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Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 10:20 am
by wwei23
I know of four so far:
Code: Select all
x = 16, y = 12, rule = B3/S23
2o2b2o4b2ob2o$ob2obo4bobobo6$6bo$7o3b2o2b2o$o10bo2bo$o3bo6bo2bo$5o7b2o
!
Re: Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 10:46 am
by toroidalet
another one:
Code: Select all
x = 5, y = 3, rule = B3/S23
o3bo$bobo$obobo!
Re: Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 10:48 am
by BlinkerSpawn
I'm not so certain that the bottom two count in the first post count as "internally consistent" but that's just me looking at the problem.
Re: Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 1:34 pm
by wwei23
The conjecture refers to patterns where all the cells survive on to Generation 1. So it would look like this:
Code: Select all
x = 7, y = 3, rule = B3/S23
bo3bo$obobobo$bobobo!
Re: Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 2:23 pm
by praosylen
A counterexample on both sides:
Code: Select all
x = 6, y = 2, rule = B3/S23
2ob3o$2o2bo!
Re: Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 2:27 pm
by wwei23
It's nice, but the right side can be stabilized with a pre-block and the top can be stabilized with a much larger induction coil(Though the left can't be stabilized. You tied my low-cell record, congratulations!).
Code: Select all
x = 34, y = 16, rule = B3/S23
4b2o2b2o14b2o2b2o$2o3bo2bo3b2o6b2o3bo2bo3b2o$o2bo6bo2bo6bo2bo6bo2bo$2b
3o4b3o10b3o4b3o$6b2o18b2o$4b2o2b2o14b2o2b2o$3bo2b2o2bo12bo2b2o2bo$4b2o
2b2o14b2o2b2o$6b2o18b2o$4bo4bo14bo4bo$4b6o14b6o2$4b2ob3o14b2ob3o$4b2o
2bo2bo12b2o2bo$10b2o18bo$29b2o!
Re: Coolout Conjecture Counterexamples
Posted: June 24th, 2017, 2:42 pm
by wwei23
Found a much smaller induction coil:(Left side can't be stabilized.):
Code: Select all
x = 21, y = 8, rule = B3/S23
7b2o10b2o$o2bobo2bo3bo2bobo2bo$4ob3o4b4ob3o2$2ob3o6b2ob3o$2o2bo7b2o2bo
2bo$6bo11b2o$5b2o!
Re: Coolout Conjecture Counterexamples
Posted: June 26th, 2017, 10:38 am
by wwei23
Try apgsearching B3-i/S23 and close variants, that's how I found one counterexample.
Re: Coolout Conjecture Counterexamples
Posted: June 27th, 2017, 9:47 am
by wwei23
Three more:
Code: Select all
x = 21, y = 7, rule = B3/S23
3bo$2b3o5b3o3b2ob2o$bo3bo3bobobo3bobo$2o3b2ob2o3b2o3bo$bo3bo3bo3bo4bo$
2b3o5b3o4bobo$3bo7bo4b2ob2o!
Edit: the third one has a smaller counterexample in it:
Code: Select all
x = 6, y = 3, rule = B3/S23
2b2o$2o2b2o$o4bo!
EDIT OF EDIT:
The first and second are really the same with a smaller counterexample in it:
Code: Select all
x = 7, y = 5, rule = B3/S23
bo3bo$2o3b2o$bo3bo$2b3o$3bo!
Re: Coolout Conjecture Counterexamples
Posted: July 3rd, 2017, 7:27 pm
by wwei23
Two more:
Code: Select all
x = 16, y = 6, rule = LifeHistory
2.A$.3A$A3.A6.A3.A$A3.A6.2A.2A$.3A9.A$2.A!
Re: Coolout Conjecture Counterexamples
Posted: July 3rd, 2017, 8:29 pm
by BlinkerSpawn
I finally got around to updating the
Coolout Conjecture wiki entry but I'm not sure about how it looks.
Does it look ok to everyone else?
Re: Coolout Conjecture Counterexamples
Posted: July 3rd, 2017, 8:43 pm
by wwei23
I think that you are completely right. It can easily be shown that 2 by 5 is the smallest counterexample-there are only 512+256+64+4=836 patterns to search, easy-even by hand!
