Unproven conjectures

[Sokwe]

184-cell unsynthesizable still life:

Code: Select all

x = 22, y = 25, rule = B3/S23
10b2o$9bo2bo$7bo2bobo$b2o4b2obo2b2o$bo2b2obo2b2obo2b2o$2b3obo2bo2bo2b
2obo$7b2obo2b2o4bo$4b2obo2b2obo2b2obo$2b3obo2bo2bo2b2obo$bo5b2obo2b2o$
2bob2obo2b2obo2b2o$b2ob2o2bo2bo2bob2o$7b2obo2b2o$b3ob2o2bob2o2bob3o$o
2b2obo2bo2bo2b2obobo$o6b2obo2b2o6bo$b3ob2o2bob2o2bob4o$3b2obo2bo2bo2b
2obo$7b2obo2b2o$2bob2obo2b2obo2b2o$bob2obo2bo2bo2b2obo$bo5b2obo2b2o4bo
$2bob2obo2b2obo2b3o$3b2obo2bo2bo2b2o$10b2o!

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

Code: Select all

x = 82, y = 25, rule = LifeHistory
10.2A28.2A28.2A$9.A2.A26.A2.A26.A2.A$7.A2.A.A24.A2.A.A24.A2.A.A$.2A3.
D2CDCD.2A16.2A3.D2CDCD.2A16.2A3.D2CDCD.2A$.A2.2CDC2D2CDC2D2C13.A2.2CD
C2D2CDC2D2C13.A2.2CDC2D2CDC2D2C$2.2ACDC2DC2DC2D2CDA13.2ACDC2DC2DC2D2C
DA13.2ACDC2DC2DC2D2CDA$4.3D2CDC2D2C3D.A14.3D2CDC2D2C3D.A14.3D2CDC2D2C
3D.A$4.2CDC2D2CDC2D2C.A14.2CDC2D2CDC2D2C.A14.2CDC2D2CDC2D2C.A$2.2ACDC
2DC2DC2D2C.A13.2ACDC2DC2DC2D2C.A13.2ACDC2DC2DC2D2C.A$.A2.3D2CDC2D2C3D
13.A2.3D2CDC2D2C3D13.A2.3D2CDC2D2C3D$2.A.2CDC2.2CDC2D2C14.A.2CDCD.2CD
C2D2C14.A.2CDCD.2CDC2D2C$.ACD2C2DC2DC2DCD2C13.ACD2C2DC2DC2DCD2C13.ACD
2C2DC2DC2DCD2C$2.5D2CDC2D2C2D15.5D2CDC2D2C.D15.5D2CDC2D2C.D$.A2CD2C2D
CD2C2DCDC2A11.A2CD2C2DCD2C2DCDC2A11.A2CD2CD.CD2C2DCDC2A$A2.2CDC2DC.DC
2D2CDA.A9.A2.2CDC2DC.DC2D2CDA.A9.A2.2CDC2DC2DC2D2CDA.A$A2.4D2CDC2D2C
2D4.A8.A2.4D2CDC2D2C2D4.A8.A2.4D2CDC2D2C2D4.A$.2ACD2C2DCD2C2DCDC3A10.
2ACD2C2DCD2C2DCDC3A10.2ACD2C2DCD2C2DCDC3A$3.2CDC2DC2DC2D2CDA14.2CDC2D
C2DC2D2CDA14.2CDC2DC2DC2D2CDA$3.4D2CDC2D2C2D16.4D2CDC2D2C2D16.4D2CDC
2D2C2D$2.AD2CDC2D2CDC2D2C14.AD2CDC2D2CDC2D2C14.AD2CDC2D2CDC2D2C$.A.2C
DC2DC2DC2D2C.A12.A.2CDC2DC2DC2D2C.A12.A.2CDC2DC2DC2D2C.A$.A3.2D2CDC2D
2C2D2.A11.A3.2D2CDC2D2C2D2.A11.A3.2D2CDC2D2C2D2.A$2.A.ACDC2D2CDC2DC2A
13.A.ACDC2D2CDC2DC2A13.A.ACDC2D2CDC2DC2A$3.2A.A2.A2.A2.2A16.2A.A2.A2.
A2.2A16.2A.A2.A2.A2.2A$10.2A28.2A28.2A!

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. For example, here is a self-forcing patch with 274 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$4.3D2CDC2D2C3D.A$4.2CDC2D2CDC2D2C.A$2.2ACDC2DC2DC2D2C.A$.A2.3D
2CDC2D2C3D$2.A.2CDC2DCADC2D2C$.ACD2C2DC2DC2DCD2C$2.5D2CDC2D2C.D$.A2CD
2C2DCD2C2DCDC2A$A2.2CDC2DC2DC2D2CDA.A$A2.4D2CDC2D2C2D4.A$.2ACD2C2DCD
2C2DCDC3A$3.2CDC2DC2DC2D2CDA$3.4D2CDC2D2C2D$2.AD2CDC2D2CDC2D2C$.A.2CD
C2DC2DC2D2C.A$.A3.2D2CDC2D2C2D2.A$2.A.ACDC2D2CDC2DC2A$3.2A.A2.A2.A2.
2A$10.2A!

Removing any one of the three interior cells present in the 273-cell solutions causes the patch to no longer be self-forcing, and I didn't find a way to reduce this 274-cell solution to a 273-cell solution. Of course, I didn't try everything. I suspect an automated search would find smaller self-forcing subpatches of the original patch.

[confocaloid]

There are no still lives other than the block where all live cells have 3 live neighbors.
There are infinitely many with all-but-4 (long^n barges), all-but-3 (long^n boats), and all-but-2 (long^n ships).
Is there a still life where all but 1 live cell has 3 live neighbors? Are there infinitely many?

I suspect that it's possible to extend the existing proof (see below), to eliminate the possibility "all but 1 live cell has 3 live neighbors" (by showing that there must be at least two bits requiring the survival condition S2, one somewhere "up the edge of the bounding diamond" and one somewhere "down the edge of the bounding diamond").

There are no still lives other than the block where all live cells have 3 live neighbors.

Proven for all finite strict still-lifes, yes. An infinite barge is a counterexample, though, as is bi-block for pseudo still-lives.

Crossposting a proof from another thread.

Block is [...] the only finite strict still life where all living cells have three neighbors. [...] Any such still life would remain a still life in B3/S3. A proof that there are no finite strict still lifes other than the block in B3/S3 was cross-posted from Discord in August 2023, settling an open problem.

Here's a proof from rachel at Discord that there are no 3-cell spaceships in B2a that travel at a speed (X,Y)/P with X+Y = P and X,Y ≠ 0.
[...]
[...]
They also proved that there are no still lifes other than the block in B3/S3 (actually, B3aijnq/S3):

the trick is to ignore all blocks in the hypothetical SL. then we just look at an edge of the bounding diamond of the rest (for ease of writing, let's specify that it's the top right edge).
first, notice that we can't have o$2o on that edge, since that would make a B3a outside of the bounding diamond of the blockless part of the SL, so we could only prevent that with blocks, which would force an infinite amount of blocks via B3nq.
then notice that if there was a 2o$o on the edge, then (since we're not making a block) the top left cell could only be stabilized through S3k, and if the top left cell of that S3k didn't have a live cell above it, that would cause a B3j which once again can't be stopped by finitely many blocks. so the top left cell of the S3k must have a live cell c1 above it. now to stabilize c1 without overpopulating the cell below c1, we need a live cell c2 to the top left of c1, and then to prevent o$2o on the edge, we need a live cell to the bottom left of c1. but then to stabilize c2 without overpopulating c1, we either make another 2o$o further up the edge of the bounding diamond, or we have a live cell c3 to the top left of c2, and in the latter case, again to prevent o$2o, we need a live cell to the bottom left of c2. to stabilize c3 without overpopulating c2, we then need to either make another 2o$o further up the edge of the bounding diamond, or we have a live cell c4 to the top left of c3, and in the latter case, again to prevent o$2o, we need a live cell to the bottom left of c3. this continues on, and since our still life is finite, we're eventually forced to choose the first option and make another 2o$o further up the edge of the bounding diamond. so a 2o$o forces another one further up, but that will force yet another, etc. and our SL will be infinite. so we can't have a 2o$o on the edge at all.
now, since there can't be an o$2o or a 2o$o on the edge, this means that if any live cell on the edge had a live VN neighbor, then it could only have two neighbors, and that's a problem because we only have S3. but no VN neighbors leaves only S3c as a possibility, and each S3c on the edge forces another S3c further up on the edge, so we once again get an infinite SL.
so no matter what, the blockless part of the SL would have to be infinite or empty. and if it's empty, then... well the SL itself would just be a bunch of blocks, it doesn't count as a separate SL.

