ConwayLife.com - A community for Conway's Game of Life and related cellular automata
Home  •  LifeWiki  •  Forums  •  Download Golly

Still life puzzles

For general discussion about Conway's Game of Life.

Re: Still life puzzles

Postby A for awesome » August 6th, 2017, 12:57 pm

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/S34
10b2ob2ob2o$9bob6obo$8bobo6bobo$4b2obob2ob4ob2obob2o$3bob4o2b2o2b2o2b
4obo$3b2o4b2o2b2o2b2o4b2o$4bob2obob2o2b2obob2obo$3b2ob2ob3ob2ob3ob2ob
2o$2bobo3bo3bo2bo3bo3bobo$bobob3ob2ob4ob2ob3obobo$ob2obobob2obo2bob2ob
obob2obo$2o2bob2o4bo2bo4b2obo2b2o$bob2obob12obob2obo$2obobobobo2bo2bo
2bobobobob2o$2obobobobo2bo2bo2bobobobob2o$bob2obob12obob2obo$2o2bob2o
4bo2bo4b2obo2b2o$ob2obobob2obo2bob2obobob2obo$bobob3ob2ob4ob2ob3obobo$
2bobo3bo3bo2bo3bo3bobo$3b2ob2ob3ob2ob3ob2ob2o$4bob2obob2o2b2obob2obo$
3b2o4b2o2b2o2b2o4b2o$3bob4o2b2o2b2o2b4obo$4b2obob2ob4ob2obob2o$8bobo6b
obo$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
User avatar
A for awesome
 
Posts: 1620
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

Re: Still life puzzles

Postby wwei23 » August 6th, 2017, 1:15 pm

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
User avatar
wwei23
 
Posts: 819
Joined: May 22nd, 2017, 6:14 pm
Location: The (Life?) Universe

Re: Still life puzzles

Postby A for awesome » August 6th, 2017, 1:28 pm

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/S34
2b2ob2o$bob3obo$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
User avatar
A for awesome
 
Posts: 1620
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

Re: Still life puzzles

Postby mniemiec » August 6th, 2017, 1:53 pm

wwei23 wrote:SHOW ME I AM SO EXCITED

My web page devoted to B34/S34: http://codercontest.com/mniemiec/rule34.htm
mniemiec
 
Posts: 923
Joined: June 1st, 2013, 12:00 am

Re: Still life puzzles

Postby dvgrn » August 6th, 2017, 1:57 pm

A for awesome wrote:Here's 36:
x = 8, y = 8, rule = B34/S34
2b2ob2o$bob3obo$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/S34
2b2ob2o$bob3obo$obo3bobo$2ob3ob2o$bobobobo$2ob3ob2o$obo3bobo$bob3obo$
2b2ob2o!

Probably a similar guess can find the 50- and 51-bit variants...?
User avatar
dvgrn
Moderator
 
Posts: 4696
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI

Re: Still life puzzles

Postby A for awesome » August 6th, 2017, 2:22 pm

50 and 51:
x = 44, y = 8, rule = B34/S34
b2obobob2o8b2obobob2o7b2ob2ob2o$ob7obo6bob7obo5bob6obo$2o7b2o6b2o7b2o
5b2o6bobo$bob5obo8bob2ob2obo7bob5ob2o$2ob2ob2ob2o6b2ob5ob2o5b2ob2ob2ob
o$obo5bobo6bobo5bobo5bobo6b2o$bob5obo8bob5obo7bob6obo$2b2obob2o10b2obo
b2o9b2ob2ob2o!
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
User avatar
A for awesome
 
Posts: 1620
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

Re: Still life puzzles

Postby wwei23 » August 6th, 2017, 7:52 pm

Can a still life be made of just T-tetrominoes?
x = 6, y = 6, rule = Life
2b3o$o2bo$2o3bo$o3b2o$2bo2bo$b3o!
User avatar
wwei23
 
Posts: 819
Joined: May 22nd, 2017, 6:14 pm
Location: The (Life?) Universe

Re: Still life puzzles

Postby mniemiec » August 6th, 2017, 8:46 pm

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
 
Posts: 923
Joined: June 1st, 2013, 12:00 am

Re: Still life puzzles

Postby Rhombic » August 7th, 2017, 5:59 am

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?
SoL : FreeElectronics : DeadlyEnemies : 6a-ite
what is “sesame oil”?
User avatar
Rhombic
 
