[Game] Most number of minpop 4-cell oscillators RELOADED

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H. H. P. M. P. Cole
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[Game] Most number of minpop 4-cell oscillators RELOADED

Post by H. H. P. M. P. Cole » October 22nd, 2023, 1:46 am

1001st topic! Mods, can this be moved to the OCA board?

This game is similar to this thread, but better. This time I allow LtL rules. These results are categorized by the number of cells they have in their neighbourhood. Hence R1 Moore and R2 Cross are in the same category but R2 Cross and R3 Cross are not.

Minpop 4-cell spaceships are allowed now!

Here are the previous rules:

RULES
- No infinite series of only 4-cell objects (fixed because of a loophole Haycat found)
- No rules where the only 4-cell stable objects are still-lifes (so no B/S012345678)
- No objects (SLs, oscs, spaceships, and growing patterns that never stop) with minpop lower than 4 in the rule
- No explosive rules
- No 4-cell predecessors of other non-minpop-4 oscillators

To this I add three more rule-making rules:
- Rule HAS to be isotropic
- Rule must not be B0 (due to a loophole toroidalet found)
- The number of points you have is governed by the number of minpop 4-cell oscillators you have in your rule (due to a loophole Haycat found)

For 8 cells in the neighbourhood there is my 13-oscillator Linea variant in this thread:

Code: Select all

x = 22, y = 35, rule = B2-an3ceqry4c5a/S1c2-en3-n5a6a7c
2o4b2o4bo5b3o$2o5b2o4b2o4bo$13bo4$2bo12bo3bobo$2o5b2o4b2o$o5bo2bo2bo6b
obo5$bo$o$3bo$2bo3$bo$3bo2bobo$o6bo$2bo5bo4$obo$bobo5$2bo$o3bo$2bo!
For 12 cells in the neighbourhood here is a CA with 40. You might know this as my CA Factorio.

Code: Select all

x = 149, y = 80, rule = R3,C2,S2,B3,N+
2o13bobo13bo2bo$2o13bobo13bo2bo9$15bobo13bo2bo2$15bobo13bo2bo9$31bo2b
o3$31bo2bo17$2b2o11b2o14b2o12bobo15bobo15b2o15b2o10bobo9b2o12b2o8bobo
$bo2bo9bo3bo10bo4bo9bo3bo13bobo14b2o16bobo9bo2bo7bo2bo11b2o8b2o9$2b2o
11b2o14b2o12bobo15bobo15b2o15b2o10bobo9b2o12b2o8bobo2$bo2bo9bo3bo10bo
4bo9bo3bo13bobo14b2o16bobo9bo2bo7bo2bo11b2o8b2o9$2b2o11b2o14b2o12bobo
15bobo15b2o15b2o10bobo9b2o12b2o8bobo3$bo2bo9bo3bo10bo4bo9bo3bo13bobo14b
2o16bobo9bo2bo7bo2bo11b2o8b2o14$bo2b3o!
The minrule of this collection in Durnak-Cole notation is:

*coughs*

B3aa.ab.ac.ad.ae.af.ag.ah.ai.aj.an.ao.ap.aq.ar.as.at/S2-a.b.j.k.m.n.o

Please let me know if there are any mistakes in this rulestring.

Use CAViewer notation for R2 Knight and R2 Cross. Use Hensel notation for Far Edges/Corners.

Here is a preliminary result:

With 4 cells in the neighbourhood the challenge is unsatisfiable. Here's why:

The only isotropic 4-cell neighbourhoods are (up to expansion) the R1 vN and Saltire neighbourhoods. The Saltire neighbourhood is equivalent to the vN neighbourhood rotated 45 degrees.

B0 and B1 are disallowed. S0 and S1 are also disallowed.
B2e is disallowed due to the duoplet being an oscillator. Adding S0 creates the dot. Hence B2e is disallowed.
S2e creates a block.
B2i/S2e does not create new 4-cell objects, so do B3/S2e or B4/S2e.
S3 and S4 need S1 to create a new SL, or B2e for a new oscillator. Both are already disallowed.