Edit:
I meant in a smaller bounding box.
Re: Coolout Conjecture Counterexamples
Posted: July 3rd, 2017, 9:55 pm
by gmc_nxtman
wwei23 wrote:I think that you are completely right. It can easily be shown that 2 by 5 is the smallest counterexample-there are only 512+256+64+4=836 patterns to search, easy-even by hand!
It is true that 2*5 is the smallest but there are actually 1024 possible patterns in a 2*5 bounding box.
Re: Coolout Conjecture Counterexamples
Posted: July 3rd, 2017, 11:19 pm
by BlinkerSpawn
gmc_nxtman wrote:wwei23 wrote:I think that you are completely right. It can easily be shown that 2 by 5 is the smallest counterexample-there are only 512+256+64+4=836 patterns to search, easy-even by hand!
It is true that 2*5 is the smallest but there are actually 1024 possible patterns in a 2*5 bounding box.
True, but some can be easily shown to be unfeasible even before evaluation. Not all the patterns need to be looked at.
Also, I showed this not by looking at 2*5 patterns but by showing that there were no 2*4 or 3*3 patterns to beat the 2*5 provided; see the first reference.
Re: Coolout Conjecture Counterexamples
Posted: July 22nd, 2017, 11:39 am
by wwei23
Three more:
Code: Select all
x = 25, y = 6, rule = LifeHistory
5A5.5A5.5A$A3.A5.A3.A5.A3.A$A3.A5.A3.A5.A3.A$5A5.A3.A5.A3.A$10.5A5.A
3.A$20.5A!
At first it seems like it is nonextendable for 3n+1, but it is:
Code: Select all
x = 5, y = 7, rule = LifeHistory
5A$ABDBA$AB.BA$A3.A$AB.BA$ABDBA$5A!
Re: Coolout Conjecture Counterexamples
Posted: August 7th, 2017, 10:45 am
by wwei23
Skew!
Code: Select all
x = 8, y = 8, rule = B3/S23
4bo$3bobo$b3obo$o4b2o$b2o4bo$2bob3o$2bobo$3bo!
Re: Coolout Conjecture Counterexamples
Posted: August 14th, 2017, 7:42 pm
by wwei23
Another:
Code: Select all
x = 6, y = 9, rule = B3/S23
5bo$3b3o$2bo$o2bo$4o2$3o$o2bo$2b2o!
Infinitely extendible:
Code: Select all
x = 37, y = 6, rule = B3/S23
o2bo6bo3bo5bo4bo4bo5bo$4o6b5o5b6o4b7o3$4o6b5o5b6o4b7o$o2bo6bo3bo5bo4bo
4bo5bo!
Re: Coolout Conjecture Counterexamples
Posted: August 16th, 2017, 5:06 pm
by fluffykitty
I think a clarification on "internally consistent" is required. I interpret it as meaning "all non-border cells do not change state".
Re: Coolout Conjecture Counterexamples
Posted: August 17th, 2017, 9:18 am
by wwei23
My interpretation is "no living cells die at the next generation."
Re: Coolout Conjecture Counterexamples
Posted: August 17th, 2017, 10:12 am
by dvgrn
wwei23 wrote:My interpretation is "no living cells die at the next generation."
I think the
original definition is more subtle than that. It seems as if it would be okay for living cells in a candidate counterexample to die at T=1. There just has to be a way to save each such individual cell from dying, by placing neighbor cells outside the boundary of the candidate pattern.
Internally consistent with being part of a still life definitely implies that non-boundary ON cells can't turn OFF at T=1 -- but also that non-boundary OFF cells can't turn ON at T=1.
Coolout Conjecture counterexamples show that very often, even though you can stabilize each individual cell considered separately, you can't always stabilize an arbitrary combination. Given that we have a minimum 2x5 counterexample, it's easy to generate an arbitrary number of additional larger counterexamples, but that doesn't add anything to the proof.
Re: Coolout Conjecture Counterexamples
Posted: August 17th, 2017, 12:27 pm
by fluffykitty
I wonder if there's a finite set of "minimal unstabilizable patterns" where every unstabilizable pattern contains at least one. Kind of like how K5 and K3,3 are the forbidden minors of planar graphs.