Posts: 985
Joined: June 1st, 2013, 5:41 pm

Re: Still life puzzles

Postby mniemiec » August 7th, 2017, 9:17 am

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
 
Posts: 923
Joined: June 1st, 2013, 12:00 am

Re: Still life puzzles

Postby wwei23 » August 7th, 2017, 9:43 am

mniemiec wrote:61

9.
User avatar
wwei23
 
Posts: 819
Joined: May 22nd, 2017, 6:14 pm
Location: The (Life?) Universe

Re: Still life puzzles

Postby muzik » August 27th, 2017, 11:32 am

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
waiting for apgsearch to support one-dimensional rules
muzik
 
Posts: 2774
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

Re: Still life puzzles

Postby mniemiec » August 27th, 2017, 11:47 am

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
 
Posts: 923
Joined: June 1st, 2013, 12:00 am

Re: Still life puzzles

Postby muzik » August 27th, 2017, 11:50 am

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".
waiting for apgsearch to support one-dimensional rules
muzik
 
Posts: 2774
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

Re: Still life puzzles

Postby Hunting » August 12th, 2018, 8:49 am

wwei23 wrote:I found a potential induction coil. Now we just need to stabilize it.
x = 5, y = 10, rule = B3/S23
2bo$b3o$o3bo$2ob2o$bobo$bobo$2ob2o$o3bo$b3o$2bo!

Edit:
Never mind:
Because all the living cells of the seed have three living neighbors, no cells can be on adjacent to any of them. These cells are shown in blue:
x = 7, y = 12, rule = LifeHistory
2.3B$.2BA2B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A
2B$.2BA2B$2.3B!

Therefore, none of the red cells can be on, because they will cause a birth at gray. But all other surrounding cells except for the seed cells are blue, so the red cells must be off:
x = 7, y = 12, rule = LifeHistory
2.3B$D2BA2BD$BF3AFB$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$BF
3AFB$D2BA2BD$2.3B!

So all cells forced off are shown in blue:
x = 7, y = 12, rule = LifeHistory
2.3B$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A
2B$3BA3B$2.3B!

The red cells have three on neighbors, four forced off neighbors, and one unset neighbor. To prevent a birth at red, the unset neighbor must be forced on:
x = 7, y = 12, rule = LifeHistory
.A3BA$2BDAD2B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B
3A2B$2BDAD2B$.A3BA!

So this is the pattern so far:
x = 7, y = 12, rule = LifeHistory
.A3BA$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A
2B$3BA3B$.A3BA!

Because the red cells would cause a birth at gray, and all other neighbors are either seed cells, cells forced on, or blue, the red cells must be off:
x = 7, y = 12, rule = LifeHistory
DA3BAD$BFBABFB$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$
2B3A2B$BFBABFB$DA3BAD!

All blue cells must be off:
x = 7, y = 12, rule = LifeHistory
BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B
3A2B$3BA3B$BA3BAB!

Because each of the gray cells are on, they must have three living neighbors:
x = 7, y = 12, rule = LifeHistory
BF3BFB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B
3A2B$3BA3B$BF3BFB!

And they have exactly three unset neighbors each, in red:
x = 7, y = 14, rule = LifeHistory
3D.3D$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3B
AB$2B3A2B$3BA3B$BA3BAB$3D.3D!

So we can force them to be on:
x = 7, y = 14, rule = LifeHistory
3A.3A$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3B
AB$2B3A2B$3BA3B$BA3BAB$3A.3A!

The red cells are off, with three on neighbors:
x = 7, y = 14, rule = LifeHistory
3A.3A$BABDBAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA
3BAB$2B3A2B$3BA3B$BABDBAB$3A.3A!

And one gray unset neighbor:
x = 7, y = 14, rule = LifeHistory
3AF3A$BABDBAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA
3BAB$2B3A2B$3BA3B$BABDBAB$3AF3A!

To prevent a birth at red, the gray cells must be on:
x = 7, y = 14, rule = LifeHistory
7A$BABDBAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB
$2B3A2B$3BA3B$BABDBAB$7A!

Because some forced on cells have three living neighbors, none of their surrounding neighbors can be on, or else they die:
x = 7, y = 16, rule = LifeHistory
7B$7A$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3B
AB$2B3A2B$3BA3B$BA3BAB$7A$7B!

But that leaves two forced on cells shown in red that have only two neighbors. And because of their surrounding forced on cells that have three, they have no unset neighbors to be forced on:
x = 7, y = 16, rule = LifeHistory
7B$3AD3A$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$B
A3BAB$2B3A2B$3BA3B$BA3BAB$3AD3A$7B!