Hence this challenge for 4 cells in the neighbourhood is unsatisfiable. Q.E.D.

Try your luck on this challenge if you have CAViewer, as if the original version wasn't hard enough!
Last edited by H. H. P. M. P. Cole on October 30th, 2023, 2:56 am, edited 4 times in total.
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Re: [Game] Most number of minpop 4-cell oscillators RELOADED

Post by Haycat2009 » October 22nd, 2023, 10:08 am

H. H. P. M. P. Cole wrote:
October 22nd, 2023, 1:46 am
1001st topic! Mods, can this be moved to the OCA board?

This game is similar to this thread, but better. This time I allow LtL rules. These results are categorized by the number of cells they have in their neighbourhood. Hence R1 Moore and R2 Cross are in the same category but R2 Cross and R3 Cross are not.

Here are the previous rules:

RULES
- No infinite series
- No rules where the only 4-cell stable objects are still-lifes (so no B/S012345678)
- No objects (SLs, oscs, spaceships, and growing patterns that never stop) with minpop lower than 4 in the rule
- No explosive rules
- No 4-cell predecessors of other non-minpop-4 oscillators

To this I add two more rule-making rules:
- Rule HAS to be isotropic
- Rule must not be B0 (due to a loophole toroidalet found)

For 8 cells in the neighbourhood there is my 13-oscillator Linea variant in this thread:

Code: Select all

x = 22, y = 35, rule = B2-an3ceqry4c5a/S1c2-en3-n5a6a7c
2o4b2o4bo5b3o$2o5b2o4b2o4bo$13bo4$2bo12bo3bobo$2o5b2o4b2o$o5bo2bo2bo6b
obo5$bo$o$3bo$2bo3$bo$3bo2bobo$o6bo$2bo5bo4$obo$bobo5$2bo$o3bo$2bo!
For 12 cells in the neighbourhood here is a CA with 40. You might know this as my CA Factorio.

Code: Select all

x = 149, y = 80, rule = R3,C2,S2,B3,N+
2o13bobo13bo2bo$2o13bobo13bo2bo9$15bobo13bo2bo2$15bobo13bo2bo9$31bo2b
o3$31bo2bo17$2b2o11b2o14b2o12bobo15bobo15b2o15b2o10bobo9b2o12b2o8bobo
$bo2bo9bo3bo10bo4bo9bo3bo13bobo14b2o16bobo9bo2bo7bo2bo11b2o8b2o9$2b2o
11b2o14b2o12bobo15bobo15b2o15b2o10bobo9b2o12b2o8bobo2$bo2bo9bo3bo10bo
4bo9bo3bo13bobo14b2o16bobo9bo2bo7bo2bo11b2o8b2o9$2b2o11b2o14b2o12bobo
15bobo15b2o15b2o10bobo9b2o12b2o8bobo3$bo2bo9bo3bo10bo4bo9bo3bo13bobo14b
2o16bobo9bo2bo7bo2bo11b2o8b2o14$bo2b3o!
The minrule of this collection in Durnak-Cole notation is:

*coughs*

B3aa.ab.ac.ad.ae.af.ag.ah.ai.aj.an.ao.ap.aq.ar.as.at/S2-a.b.j.k.m.n.o

Please let me know if there are any mistakes in this rulestring.

Use CAViewer notation for R2 Knight and R2 Cross. Use Hensel notation for Far Edges/Corners.

Here is a preliminary result:

With 4 cells in the neighbourhood the challenge is unsatisfiable. Here's why:

The only isotropic 4-cell neighbourhoods are (up to expansion) the R1 vN and Saltire neighbourhoods. The Saltire neighbourhood is equivalent to the vN neighbourhood rotated 45 degrees.

