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## Still life puzzles

For general discussion about Conway's Game of Life.

### Re: Still life puzzles

wwei23 wrote:
dvgrn wrote:One more question: does this glider... strike in the "same spot" as these *WSSes?

Basically first interaction must be the same.

Hmm. Okay, that makes it a little tougher. Now I'm fairly sure that no such eater currently exists, and also that the easiest way to find one would be to build something very large that recovers very slowly.

If I had to construct one of these, I'd probably look for a small constellation that exploded when struck by any *WSS, as the pond does -- but that also exploded in a different way when struck by a glider (with the same initial interaction).

Then it's "just" a matter of adding catalysts to channel the two explosions, getting a different signal output in each case, and using each signal appropriately to send glider salvos back to clean up any leftover junk and rebuild the initial constellation.

It would certainly be much nicer to find a bait object that explodes the same way no matter what hits it, or even better a true four-way omnivorous eater -- generated by Bellman, let's say. But it seems as if that's likely to be a lot harder to find. Gliders just plain have a lot less "weight" behind them than the weightships do.

dvgrn
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### Re: Still life puzzles

wwei23 wrote:Do any still lifes exist from 9 to 19 cells so that every living cell has exactly two living neighbors and if not, then why(I've also apgsearched and turned up empty-handed.)?

No. I just did some searches in rule B3/S23 (where all still-lifes must necessarily be of that form), and the counts for various populations are:
4=1; 6=2; 7=1; 8=2; 20=4; 21=1; 22=34; 23=34; 24=57; 25=25; 26=97; 27=88; 28=165; 29=91; 30=129; 31=121; 32=265.

The reason there is a huge gap between 8 and 20 is that this constraint requires that all still-lifes be formed of one or more loops of living cells. This can be stable loops like beehives and ponds, unstable loops that require external stabilization, or L-trominos which also require stabilization. The smallest island that requires such stabilization is 8 bits (a long beehive) but that would require at least another copy of itself on both sides. A 10-bit long long beehive requires at least two beehives, for 22. A 12-bit long beehive can be supported by 2 tubs, for 20, but at that point, we already have the small lake.
wwei23 wrote:Other than some agars, is the block the only still life where every living cell has exactly three neighbors? If so, then why?

A brute-force search reveals none (other than the block) up to 36 bits. If you try to construct one manually, there are many that work as wicks or agars, but none appear to have finite stabilizations. It may be possible to prove this, although it appears that such a proof would be fairly complicated.

Some years ago, I went through a similar process for the rule B34/S34, in which P2 oscillators are plentiful, but the block appears to be the only still-life. I was able to show that other than the block, all still-lifes must necessarily have an exterior that consists of crenelations (straight lines with 1-bit protrusions every 2 or 3 cells), with Life ships or long ships at each corner. No other forms are possible. The smallest such still-life is 36 bits, and an exhaustive search did indeed find that one, plus one at 44; the next ones are two at 50 and one at 51.
mniemiec

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### Re: Still life puzzles

mniemiec wrote:Some years ago, I went through a similar process for the rule B34/S34, in which P2 oscillators are plentiful, but the block appears to be the only still-life. I was able to show that other than the block, all still-lifes must necessarily have an exterior that consists of crenelations (straight lines with 1-bit protrusions every 2 or 3 cells), with Life ships or long ships at each corner. No other forms are possible.

What about something like this? Maybe I'm misunderstanding.
x = 28, y = 28, rule = B34/S3410b2ob2ob2o$9bob6obo$8bobo6bobo$4b2obob2ob4ob2obob2o$3bob4o2b2o2b2o2b4obo$3b2o4b2o2b2o2b2o4b2o$4bob2obob2o2b2obob2obo$3b2ob2ob3ob2ob3ob2ob2o$2bobo3bo3bo2bo3bo3bobo$bobob3ob2ob4ob2ob3obobo$ob2obobob2obo2bob2obobob2obo$2o2bob2o4bo2bo4b2obo2b2o$bob2obob12obob2obo$2obobobobo2bo2bo2bobobobob2o$2obobobobo2bo2bo2bobobobob2o$bob2obob12obob2obo$2o2bob2o4bo2bo4b2obo2b2o$ob2obobob2obo2bob2obobob2obo$bobob3ob2ob4ob2ob3obobo$2bobo3bo3bo2bo3bo3bobo$3b2ob2ob3ob2ob3ob2ob2o$4bob2obob2o2b2obob2obo$3b2o4b2o2b2o2b2o4b2o$3bob4o2b2o2b2o2b4obo$4b2obob2ob4ob2obob2o$8bobo6bobo$9bob6obo$10b2ob2ob2o! x₁=ηx V ⃰_η=c²√(Λη) K=(Λu²)/2 Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt) $$x_1=\eta x$$ $$V^*_\eta=c^2\sqrt{\Lambda\eta}$$ $$K=\frac{\Lambda u^2}2$$ $$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$ http://conwaylife.com/wiki/A_for_all Aidan F. Pierce A for awesome Posts: 1611 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 ### Re: Still life puzzles mniemiec wrote:Some years ago, I went through a similar process for the rule B34/S34, in which P2 oscillators are plentiful, but the block appears to be the only still-life. I was able to show that other than the block, all still-lifes must necessarily have an exterior that consists of crenelations (straight lines with 1-bit protrusions every 2 or 3 cells), with Life ships or long ships at each corner. No other forms are possible. The smallest such still-life is 36 bits, and an exhaustive search did indeed find that one, plus one at 44; the next ones are two at 50 and one at 51. SHOW ME I AM SO EXCITED Please, stop spam searching Life. wwei23 Posts: 754 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### Re: Still life puzzles wwei23 wrote: mniemiec wrote:Some years ago, I went through a similar process for the rule B34/S34, in which P2 oscillators are plentiful, but the block appears to be the only still-life. I was able to show that other than the block, all still-lifes must necessarily have an exterior that consists of crenelations (straight lines with 1-bit protrusions every 2 or 3 cells), with Life ships or long ships at each corner. No other forms are possible. The smallest such still-life is 36 bits, and an exhaustive search did indeed find that one, plus one at 44; the next ones are two at 50 and one at 51. SHOW ME I AM SO EXCITED Here's 36: x = 8, y = 8, rule = B34/S342b2ob2o$bob3obo$obo3b2o$2ob2obo$bob2ob2o$2o3bobo$ob3obo$b2ob2o!

