## Coolout Conjecture Counterexamples

For general discussion about Conway's Game of Life.

### Coolout Conjecture Counterexamples

I know of four so far:
x = 16, y = 12, rule = B3/S232o2b2o4b2ob2o$ob2obo4bobobo6$6bo$7o3b2o2b2o$o10bo2bo$o3bo6bo2bo$5o7b2o!
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

another one:
x = 5, y = 3, rule = B3/S23o3bo$bobo$obobo!
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toroidalet

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### Re: Coolout Conjecture Counterexamples

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.
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### Re: Coolout Conjecture Counterexamples

The conjecture refers to patterns where all the cells survive on to Generation 1. So it would look like this:
x = 7, y = 3, rule = B3/S23bo3bo$obobobo$bobobo!
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

A counterexample on both sides:
x = 6, y = 2, rule = B3/S232ob3o$2o2bo! 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: 1440 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 ### Re: Coolout Conjecture Counterexamples 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!). x = 34, y = 16, rule = B3/S234b2o2b2o14b2o2b2o$2o3bo2bo3b2o6b2o3bo2bo3b2o$o2bo6bo2bo6bo2bo6bo2bo$2b3o4b3o10b3o4b3o$6b2o18b2o$4b2o2b2o14b2o2b2o$3bo2b2o2bo12bo2b2o2bo$4b2o2b2o14b2o2b2o$6b2o18b2o$4bo4bo14bo4bo$4b6o14b6o2$4b2ob3o14b2ob3o$4b2o2bo2bo12b2o2bo$10b2o18bo$29b2o! Replicator! x = 3, y = 3, rule = B3/S234y2bo$3o$bo! wwei23 Posts: 642 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### Re: Coolout Conjecture Counterexamples Found a much smaller induction coil:(Left side can't be stabilized.): x = 21, y = 8, rule = B3/S237b2o10b2o$o2bobo2bo3bo2bobo2bo$4ob3o4b4ob3o2$2ob3o6b2ob3o$2o2bo7b2o2bo2bo$6bo11b2o$5b2o! Replicator! x = 3, y = 3, rule = B3/S234y2bo$3o$bo! wwei23 Posts: 642 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### Re: Coolout Conjecture Counterexamples Try apgsearching B3-i/S23 and close variants, that's how I found one counterexample. Replicator! x = 3, y = 3, rule = B3/S234y2bo$3o$bo! wwei23 Posts: 642 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### Re: Coolout Conjecture Counterexamples Three more: x = 21, y = 7, rule = B3/S233bo$2b3o5b3o3b2ob2o$bo3bo3bobobo3bobo$2o3b2ob2o3b2o3bo$bo3bo3bo3bo4bo$2b3o5b3o4bobo$3bo7bo4b2ob2o! Edit: the third one has a smaller counterexample in it: x = 6, y = 3, rule = B3/S232b2o$2o2b2o$o4bo! EDIT OF EDIT: The first and second are really the same with a smaller counterexample in it: x = 7, y = 5, rule = B3/S23bo3bo$2o3b2o$bo3bo$2b3o$3bo! Last edited by wwei23 on July 8th, 2017, 7:57 pm, edited 1 time in total. Replicator! x = 3, y = 3, rule = B3/S234y2bo$3o$bo! wwei23 Posts: 642 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### Re: Coolout Conjecture Counterexamples Two more: x = 16, y = 6, rule = LifeHistory2.A$.3A$A3.A6.A3.A$A3.A6.2A.2A$.3A9.A$2.A!
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

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?
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### Re: Coolout Conjecture Counterexamples

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.
Last edited by wwei23 on July 4th, 2017, 8:11 pm, edited 1 time in total.
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

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.

gmc_nxtman

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### Re: Coolout Conjecture Counterexamples

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.
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Three more:
x = 25, y = 6, rule = LifeHistory5A5.5A5.5A$A3.A5.A3.A5.A3.A$A3.A5.A3.A5.A3.A$5A5.A3.A5.A3.A$10.5A5.A3.A$20.5A! At first it seems like it is nonextendable for 3n+1, but it is: x = 5, y = 7, rule = LifeHistory5A$ABDBA$AB.BA$A3.A$AB.BA$ABDBA$5A! Replicator! x = 3, y = 3, rule = B3/S234y2bo$3o$bo! wwei23 Posts: 642 Joined: May 22nd, 2017, 6:14 pm Location: The (Life?) Universe ### Re: Coolout Conjecture Counterexamples Skew! x = 8, y = 8, rule = B3/S234bo$3bobo$b3obo$o4b2o$b2o4bo$2bob3o$2bobo$3bo!
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

Another:
x = 6, y = 9, rule = B3/S235bo$3b3o$2bo$o2bo$4o2$3o$o2bo$2b2o! Infinitely extendible: x = 37, y = 6, rule = B3/S23o2bo6bo3bo5bo4bo4bo5bo$4o6b5o5b6o4b7o3$4o6b5o5b6o4b7o$o2bo6bo3bo5bo4bo4bo5bo!
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

I think a clarification on "internally consistent" is required. I interpret it as meaning "all non-border cells do not change state".
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fluffykitty

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### Re: Coolout Conjecture Counterexamples

My interpretation is "no living cells die at the next generation."
Replicator!
x = 3, y = 3, rule = B3/S234y2bo$3o$bo!

wwei23

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### Re: Coolout Conjecture Counterexamples

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.
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### Re: Coolout Conjecture Counterexamples

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.
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fluffykitty

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