B0 and B1 are disallowed. S0 and S1 are also disallowed.
B2e is disallowed due to the duoplet being an oscillator. Adding S0 creates the dot. Hence B2e is disallowed.
S2e creates a block.
B2i/S2e does not create new 4-cell objects, so do B3/S2e or B4/S2e.
S3 and S4 need S1 to create a new SL, or B2e for a new oscillator. Both are already disallowed.

Hence this challenge for 4 cells in the neighbourhood is unsatisfiable. Q.E.D.

Try your luck on this challenge if you have CAViewer, as if the original version wasn't hard enough!
40 is going to be very hard to beat. Do different phrases of the same pattern count? That is another loophole.
~ Haycat Durnak, a hard-working editor
Also, support Conway and Friends story mode!
I mean no harm to those who have tested me. But do not take this for granted.

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H. H. P. M. P. Cole
Posts: 90
Joined: July 15th, 2023, 9:36 pm
Location: Wherever Haycat2009 is

Re: [Game] Most number of minpop 4-cell oscillators RELOADED

Post by H. H. P. M. P. Cole » October 22nd, 2023, 3:51 pm

Haycat2009 wrote:
October 22nd, 2023, 10:08 am
40 is going to be very hard to beat. Do different phrases of the same pattern count? That is another loophole.
No. If all patterns are mod 1 then counting would be easier. As in the previous game, the number of points you have is governed by the number of minpop 4-cell oscillators/spaceships in your rule.

I edited the rule-making rules to account for your loophole.

EDIT: Made a wiki page: https://conwaylife.com/wiki/User:H._H._ ... scillators
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get_Snacked
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Re: [Game] Most number of minpop 4-cell oscillators RELOADED

Post by get_Snacked » November 9th, 2023, 9:27 am

infinite:

Code: Select all

x = 3, y = 25, rule = B2cei7e/S012aik3ir
bo22$obo2$bo!
1983263225470666662666647618

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rutabaga
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Re: [Game] Most number of minpop 4-cell oscillators RELOADED

Post by rutabaga » November 15th, 2023, 2:02 pm

get_Snacked wrote:
November 9th, 2023, 9:27 am
infinite:

Code: Select all

x = 3, y = 25, rule = B2cei7e/S012aik3ir
bo22$obo2$bo!
unfortunately, no infinite is allowed

Code: Select all

x=4, y=9, rule=1e234j/3-j/6
.2A3$2A$A.EA$.BEA$2.C$.B2C$2.2A!
i'm back! here have a p12

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H. H. P. M. P. Cole
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Joined: July 15th, 2023, 9:36 pm
Location: Wherever Haycat2009 is

Re: [Game] Most number of minpop 4-cell oscillators RELOADED

Post by H. H. P. M. P. Cole » February 7th, 2024, 9:45 pm

Time to revive this thread.

I have devised a method which allows one to solve the HROT version of this game. This method is far from ideal as it only considers SLs and constant population p2s. (If you are interested this method involves graph theory.)

Using this method, I have come up with a HROT rule with 5 SLs and 4 p2s. You can check that it follows the constraints stated at the start of this thread.

Code: Select all

x = 36, y = 10, rule = R3,C2,S2,B3,NW0001000000000000010001010101000100000000000001000
bobo6bobo5b4o5b2o5bo2bo$o3bo4bobo15b2o5bo2bo5$bobo6bobo5b2ob2o11bo2bo
3$o3bo4bobo22bo2bo!
I'm pretty sure there is one with more, but I'm not sure yet. If anyone can find a rule with more minpop-4 cell objects, let me know!

Edit: There is no hard limit to the number of 4-cell objects in this game.

PROOF:
Let w be the maximum width of the 4-cell objects in some rule. Now layer the neighbourhood on itself, but one copy has a scale factor of w+1, and a weight of 5. Taking the union of the original B/S conditions and the multiplied-by-5 B/S conditions creates a rule with 2n 4-cell objects where n is the number of 4-cell objects in the original rule. One can do this by induction: multiplying weights by 5 each time, giving weights of 1, 5, 125, 625, etc.
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