edit: i think i didn't explain well enough why the top left cell of the 2o$o must be stabilized with an S3k (and not with e.g. an S3j). it's because the 2o$o is forced to be part of a 2o$obo$bo, so the bottom left cell of the 2o$o already has 3 neighbors. this only works because we're on the edge of the bounding diamond, otherwise the 2o$obo$bo wouldn't be forced.

[Sokwe]

184-cell unsynthesizable still life:

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

Code: Select all

...

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:

[Tawal]

I think this thread needs a summary of unproven conjectures at the 1rst post like dvgrn did at #4

[confocaloid]

I think this thread needs a summary of unproven conjectures at the 1rst post like dvgrn did at #4

Summaries don't always work well. Sometimes the thread itself can serve as a "summary" (while trying to maintain a summary post quickly becomes somewhat of a headache).

Sometimes people post their own (wish)lists of unsolved problems/conjectures here and there.

Edit: for example, check these links:

[...] i mean, our previous goals were "smallest inf growth?" and "is omniperiodic?" but they've been resolved, [...]

• Incomplete search patterns - try to complete
• Unsolved life problems
• Technological Challenges
• Unproven conjectures
• Life has been proven omniperiodic. What next?
• Where we go ?

[WhiteHawk]

Conjecture: Pentapole is the only 11-cell oscillator (I don't think this has been proven yet)

Related: There are only eight oscillators with minipop less than 12 cells

[Hdjensofjfnen]

Consider any still life and an OFF cell in the same universe in generation 0. Now let the still life be destroyed by any combination of at most N gliders originating from infinity. We define the OFF cell to be in the N-shadow of the still life if, no matter the combination of gliders, the OFF cell must turn ON in some generation.

For instance, the center OFF cell in the boat is in the boat's 1-shadow, as no one-glider destruction of the boat avoids turning it ON in at least one generation:

Code: Select all

x = 101, y = 166, rule = LifeSuper
.A$2.A3.19L16.19L16.19L$3A3.19L16.19L16.19L$6.19L16.19L16.19L$6.19L
16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.
19L16.19L16.19L$6.8L2G9L16.8L2G9L16.8L2G9L$6.8LGRG8L16.8LGRG8L16.8LGR
G8L$6.9LG9L16.9LG9L16.9LG9L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L
16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.
19L10.3A3.19L16.19L$6.19L12.A3.19L16.19L$36.A3$94.3A$94.A$95.A5$7.A$
8.A$6.3A4$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L
16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.
19L16.19L16.19L$6.8L2G9L16.8L2G9L16.8L2G9L$6.8LGRG8L16.8LGRG8L16.8LGR
G8L$6.9LG9L16.9LG9L16.9LG9L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L
16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.19L16.19L16.19L$6.
19L16.19L16.19L$6.19L16.19L16.19L2$35.3A$37.A$36.A60.3A$97.A$98.A11$
41.19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.
19L16.19L$41.19L16.19L$41.19L16.19L$41.8L2G9L16.8L2G9L$41.8LGRG8L16.
8LGRG8L$41.9LG9L16.9LG9L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.
19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L4$35.3A
60.3A$37.A60.A$36.A62.A11$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.
19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.8L2G
9L16.8L2G9L$41.8LGRG8L16.8LGRG8L$41.9LG9L16.9LG9L$41.19L16.19L$41.19L
16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.19L$41.19L16.
19L$41.19L16.19L$98.3A$98.A$99.A$36.3A$38.A$37.A11$41.19L$41.19L$41.
19L$41.19L$41.19L$41.19L$41.19L$41.19L$41.8L2G9L$41.8LGRG8L$41.9LG9L$
41.19L$41.19L$41.19L$41.19L$41.19L$41.19L$41.19L$41.19L4$40.3A$42.A$
41.A!

