managed a complete enumeration of all structures that are generated by 1xN lines.

Posted: **May 20th, 2024, 3:19 am**

I was just recently looking at oscillators with low heat. Burloaferimeter with heat 10/7 ≈ 1.43 is the only known oscillator with heat below 2. However, oscillators with heat greater than but arbitrarily close to 4/3 ≈ 1.33 almost certainly exist. This is because the 2c/3 half-signal has heat 4/3.

Code: Select all

x = 38, y = 40, rule = B3/S23
6b2o2b2o2b2o$6b2o2b2o2b2o2$4b12o$3bo12bo$3bob12o$2obo15bo$2obo2b14o$3b
obo$3bobo2b14o$2obobobo14bo$2obobobo2b13o$3bobobobo15bo$3bobobobo2b14o
$2obobobobobo$2obobobobobo2b14o$3bobobobobobo14bo$3bobobobobobo2b13o$
2b2obobobobobobo15bo$5bobobobobobo2b14o$5bobobobobobobo$4b2obobobobobo
bo2b14o$7bobobobobobobo14bo$7bobobobobobobo2b13o$8b2obobobobobobo15bo$
11bobobobobobo2b14o$11bobobobobobobo$10b2obobobobobobo2b2o2b2o2b2o$13b
obobobobobobobo2b2o2b2o$13bobobobobobob2o$14b2obobobobo$17bobobobo$17b
obobobob2o$16b2obobobob2o$19bobobo$19bobobo$20b2obob2o$23bob2o$23bo$
22b2o!

A low-heat oscillator would work by reflecting a single half-signal to form a "loop". The difficulty is that the orientation of the half signal must be flipped on the second pass through the loop. This might require the injector and receiver ends of the wire to be completely reconstructed after each pass of the signal to work with the opposite orientation on the next pass. However many cell changes this reconstruction requires can be made up for by extending the length of the 2c/3 wire to get the overall heat arbitrarily close to the half-signal heat of 4/3.

Of course, this all assumes that half-signal injectors and receivers exist, along with a way to reconstruct the ends of the wire so that they can inject/accept an oppositely oriented signal on the second pass. It seems almost certain that these things exist, although I'm not sure how likely it is that we can find them.

So there are almost certainly oscillators with lower heat than burloaferimeter, but it's not clear if there are any oscillators with heat 4/3 or lower.

Edit: since this is the unproven conjectures thread, I'll make my conjecture explicit:

Conjecture: there are no oscillators with heat less than or equal to 4/3.

Posted: **May 22nd, 2024, 7:00 am**

b-engine wrote: May 7th, 2024, 9:32 am Conjecture: There're infinite distinct 4-glider collisions even when excluding trivial collisions by adjusting the position of a glider that collides a glider produced from glider collisions.

Probable prove: You can synthesis a 3G GPSE then have it's gliders pointed toward a glider:
Code: Select all
x = 2982, y = 2841, rule = B3/S23
95bobo$96b2o$96bo$92b3o$94bo$93bo69$2o$b2o$o2762$2979b3o$2979bo$2980b
o!
I'm thinking of making "RCT4", which has the same concept as this.

I don’t think you understand the context behind how the RCT works, it’s physically impossible to use one GPSE because the gliders need to bounce off of a GPSE and escape somewhere else

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