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!
```

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!
```

*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!