I don't know about the other ones.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

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### Re: Still life puzzles

wwei23 wrote:SHOW ME I AM SO EXCITED

My web page devoted to B34/S34: http://codercontest.com/mniemiec/rule34.htm
mniemiec

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### Re: Still life puzzles

A for awesome wrote:Here's 36:
x = 8, y = 8, rule = B34/S342b2ob2o$bob3obo$obo3b2o$2ob2obo$bob2ob2o$2o3bobo$ob3obo$b2ob2o! A four-way symmetric version of this is 44 bits, so I assume that's the next smallest one: x = 9, y = 9, rule = B34/S342b2ob2o$bob3obo$obo3bobo$2ob3ob2o$bobobobo$2ob3ob2o$obo3bobo$bob3obo$2b2ob2o! Probably a similar guess can find the 50- and 51-bit variants...? dvgrn Moderator Posts: 4465 Joined: May 17th, 2009, 11:00 pm Location: Madison, WI ### Re: Still life puzzles 50 and 51: x = 44, y = 8, rule = B34/S34b2obobob2o8b2obobob2o7b2ob2ob2o$ob7obo6bob7obo5bob6obo$2o7b2o6b2o7b2o5b2o6bobo$bob5obo8bob2ob2obo7bob5ob2o$2ob2ob2ob2o6b2ob5ob2o5b2ob2ob2obo$obo5bobo6bobo5bobo5bobo6b2o$bob5obo8bob5obo7bob6obo$2b2obob2o10b2obob2o9b2ob2ob2o!
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

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Location: 0x-1

### Re: Still life puzzles

Can a still life be made of just T-tetrominoes?
x = 6, y = 6, rule = Life2b3o$o2bo$2o3bo$o3b2o$2bo2bo\$b3o!
Please, stop spam searching Life.
wwei23

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### Re: Still life puzzles

I wrote:My web page devoted to B34/S34: http://codercontest.com/mniemiec/rule34.htm

dvgrn wrote:A four-way symmetric version of this is 44 bits, so I assume that's the next smallest one: ...

A for awesome wrote:50 and 51: ...

See my page above for all of these (although I never bothered looking for larger ones), plus much more of interest on B34/S34. The only other references I could find on the internet (other than the initial brief introduction to the rule in Lifeline) was Jack Eisenmann's page:
http://www.ostracodfiles.com/34life/main.html, which contains several oscillators and many new spaceships that (as far as I am aware) were not previously known.

wwei23 wrote:Can a still life be made of just T-tetrominoes? ...

It can be done with an agar, but not with a finite still-life, as the T-tetromino is unstable on three different sides, and the only way to stabilize it is with other T-tetrominos. The entire outside of any collection of T-tetrominos must necessarily be unstable.
mniemiec

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### Re: Still life puzzles

dvgrn wrote:
A for awesome wrote:Here's 36:[codecode
[/code]

A four-way symmetric version of this is 44 bits, so I assume that's the next smallest one:

code

Probably a similar guess can find the 50- and 51-bit variants...?

What sequence do the valid numbers follow? Is there an infinite number of distinct cell count still lifes?
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### Re: Still life puzzles

Rhombic wrote:What sequence do the valid numbers follow? Is there an infinite number of distinct cell count still lifes?

There are an infinite number of still-lifes in most rules, and the numbers tend to increase roughtly with population. Unfortunately, in most rules, the dynamics of still life construction are sufficiently sophisticated that there is no obvious easy-to-use formula for them. For example, in Life, the first few counts are 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, ... As the numbers grow large, they appear to grow at a rate of approximately O(2.4^n), although this is merely an empirical observation, as statistics only exist up to 32 bits; this rate may change (probably upwards) with much higher sizes. (Some rules have finite numbers of still-lifes; e.g. B3/S0 has one and B3/S1 has two.)
Last edited by mniemiec on August 8th, 2017, 3:36 am, edited 1 time in total.
mniemiec

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### Re: Still life puzzles

mniemiec wrote:61

9.
Please, stop spam searching Life.
wwei23

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### Re: Still life puzzles

muzik wrote:I'm pretty sure that the block is the only finite pattern at all with very cell having exactly 3 neighbours.

Turns out this is a big, fat, stinking lie:
https://catagolue.appspot.com/census/bs3/D8_1
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### Re: Still life puzzles

muzik wrote:I'm pretty sure that the block is the only finite pattern at all with very cell having exactly 3 neighbours.
muzik wrote:Turns out this is a big, fat, stinking lie:
https://catagolue.appspot.com/census/bs3/D8_1

Shouldn't you be searching b3s3 instead of bs3?
mniemiec

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### Re: Still life puzzles

mniemiec wrote:
muzik wrote:I'm pretty sure that the block is the only finite pattern at all with very cell having exactly 3 neighbours.
muzik wrote:Turns out this is a big, fat, stinking lie:
https://catagolue.appspot.com/census/bs3/D8_1

Shouldn't you be searching b3s3 instead of bs3?

I said "finite pattern" instead of just "still life".
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