## Still life puzzles

For general discussion about Conway's Game of Life.

### 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: 1809 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 wwei23 Posts: 935 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

A for awesome

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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$2b2ob2o! Probably a similar guess can find the 50- and 51-bit variants...? dvgrn Moderator Posts: 5634 Joined: May 17th, 2009, 11:00 pm Location: Madison, WI ### Re: Still life puzzles 50 and 51: x = 44, y = 8, rule = B34/S34b2obobob2o8b2obobob2o7b2ob2ob2o$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

A for awesome

Posts: 1809
Joined: September 13th, 2014, 5:36 pm
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! wwei23 Posts: 935 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### 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 Posts: 1022 Joined: June 1st, 2013, 12:00 am ### 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? SoL : FreeElectronics : DeadlyEnemies : 6a-ite : Rule X3VI what is “sesame oil”? Rhombic Posts: 1056 Joined: June 1st, 2013, 5:41 pm ### 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 Posts: 1022 Joined: June 1st, 2013, 12:00 am ### Re: Still life puzzles mniemiec wrote:61 9. wwei23 Posts: 935 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### 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 Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace! muzik Posts: 3301 Joined: January 28th, 2016, 2:47 pm Location: Scotland ### 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 Posts: 1022 Joined: June 1st, 2013, 12:00 am ### 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". Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace! muzik Posts: 3301 Joined: January 28th, 2016, 2:47 pm Location: Scotland ### Re: Still life puzzles wwei23 wrote:I found a potential induction coil. Now we just need to stabilize it. x = 5, y = 10, rule = B3/S232bo$b3o$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 = LifeHistory2.3B$.2BA2B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$.2BA2B$2.3B! Therefore, none of the red cells can be on, because they will cause a birth at gray. But all other surrounding cells except for the seed cells are blue, so the red cells must be off: x = 7, y = 12, rule = LifeHistory2.3B$D2BA2BD$BF3AFB$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$BF3AFB$D2BA2BD$2.3B!

So all cells forced off are shown in blue:
x = 7, y = 12, rule = LifeHistory2.3B$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$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$2B3A2B$2BDAD2B$.A3BA!

So this is the pattern so far:
x = 7, y = 12, rule = LifeHistory.A3BA$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$.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 = LifeHistoryDA3BAD$BFBABFB$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$BFBABFB$DA3BAD!

All blue cells must be off:
x = 7, y = 12, rule = LifeHistoryBA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BA3BAB! Because each of the gray cells are on, they must have three living neighbors: x = 7, y = 12, rule = LifeHistoryBF3BFB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BF3BFB!

And they have exactly three unset neighbors each, in red:
x = 7, y = 14, rule = LifeHistory3D.3D$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BA3BAB$3D.3D! So we can force them to be on: x = 7, y = 14, rule = LifeHistory3A.3A$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BA3BAB$3A.3A!

The red cells are off, with three on neighbors:
x = 7, y = 14, rule = LifeHistory3A.3A$BABDBAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BABDBAB$3A.3A! And one gray unset neighbor: x = 7, y = 14, rule = LifeHistory3AF3A$BABDBAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BABDBAB$3AF3A!

To prevent a birth at red, the gray cells must be on:
x = 7, y = 14, rule = LifeHistory7A$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 = LifeHistory7B$7A$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$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 = LifeHistory7B$3AD3A$BA3BAB$3BA3B$2B3A2B$BA3BAB$B2AB2AB$2BABA2B$2BABA2B$B2AB2AB$BA3BAB$2B3A2B$3BA3B$BA3BAB$3AD3A$7B! And therefore the initial pattern, the seed, cannot be stabilized. I proved that x = 10, y = 8, rule = LifeHistory6.A$2.A2.3A$.4A3.A$A4.4A$2A3.A3.A$A3.A3.2A$.3A.2A.A$2.A3.2A!

Cannot stabilize.
x = 12, y = 10, rule = LifeHistory6.3B$2.5BA2BD$D2BA2B3AFB$BF4A3BAB$BA4B4A2B$B2A3BA3BAB$BA3BA3B2AB$BF3AB2ABA2B$D2BABFB2AFB$2.3BD4BD! So: x = 12, y = 10, rule = LifeHistory6.3B$2.5BA3B$3BA2B3A2B$2B4A3BAB$BA4B4A2B$B2A3BA3BAB$BA3BA3B2AB$2B3AB2ABA2B$3BA3B2A2B$2.9B!

x = 12, y = 10, rule = LifeHistory6.3B$2.5BA3B$3BABD3A2B$2B4A3BAB$BA4B4A2B$B2A3BA3BAB$BA3BA3B2AB$2B3AB2ABA2B$3BA3B2A2B$2.9B! Red cell have no unset neighbor. This post was brought to you by the Element of Magic. Plz correct my grammar mistakes. I'm still studying English. Working on: Nothing. Favorite gun ever: #C Favorite Gun. Found by me.x = 4, y = 6, rule = B2e3i4at/S1c23cijn4ao2bo$4o3$4o$o2bo!
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### Re: Still life puzzles

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

x = 5, y = 22, rule = B3/S232bo$bobo$2bo2$5o$o3bo$2bo$b3o$o3bo$2ob2o$bobo$bobo$2ob2o$o3bo$b3o$2bo$o3bo$5o2$2bo$bobo$2bo! What do you do with ill crystallographers? Take them to the mono-clinic! calcyman Posts: 2031 Joined: June 1st, 2009, 4:32 pm ### Re: Still life puzzles calcyman wrote: wwei23 wrote:And therefore the initial pattern, the seed, cannot be stabilized. x = 5, y = 22, rule = B3/S232bo$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.

wwei23

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### Re: Still life puzzles

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
she/they // Please stop using my full name. Refer to me as dani.

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### Re: Still life puzzles

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

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### Re: Still life puzzles

wwei23 wrote:Can a still life be made of just T-tetrominoes?
x = 6, y = 6, rule = Life2b3o$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. My favorite oscillator of all time: x = 7, y = 5, rule = B3/S2-i3-y4i4b3o$6bo$o3b3o$2o\$bo!

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