Sokwe wrote: ↑September 3rd, 2024, 3:33 am

400spartans wrote: ↑September 2nd, 2024, 1:20 am

184-cell unsynthesizable still life:

Here are three different self-forcing patches for this still life with 273 specified cells:

However, their intersection is not self-forcing. I found these by playing around with JLS and removing one cell at a time. If the patch stayed self-forcing, I kept that modification. There's obviously a limitation to this method, as it depends on the order you remove the cells.

Out of curiosity, I decided to investigate this a bit further. Modifications were done by hand and searches done with JLS, so it's possible I made a slight mistake somewhere. I started with the following simply connected self-forcing patch:

Code: Select all

```
x = 16, y = 20, rule = LifeHistory
4.D2CDCD$2.2CDC2D2CDC2D2C$2.CDC2DC2DC2D2CD$2.3D2CDC2D2C3D$2.2CDC2D2CD
C2D2C$2.CDC2DC2DC2D2C$2.3D2CDC2D2C3D$2.2CDC2D2CDC2D2C$CD2C2DC2DC2DCD
2C$5D2CDC2D2C2D$2CD2C2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4D2CDC2D2C2D$.CD2C
2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4D2CDC2D2C2D$.D2CDC2D2CDC2D2C$.2CDC2DC
2DC2D2C$3.2D2CDC2D2C2D$3.CDC2D2CDC2DC!
```

For each cell in this patch, I checked to see if the patch without that cell was still self-forcing. The intersection of these self forcing patches is the following (removed on/off cells marked with green/blue):

Code: Select all

```
x = 16, y = 20, rule = LifeHistory
4.D2CDCD$2.2CDC2D2CDC2D2C$2.CDC2DC2DCDB2CD$2.3DACDABD2C3D$2.2CBA2D2CD
C2D2C$2.CDC2DCBDC2D2C$2.3D2CDCDBAC3D$2.CADC2BCABA2D2C$CD2C2DC2DCBDCDA
C$5D2CBCDB2CBD$2CD2CDBCD2C2DCDC$.2CDC2DCBDCDB2CD$.4D2CDC2D2C2D$.CD2C
2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4DACDC2D2C2D$.D2CDC2D2CDC2D2C$.2CDC2DC
2DC2D2C$3.2D2CDC2D2C2D$3.CDC2D2CDC2DC!
```

Of course, this new patch is not self-forcing (in fact, with these cells removed none of the remaining cells are forced), but I think that any self-forcing subpatch of the original patch would need to include this patch. Since I removed cells one at a time, you might think that there could be a self-forcing subpatch that requires removing more than one cell at once, but I think I've eliminated that possibility. I noticed that every time a cell removal caused the remaining patch to not be self-forcing, at least one of the following four corners was no longer forced:

Code: Select all

```
x = 16, y = 20, rule = LifeHistory
4.6B$2.E12BE$2.14B$2.14B$2.14B$2.13B$2.14B$2.14B$16B$15B$16B$.15B$.
14B$.15B$.15B$.14B$.15B$.14B$3.12B$3.E10BE!
```

If these four corners turned out to be necessarily included in any self-forcing subpatch, then so too would be the cells in the above intersection of self-forcing patches. I started by removing the top left corner and used JLS to see that the following cells remained forced (removed on/off cells marked with green/blue):

Code: Select all

```
x = 16, y = 20, rule = LifeHistory
4.D2CDCB$2.2ADCBD2ADC2D2C$2.ABA2DC2DC2D2CD$2.3D2CDC2D2C3D$2.2CDC2D2CD
C2D2C$2.CDC2DC2DC2D2C$2.3D2CDC2D2C3D$2.2CDC2D2CDC2D2C$CD2C2DC2DC2DCD
2C$5D2CDC2D2C2D$2CD2C2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4D2CDC2D2C2D$.CD2C
2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4D2CDC2D2C2D$.D2CDC2D2CDC2D2C$.2CDC2DC
2DC2D2C$3.2D2CDC2D2C2D$3.CDC2D2CDC2DC!
```

I then considered that this remaining patch might be self-forcing, but JLS showed that with the remaining patch, the only forced cells were the following (shape of original self-forcing patch shown in blue):

Code: Select all

```
x = 16, y = 20, rule = LifeHistory
4.6B$2.14B$2.10BD3B$2.5BD8B$2.4BD3BD5B$2.7BD2BD2B$2.DBD5BD5B$2.4B2DBC
BC2D2B$8BD2BD4B$2B2D5BD2BCDB$4BC2D5BDBDB$.2BD7B2DC2B$.D2BD2BDCD2BC2D$
.4BD8BDB$.5BD9B$.3BDCBDBD3BDB$.D8BDBD3B$.14B$3.4BD7B$3.7BD4B!
```

This is obviously not self-forcing, so I reasoned that the top left cell must be included in any self-forcing subpatch. I then removed the top right and bottom right corners, but this caused the top left corner to no longer be forced, so these were also necessary cells. I then removed the bottom left corner and found that the following cells were forced:

Code: Select all

```
x = 16, y = 20, rule = LifeHistory
4.D2CDCD$2.2CDC2D2CDC2D2C$2.CDC2DC2DC2D2CD$2.3D2CDC2D2C3D$2.2CDC2D2CD
C2D2C$2.CDC2DC2DC2D2C$2.3D2CDC2D2C3D$2.2CDC2D2CDC2D2C$CD2C2DC2DC2DCD
2C$5D2CDC2D2C2D$2CD2C2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4D2CDC2D2C2D$.CD2C
2DCD2C2DCDC$.2CDC2DC2DC2D2CD$.4D2CDC2D2C2D$.D2CDC2D2CDC2D2C$.2CDC2DC
2DC2D2C$3.2D2ADC2D2A2D$3.ADA2B2ADA2BC!
```

Running JLS on this patch quickly revealed that the bottom right corner was no longer forced, so each of the four corners, and thus all of the cells in the intersection, are necessary.

Finally, I played around a bit to get this self-forcing patch with 271 specified cells:

Code: Select all

```
x = 22, y = 25, rule = LifeHistory
10.2A$9.A2.A$7.A2.A.A$.2A3.D2CDCD.2A$.A2.2CDC2D2CDC2D2C$2.2ACDC2DC2DC
2D2CDA$4.3DACDC2D2C3D.A$4.2CDC2D2CDC2D2C.A$2.2ACDC2DC2DC2D2C.A$.A2.3D
2CDC2D2C3D$2.A.2CDC2D2C.C2D2C$.ACD2C2DC2DC2DCD2C$2.5D2CDC2D2C.D$.A2CD
2C2DCD2C2DCDC2A$A2.2CDC2DC.DC2D2CDA.A$A2.4D2CDC2D2C2D4.A$.2ACD2C2DCD
2C2DCDC3A$3.2CDC2DC2DC2D2CDA$3.4DACDC2D2C2D$2.AD2CDC2D2CDC2D2C$.A.2CD
C2DC2DC2D2C.A$.A3.2D2CDC2D2C2D2.A$2.A.ACDC2D2CDC2DC2A$3.2A.A2.A2.A2.
2A$10.2A!
```

There may be slightly smaller self-forcing patches. I don't see a way to find a minimally-sized patch without automation.

Here is a zip file containing the JLS search file validating the 271-specified-cells self-forcing patch: