confocaloid wrote: January 22nd, 2025, 12:42 pm
I think these are the highest-population still-life patterns in square-shaped regions (see also
Density):
This looks correct to me. I also used an ilp solver (which I also used, or used an earlier version of, to find the mice and methuselahs in
here) to find the highest population still-lifes in squares of sizes 13-16 [Edit May 23: also 17]:
Code:
Select all
x = 17, y = 83, rule = B36/S23
2ob2obob2ob2o$2obob3obob2o$3bo5bo$3ob3obob3o$o2bo2bob2o2bo$bobobo5bo$
2ob2ob4ob2o$bo4bo2bobo$o2b2obo2bo2bo$3obob3ob3o$3bo5bo$2obob3obob2o$2o
b2obob2ob2o3$2ob2ob2ob2ob2o$2ob2ob2ob2ob2o2$2ob2ob2ob2ob2o$2ob2ob2ob2o
b2o2$2ob2ob2ob2ob2o$2ob2ob2ob2ob2o2$2ob2ob2ob2ob2o$2ob2ob2ob2ob2o2$2o
b2ob2ob2ob2o$2ob2ob2ob2ob2o3$o2bob2ob2ob2obo$5obob2obob2o$6bo4bo$2ob3o
b3obob2o$bobo2bo2bobob2o$o2bo2bobo2bo$3ob3ob3ob3o$3bo3bo3bo2bo$2obobo
bobobobo$bobob2obob2ob2o$o2bo4bo$3ob4ob6o$3bo4bo5bo$2obob2obob2obo$2o
b2obob2obob2o3$2ob2ob2obob2ob2o$2ob2obob3obob2o$6bo5bo$5ob2ob3ob3o$o3b
o2bobo2bo2bo$bobo2bo2bo2bobo$2ob2obob2ob2ob2o$bo2bobo2bo2bo2bo$o2bo2b
2obobo2bo$2ob2obo2bob2ob2o$bobo2bob2o2bobo$o3bobo2bobo3bo$5ob2obob5o$
9bo$2ob2ob2obob2ob2o$2ob2ob2ob2obob2o3$2ob2obob2obob2obo$bobob2obob2o
bob2o$o2bo4bo4bo$3ob4ob4ob3o$3bo4bo4bo2bo$2obob2obob2obobo$bobob2obob
2obob2o$o2bo4bo4bo$3ob4ob4ob3o$3bo4bo4bo2bo$2obob2obob2obobo$bobob2ob
ob2obob2o$o2bo4bo4bo$3ob4ob4ob3o$3bo4bo4bo2bo$2obob2obob2obobo$ob2obo
b2obob2ob2o!
I also found a still-life with 148 cells in a 17x17 square, but it is not yet proven maximal, so there might be one with 149 cells. The ilp solver already rejected any 150 cell options, I'll update here when I see if it finds a 149 cell solution or rejects the option:
[Edit May 23: It's maximal, copied to the pattern above]
This is particularly interesting since it is denser than the
block agar, which is the densest for all smaller 3n+2 side lengths. Unlike a portion of the block agar with the same side lengths, and like all other maximal population still lifes in squares with side length at least 2, it has population larger than half the area of the square in which it resides, which is 144.5 in this case. This, and the fact that the simplest
chicken wire also works in highlife, makes me conjecture that
like in life, the maximal density of a highlife still life in any square is at least 1/2.
Another thing I found with the ilp solver is a way to stabilize the UWSS, similarly to
this in life:
Code:
Select all
x = 20, y = 16, rule = B36/S23
16bo$b5o3bo7bo$o5bo2b4o4b2o$7bob4o3b2o$o9bo5bo$8bo3b3o$8bo4bo3b3o$9b4o
bobo2bo$6bobo10bo$7bob2ob2obo2bo$7bo2b2ob2o$b4obobobo3b2obo$o4bo4b2o3b
4o$5bo10b2o$o3bo11bo$2bo!
It has a mod of 2 and a minimum population of 82 cells.
Edit May 28th: This is the minimal population for a mod 2 spaceship that extends the UWSS from behind and is contained with a band of width 21.
Like the similar life spaceship this one can also support a MWSS instead of an UWSS:
Code:
Select all
x = 22, y = 16, rule = B36/S23
16bo$b5o3bo7bo$o5bo2b4o4b2o$7bob4o3b2o$o9bo5bo$8bo3b3o$8bo4bo3b5o$9b4o
bobo4bo$6bobo12bo$7bob2ob2obo4bo$7bo2b2ob2o$b4obobobo3b2obo$o4bo4b2o3b
4o$5bo10b2o$o3bo11bo$2bo!
In addition, each end can be lengthened individually to make either a mod and period 4 spaceship or a larger mod 2 spaceship:
Code:
Select all
x = 23, y = 76, rule = B36/S23
17bo$b6o3bo7bo$o6bo2b4o4b2o$8bob4o3b2o$o10bo5bo$2bo6bo3b3o$9bo4bo3b3o
$10b4obobo2bo$7bobo10bo$8bob2ob2obo2bo$8bo2b2ob2o$2b4obobobo3b2obo$bo
4bo4b2o3b4o$6bo10b2o$bo3bo11bo$3bo5$17bo$b6o3bo7bo$o6bo2b4o4b2o$8bob4o
3b2o$o10bo5bo$2bo6bo3b3o$9bo4bo3b3o$10b4obobo2bo$7bobo10bo$8bob2ob2ob
o2bo$8bo2b2ob2o$b5obobobo3b2obo$o5bo4b2o3b4o$6bo10b2o$o4bo11bo$2b2o5$
17bo$b6o3bo7bo$o6bo2b4o4b2o$8bob4o3b2o$o10bo5bo$2bo6bo3b3o$9bo4bo3b5o
$10b4obobo4bo$7bobo12bo$8bob2ob2obo4bo$8bo2b2ob2o$2b4obobobo3b2obo$bo
4bo4b2o3b4o$6bo10b2o$bo3bo11bo$3bo5$17bo$b6o3bo7bo$o6bo2b4o4b2o$8bob4o
3b2o$o10bo5bo$2bo6bo3b3o$9bo4bo3b5o$10b4obobo4bo$7bobo12bo$8bob2ob2ob
o4bo$8bo2b2ob2o$b5obobobo3b2obo$o5bo4b2o3b4o$6bo10b2o$o4bo11bo$2b2o!