And therefore the initial pattern, the seed, cannot be stabilized.

I proved that
x = 10, y = 8, rule = LifeHistory
6.A$2.A2.3A$.4A3.A$A4.4A$2A3.A3.A$A3.A3.2A$.3A.2A.A$2.A
3.2A!

Cannot stabilize.
x = 12, y = 10, rule = LifeHistory
6.3B$2.5BA2BD$D2BA2B3AFB$BF4A3BAB$BA4B4A2B$B2A3BA3BAB$BA3BA3B2AB$BF3A
B2ABA2B$D2BABFB2AFB$2.3BD4BD!

So:
x = 12, y = 10, rule = LifeHistory
6.3B$2.5BA3B$3BA2B3A2B$2B4A3BAB$BA4B4A2B$B2A3BA3BAB$BA3BA3B2AB$2B3AB
2ABA2B$3BA3B2A2B$2.9B!

x = 12, y = 10, rule = LifeHistory
6.3B$2.5BA3B$3BABD3A2B$2B4A3BAB$BA4B4A2B$B2A3BA3BAB$BA3BA3B2AB$2B3AB
2ABA2B$3BA3B2A2B$2.9B!


Red cell have no unset neighbor.
Hunting
 
Posts: 82
Joined: September 11th, 2017, 2:54 am

Re: Still life puzzles

Postby calcyman » August 12th, 2018, 8:59 am

wwei23 wrote:And therefore the initial pattern, the seed, cannot be stabilized.


x = 5, y = 22, rule = B3/S23
2bo$bobo$2bo2$5o$o3bo$2bo$b3o$o3bo$2ob2o$bobo$bobo$2ob2o$o3bo$b3o$2bo$
o3bo$5o2$2bo$bobo$2bo!
What do you do with ill crystallographers? Take them to the mono-clinic!
User avatar
calcyman
 
Posts: 1642
Joined: June 1st, 2009, 4:32 pm

Re: Still life puzzles

Postby wwei23 » August 12th, 2018, 9:06 am

calcyman wrote:
wwei23 wrote:And therefore the initial pattern, the seed, cannot be stabilized.


x = 5, y = 22, rule = B3/S23
2bo$bobo$2bo2$5o$o3bo$2bo$b3o$o3bo$2ob2o$bobo$bobo$2ob2o$o3bo$b3o$2bo$
o3bo$5o2$2bo$bobo$2bo!

I was talking about in B3/S3.
User avatar
wwei23
 
Posts: 819
Joined: May 22nd, 2017, 6:14 pm
Location: The (Life?) Universe

Re: Still life puzzles

Postby danny » August 12th, 2018, 5:40 pm

wwei23 wrote:I was talking about in B3/S3.

Then why is this not in OCA? Why did you use LifeHistory?

Also no still lifes besides the block exist in B3/S3 anyway
get down with the sickness.
User avatar
danny
 
Posts: 593
Joined: October 27th, 2017, 3:43 pm
Location: i love to eat bees

Re: Still life puzzles

Postby mniemiec » August 12th, 2018, 7:56 pm

wwei23 wrote:I was talking about in B3/S3.

danny wrote:Then why is this not in OCA? Why did you use LifeHistory? Also no still lifes besides the block exist in B3/S3 anyway

When talking about still lifes, conditions don't matter. LifeHistory provides the ability to show different states, which is useful in illustrating a step-by-step proof, as was done above.
mniemiec
 
Posts: 923
Joined: June 1st, 2013, 12:00 am

Re: Still life puzzles

Postby Hdjensofjfnen » August 13th, 2018, 8:25 pm

wwei23 wrote:Can a still life be made of just T-tetrominoes?
x = 6, y = 6, rule = Life
2b3o$o2bo$2o3bo$o3b2o$2bo2bo$b3o!


No, it would have to be an infinite pattern of some sort. It has to be stabilised in 3 directions. To use an analogy, the block is the only still life in which every cell has 3 neighbors.
Life is hard. Deal with it.
This is my new favorite spaceship:
x = 8, y = 4, rule = B3-ek/S023
bobo$2obo3bo$bobo$3bo!
User avatar
Hdjensofjfnen
 
Posts: 519
Joined: March 15th, 2016, 6:41 pm
Location: (394234, -234231)

Previous

Return to General Discussion

Who is online

Users browsing this forum: wildmyron and 2 guests