Unproven conjectures

[g0t0]

Conjecture: there are not oscillator with period >2 in B8/S<any> rules(accepting infinity patterns such as agar)

Edit:
Overturn! this is a p6 oscillator:

Code: Select all

x = 6, y = 3, rule = B8/S234568
6o$ob4o$6o!

[Disaster16439]

Conjecture: there are not oscillator with period >2 in B8/S<any> rules(accepting infinity patterns such as agar)

Edit:
Overturn! this is a p6 oscillator:

Code: Select all

x = 6, y = 3, rule = B8/S234568
6o$ob4o$6o!

Could the conjecture still hold if we add the restriction that the period is odd?

[EvinZL]

Conjecture: there are not oscillator with period >2 in B8/S<any> rules(accepting infinity patterns such as agar)

Edit:
Overturn! this is a p6 oscillator:

Code: Select all

x = 6, y = 3, rule = B8/S234568
6o$ob4o$6o!

Could the conjecture still hold if we add the restriction that the period is odd?

p3:

Code: Select all

x = 5, y = 5, rule = B8/S0123456-c7c8
b3o$b3o$5o$obobo$5o!

EDIT: p5

Code: Select all

x = 14, y = 8, rule = B8/S0123-y4-ce567c8
4b7o$b7ob5o$b2ob10o$b5o2bob4o$4obo2b5o$10ob2o$5ob7o$3b7o!

p7

Code: Select all

x = 9, y = 11, rule = B8/S01234567c8
6b3o$6b3o$6b3o$6b3o$9o$2ob4obo$5ob3o$4b3obo$4b5o$4b5o$4b5o!

[NNlk05]

For every finite Game of Life pattern with n live cells, the next generation contains at most 3n−6 live cells.

I don't have a mathematical proof for this; I just know that a straight line with n cells (n > 2) has a population of 3n-6 in the next generation, and that looks pretty optimal. Has work on this been done before?

See:

i'm pretty sure it can be proven pretty easily that the maximum amount of cells from one generation to the next in B3/S23 is 3p, where p is the population at the before-generation.
here are some agars that showcase this maximum:

Code: Select all

x = 7, y = 9, rule = B3/S23:T9,9
o2bo2bo$o2bo2bo$o2bo2bo$o2bo2bo$o2bo2bo$o2bo2bo$o2bo2bo$o2bo2bo$o2bo2b
o!
[[ SHOWGENSTATS ]]

Code: Select all

x = 9, y = 9, rule = B3/S23:T9,9
bo2bo2bo$2bo2bo2bo$o2bo2bo$bo2bo2bo$2bo2bo2bo$o2bo2bo$bo2bo2bo$2bo2bo
2bo$o2bo2bo!
[[ SHOWGENSTATS ]]

ćode+x = 9, y = 9, rule = B3/S23:T9,9
bo2bo2bo$o2bo2bo$o2bo2bo$2bo2bo2bo$2bo2bo2bo$bo2bo2bo$bo2bo2bo$o2bo2b
o$o2bo2bo!
code+
now, what about extending this question to ask what the maximum amount of cells from one generation to two generations after is in B3-S23_ what about three generations, four generation, and so on_

[HartmutHolzwart]

You might not have recognized it, but I spent some time on the topic, see the patterns forum. In skew tori, there are basically three regimes that achieve maximum growth rates. The 3x3 torus with a block, the 3x1 with a stripe and 3x1 with skew 1. The last two can be mixed.

https://conwaylife.com/forums/viewtopic.php?t=7395

[g0t0]

For every finite Game of Life pattern with n live cells, the next generation contains at most 3n−6 live cells.

I don't have a mathematical proof for this; I just know that a straight line with n cells (n > 2) has a population of 3n-6 in the next generation, and that looks pretty optimal. Has work on this been done before?

S3 is seemed to be useless.

How about the same conjecture in B3/S2?

[g0t0]

Conjecture: the are no infinite growth in a row of n cells.

[g0t0]

What about a hypothetical spaceship that will certainly produce lots of gliders when destroyed? As an extreme example, we could use 0E0P metacells to emulate a chaotic spaceship from another (preferably explosive) rule, and I strongly doubt you can collide such metaships together (either in a way that simulates another rule or otherwise) without releasing gliders in every direction, making cleanup almost impossible. There is also almost certainly no way that you can clean up the released gliders using other metaships without releasing gliders in every direction. There will almost certainly be gliders that will be permanently out of reach of the RCT - no mechanism could be used to catch them.

If someone were to try this (using some futuristic supercomputer, since there is no way modern desktop computers could begin to run this), make sure that the emulated spaceship cannot cleanly annihilate copies of itself.

If this method turns out to be insufficient somehow, we might be able to engineer a spaceship that will always release gliders in every direction.

EDIT: We could also - as an example firmly outside of plausibility - make a spaceship that is so ridiculously large that any collision involving it will end up producing IceNine. (I only came up with this after seeing my post count at 137). Since IceNine, if it exists at all, is an unstoppable quadratic growth pattern, this would make it impossible for the spaceship to be cleanly destroyed at all, let alone by copies of itself.

0E0P metacell ships can hit unexpectedly and explode!

[g0t0]

Conjecture: there are not oscillator with period >2 in B8/S<any> rules(accepting infinity patterns such as agar)

Edit:
Overturn! this is a p6 oscillator:

Code: Select all

x = 6, y = 3, rule = B8/S234568
6o$ob4o$6o!

Could the conjecture still hold if we add the restriction that the period is odd?

p3:

Code: Select all

x = 5, y = 5, rule = B8/S0123456-c7c8
b3o$b3o$5o$obobo$5o!

EDIT: p5

Code: Select all

x = 14, y = 8, rule = B8/S0123-y4-ce567c8
4b7o$b7ob5o$b2ob10o$b5o2bob4o$4obo2b5o$10ob2o$5ob7o$3b7o!

p7

Code: Select all

x = 9, y = 11, rule = B8/S01234567c8
6b3o$6b3o$6b3o$6b3o$9o$2ob4obo$5ob3o$4b3obo$4b5o$4b5o$4b5o!

No INT rule.

[NNlk05]

Conjecture: the are no infinite growth in a row of n cells.

Overturn by counterexample

[rabbit]

I would assume that g0t0 is talking about full 1xN rectangles, rather than 1xN patterns of any variety.

I checked all filled boxes from 1x1 up to 1x10000 with a small apgsearch stdin script and none of them yielded linear growth. It appears that, on the whole, they tend to settle into repetitive behaviour that leaves little room for switch engines to appear, although I'm not sure if they fully repeat or not -- for example, 1x4330 is the smallest of the 1xN boxes that produces MWSSes.

[LuveelVoom]

Does there exist an unsynthesizable spaceship?

I think the best way to go about proving this would be to find an agar with a p2 generator and bottom eater, prove that it cannot be synthesized (but can be stabilized!) due to requiring infinite extension at the sides to synthesize, then stabilize it inside of a c/2o spaceship.