However, a two-glider destruction of the boat suffices to keep that cell OFF, so the cell is not in the boat's 2-shadow:

Code: Select all

x = 31, y = 27, rule = LifeSuper
.A$2.A$3A4$7.A$8.A$6.3A3.19L$12.19L$12.19L$12.19L$12.19L$12.19L$12.
19L$12.19L$12.8L2G9L$12.8LGRG8L$12.9LG9L$12.19L$12.19L$12.19L$12.19L$
12.19L$12.19L$12.19L$12.19L!

• Is there any still life with an OFF cell that is in its N-shadow for all finite N? (TLDR: is there any combination of a still life and an OFF cell in the same universe, such that the OFF cell must turn ON in any glider destruction of the still life originating from infinity?)

• We define an OFF cell satisfying the above condition as being in a still life's infinity-shadow. Is there any still life with an infinite number of cells in its infinity-shadow?

Note that the answers to both questions are automatically yes if an indestructible still life is shown to exist. However, this problem is weaker.

[confocaloid]

Consider any still life and an OFF cell in the same universe in generation 0. Now let the still life be destroyed by any combination of fewer than N gliders originating from infinity. We define the OFF cell to be in the N-shadow of the still life if, no matter the combination of gliders, the OFF cell must turn ON in some generation.
[...]

This looks like an interesting problem. However, I think your definition is inconsistent with the "middle cell of the boat" example in the rest of your post. Either the definition needs to be changed, or the example/questions need to be changed, or both.

• (1) I think you need to change the part saying
  >> "... any combination of fewer than N gliders ..."
  into "... any combination of at most N gliders ...".
  Otherwise, the case N = 1 becomes "destroyed by 0 gliders" and the case N = 2 becomes "destroyed by one glider", which is inconsistent with the boat example in the rest of your post.
• (2) If I understand correctly, with that definition (corrected per (1)), and with N = 1, the middle OFF cell in the boat is in the N-shadow of the boat, because "no matter the combination of gliders, the OFF cell must turn ON in some generation". (For N = 1, that means every 1G destruction of the boat turns the middle OFF cell ON at some point.)
  However, the example in the rest of your post claims otherwise ("the center OFF cell in the boat is not in the boat's 1-shadow").
  
  Further, with N = 2, the middle OFF cell in the boat is not in the N-shadow of the boat, because there is a 2G destruction of the boat that avoids turning that cell ON.
  However, the example in the rest of your post claims otherwise ("putting the cell into the boat's 2-shadow").

[Hdjensofjfnen]

This looks like an interesting problem. However, I think your definition is inconsistent with the "middle cell of the boat" example in the rest of your post. Either the definition needs to be changed, or the example/questions need to be changed, or both.

[...]

Thank you for reading closely! I had the formulation as "at most" originally, but changed it. I thought I was fixing an off-by-one error, when the problem was actually an inversion error (see below). It should be "at most", as you pointed out.

The cell in the boat example is actually part of the boat's 1-shadow but not its 2-shadow or any higher-order N-shadow, which you also pointed out. That one was just a brain fart to begin with.

EDIT: I also edited the associated question for consistency. It's now "Is there any still life with an OFF cell that is in its N-shadow for all finite N?"