Still Lifes made of Isolated Polyominoes

[dani]

This idea started when I found an extensible series of still lifes that are made of pre-blocks and T-tetrominoes:

Code: Select all

x = 12, y = 19, rule = LifeHistory
2.2A3.2A$2.A2.A2.A$4.3A2$2.3A3.2A$A2.A2.A2.A$2A3.3A2$.2D3C3D2C$.C2DC
2DC2DC$.2C3D3C2D2$4.3A3.2A$2.A2.A2.A2.A$2.2A3.3A2$5.3A$3.A2.A2.A$3.2A
3.2A!

I began to wonder about more sets. Pre-block and (S/Z)-tetromino is easy:

Code: Select all

x = 13, y = 6, rule = LifeHistory
2A$A2.2A$2.2A.D2C$5.2CD.2A$8.2A2.A$11.2A!

However, doing it with just S and T is hard. I'm pretty sure Yto proved it impossible, but the Discord message the "proof" was in was a bit messy so I'm not sure if it's formal.

While pondering, I had another idea of an easy extension that should lead to interesting structures. For any n, can an infinite amount of strict still lifes be made of n-ominoes that do not directly touch eachother? If not, can only one be made? Here's one for n=3 provided by user praosylen on Discord:

Code: Select all

x = 13, y = 39, rule = LifeHistory
5.2A$5.A$7.A.2A$6.2A.A2.A$4.A6.2A$4.2A.2A$2A6.A$A2.A.2A$2.2A.A$7.A.2A
$6.2A.A2.A$4.A6.2A$4.2A.2A$2A6.A$A2.A.2A$2.2A.A$7DCD2C2D$6D2CDC2DC$4D
C6D2C$4D2CD2C4D$2C6DC4D$C2DCD2C6D$2D2CDC7D$7.A.2A$6.2A.A2.A$4.A6.2A$
4.2A.2A$2A6.A$A2.A.2A$2.2A.A$7.A.2A$6.2A.A2.A$4.A6.2A$4.2A.2A$2A6.A$A
2.A.2A$2.2A.A$7.A$6.2A!

n=4 seems tricky. It's basically the same as the S&T case, but with the addition of blocks which seems like it'd do something. I haven't been able to crack it. Here are the pieces you're allowed, for reference:

Code: Select all

x = 12, y = 2, rule = LifeHistory
2A2.2A4.A$2A3.2A2.3A!

n=5 is impossible. There are no pentominoes that don't include either an overpopulated or underpopulated cell. This might be the only non-trivial impossible case if n=4 proves viable.

n=6 is very easy. You're given the gift of having access to the table induction coil. Here's one that also uses the long Z hexomino, as well as one that only uses tables:

Code: Select all

x = 50, y = 21, rule = LifeHistory
14.2A32.2A$14.A33.A$11.2A.A30.2A.A$12.A.2A30.A.2A$12.A33.A$11.2A32.2A
$9.A$9.4A25.2D2C.4A$6.C3D2.A25.3DC.A2.A$6.4C28.3DC$3.A2.3DC28.2D2C$3.
4A31.4D$6.A28.2A.4C$3.2A31.A.C2DC$3.A32.A$2A.A31.2A$.A.2A$.A28.2A.4A$
2A29.A.A2.A$31.A$30.2A!

n=7 is a little hard but with two heptominoes we can make these stable rhombic shapes:

Code: Select all

x = 68, y = 68, rule = LifeHistory
30.2A.2A$31.A.A$31.A.A$30.2A.2A$25.A2.A2.A.A$25.5A5.A2.A$34.5A$25.5A$
26.A2.A4.5A$21.2A.A9.A2.A2.A.2A$22.A.2A13.2A.A$22.A.A15.A.A$21.2A.A
15.A.2A$22.A.2A13.2A.A.DC2DC5D$17.A2.A23.5C5D$17.5A22.10D$44.5C5D$17.
5A22.C2DC2DCD2C$18.A2.A22.5D2CDCD$13.2A.A27.6DCDCD$14.A.2A26.6DCD2C$
14.A.A27.5D2CDCD.A2.A$13.2A.A37.5A$14.A.2A$9.A2.A41.5A$9.5A40.A2.A2.A
.2A$59.2A.A$9.5A46.A.A$10.A2.A46.A.2A$5.2A.A50.2A.A$6.A.2A54.A2.A$6.A
.A54.5A$5.2A.A$A2.A2.A.2A53.5A$5A53.2A.A2.A2.A$59.A.2A$5A54.A.A$A2.A
54.2A.A$5.A.2A50.A.2A$4.2A.A46.A2.A$5.A.A46.5A$5.A.2A$4.2A.A2.A2.A40.
5A$9.5A41.A2.A$50.2A.A$9.5A37.A.2A$9.A2.A2.A.2A32.A.A$14.2A.A32.2A.A$
15.A.A33.A.2A$15.A.2A27.A2.A$14.2A.A2.A2.A22.5A$19.5A$46.5A$19.5A23.A
2.A$19.A2.A2.A.2A13.2A.A$24.2A.A15.A.2A$25.A.A15.A.A$25.A.2A13.2A.A$
24.2A.A2.A2.A9.A.2A$29.5A4.A2.A$38.5A$29.5A$29.A2.A5.5A$34.A.A2.A2.A$
33.2A.2A$34.A.A$34.A.A$33.2A.2A!

You can make the rhombi thinner but I thought it looked nicer when it was fully symmetric. I'm sure there's also a much smaller solution.

