It is well-known, but up to n=9, I verified via running my fast-made mixed integer program containing real and binary variables only - in GMPL, to be published later. I am aware that this had been done by others earlier, just for fun

An excerpt from logs when determining n=9 case:

+35650619: mip = 4.300000000e+01 <= 4.400000000e+01 2.3% (425; 358842)

+35673703: mip = 4.300000000e+01 <= 4.400000000e+01 2.3% (332; 359323)

Warning: numerical instability (dual simplex, phase II)

+35697025: mip = 4.300000000e+01 <= 4.400000000e+01 2.3% (273; 359753)

+35713230: mip = 4.300000000e+01 <= tree is empty 0.0% (0; 363429)

INTEGER OPTIMAL SOLUTION FOUND

Time used: 11618.0 secs

Memory used: 502.7 Mb (527138079 bytes)Model has been successfully processedCode: Select all

`x = 9 , y = 9 , rule = B3/S23 obooboobo$ooboboboo$bbbobobbb$ooboboooo$oobobbbbo$bbbooooob$ooobbbbbb$obboboobo$bbooboboo$!`

SZUPER

user@:0:~/20200724$

**Undoctored console dump for smaller areas:**

user@errorlevel:0:~/.golly/Downloads/privat/20200724$ for((i=2;i<=8;i++));{ for((j=i;j<=i;j++));{ echo 'param Xmax := '"$i"'; param Ymax := '"$j"';' | glpsol --proxy 6 --cuts -m densest.mod -d /dev/stdin;};} | sed -nre '/^.code/,/^..code/p'

Code: Select all

```
x = 2 , y = 2 , rule = B3/S23
oo$oo$!
```

Code: Select all

```
x = 3 , y = 3 , rule = B3/S23
oob$obo$boo$!
```

Code: Select all

```
x = 4 , y = 4 , rule = B3/S23
oobb$obob$bobo$bboo$!
```

Code: Select all

```
x = 5 , y = 5 , rule = B3/S23
ooboo$ooboo$bbbbb$ooboo$ooboo$!
```

Code: Select all

```
x = 6 , y = 6 , rule = B3/S23
oobboo$obbobo$boboob$oobbbb$bboobo$bboboo$!
```

Code: Select all

```
x = 7 , y = 7 , rule = B3/S23
obooboo$oobobob$bbbobbo$ooobooo$obbobbb$boboboo$ooboobo$!
```

Code: Select all

```
x = 8 , y = 8 , rule = B3/S23
oobooboo$oobooboo$bbbbbbbb$oobooboo$oobooboo$bbbbbbbb$oobooboo$oobooboo$!
```

[APPEND#1]

up to 6x6, all rectangles in one pattern:

Code: Select all

```
x = 21, y = 153, rule = B3/S23
obobobobobobobobobobo6$obobobobobobobobobobo3$2o$2o3$obobobobobobobobo
bobo6$obobobobobobobobobobo3$2o$2o3$obobobobobobobobobobo3$2o$obo$b2o
3$obobobobobobobobobobo6$obobobobobobobobobobo3$2obo$ob2o3$obobobobobo
bobobobobo3$b2o$o2bo$b2o3$obobobobobobobobobobo3$2o$obo$bobo$2b2o3$obo
bobobobobobobobobo6$obobobobobobobobobobo3$2ob2o$2ob2o3$obobobobobobob
obobobo3$2o$o2b2o$b2obo3$obobobobobobobobobobo3$2ob2o$bobo$bobo$2ob2o
3$obobobobobobobobobobo3$2ob2o$2ob2o2$2ob2o$2ob2o3$obobobobobobobobobo
bo6$obobobobobobobobobobo3$b2ob2o$b2ob2o3$obobobobobobobobobobo3$2ob2o
$2obobo$4b2o3$obobobobobobobobobobo3$b2ob2o$obobo$obobo$b2ob2o3$obobob
obobobobobobobo3$2ob2o$bob2o$o$ob2obo$b2ob2o3$obobobobobobobobobobo3$
2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o!
```