I've been stuck on n=8 for a while. It's not too hard to make a repeating unit, but stabilizing that unit is trickier than it looks:

Code: Select all

x = 12, y = 12, rule = LifeHistory
D.D6.2A$.D8.A$D.D7.A$7.2CDA$7.DCDA$7.DCD2A$5.2ADCD$6.ADCD$6.AD2C$6.A$
6.A$6.2A!

And n=9 is very easy using just one nonomino:

Code: Select all

x = 19, y = 9, rule = LifeHistory
2.2A.2A3.2C2D.2A$3.A.A4.C3D.A$A2.A.A2.A.C2DC.A2.A$4A.4A.4C.4A$10.4D$
4A.4A.4C.4A$A2.A.A2.A.C2DC.A2.A$3.A.A4.C3D.A$2.2A.2A3.2C2D.2A!

So there are still a few questions:

1) Can every n be solved besides 0, 1, 2, and 5?
2) Does every n have either 0 or infinitely many still lifes?
3) Can n=4 and n=8 be solved?
4) Is there a general solution that can be extended to every n above a specific size? The repeating unit in n=8 can obviously be extended to higher n's.
5) Finally, can we visualize these still lifes in other ways? The analogy that immediately comes to mind is a big puzzle, and every n-omino is its own piece. Births that happen between them can be prevented from working by adding certain ridges to the pieces.

[squareroot12621]

Nevermind.

[dani]

Here's a starting point for $n=8:$

Code: Select all

x = 15, y = 12, rule = LifeHistory
4.2A$5.A7.2C$5.A.2E3.C.C$2.2CDA.E.E.E2.C$2.DCDA.E.2E3.C$2.DCD2A6.2C$2A
DCD$.ADCD$.AD2C$.A$.A$.2A!

Those are polyplets instead of polyominoes, and they're not isolated. With such loose criteria, any 8-cell still life like long ship would work, so that doesn't count.

[HotWheels9232]

Let's slow down a bit. For any n=4n+6, n is non-negative, there is at least 1.

Code: Select all

x = 31, y = 10, rule = LifeSuper
2M.2A9.2M.2A7.2M.2A$.M.A11.M.A9.M.A$.M.A11.M.A9.M.A$2M.2A9.2M.2A7.2M.
2A$15.M.A9.M.A$15.M.A9.M.A$14.2M.2A7.2M.2A$27.M.A$27.M.A$26.2M.2A!

Shifting those still lifes should make it infinitely many still lifes.

[dani]

Let's slow down a bit. For any n=4m+6, m is non-negative, there is at least 1.

Code: Select all

x = 31, y = 10, rule = LifeSuper
2M.2A9.2M.2A7.2M.2A$.M.A11.M.A9.M.A$.M.A11.M.A9.M.A$2M.2A9.2M.2A7.2M.
2A$15.M.A9.M.A$15.M.A9.M.A$14.2M.2A7.2M.2A$27.M.A$27.M.A$26.2M.2A!

Shifting those still lifes should make it infinitely many still lifes.

This is a good jumping off point. n=10 has the repeating unit solution:

Code: Select all

x = 14, y = 21, rule = LifeHistory
2A.2A$.A.A$.A.A$2A.2A$.A.A$.A.A$2A.2A$7.A$8A$A9.A$3.8A$3.A9.A$6.8A$6.
A$9.2A.2A$10.A.A$10.A.A$9.2A.2A$10.A.A$10.A.A$9.2A.2A!

I haven't come up with one that works generally yet but here's n=14, n=18, and n=22:

Code: Select all

x = 105, y = 26, rule = LifeHistory
D.D.D25.D.3D35.3D.3D$D.D.D25.D.D.D37.D3.D$D.3D25.D.3D35.3D.3D$D3.D25.
D.D.D35.D3.D$D3.D25.D.3D35.3D.3D6$A2.A2.A2.A20.A2.A2.A2.A2.A.2A41.2A$
10A3.2A15.13A.A42.A$13.A30.A25.A2.A2.A2.A2.A2.A.A$2.10A.A16.13A.2A24.
16A.2A$2.A2.A2.A2.A.2A15.A2.A2.A2.A2.A.A42.A$13.A30.A25.16A.A$13.A30.
2A24.A2.A2.A2.A2.A2.A.2A$12.2A.A2.A2.A2.A20.A41.A$13.A.10A20.A.A2.A2.
A2.A2.A27.A$13.A30.2A.13A26.2A.A2.A2.A2.A2.A2.A$12.2A3.10A18.A41.A.
16A$17.A2.A2.A2.A18.A.13A27.A$44.2A.A2.A2.A2.A2.A26.2A.16A$87.A.A2.A
2.A2.A2.A2.A$87.A$86.2A!

EDIT: If this type of construction is strict, which I have my doubts about, this should work as one still life for each n=4m+6 (where n=30, in this case):

Code: Select all

x = 25, y = 91, rule = LifeHistory
A2.A2.A2.A2.A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A2.A2.A2.A2.A2.A.A$
23.A$A2.A2.A2.A2.A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A2.A2.A2.A2.A
2.A.A$23.A$A2.A2.A2.A2.A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A2.A2.A
2.A2.A2.A.A$23.A$A2.A2.A2.A2.A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A
2.A2.A2.A2.A2.A$23.2A$A2.A2.A2.A2.A2.A2.A2.A.A$22A.A$23.2A$22A.A$A2.A
2.A2.A2.A2.A2.A2.A.A$23.2A$A2.A2.A2.A2.A2.A2.A2.A.A$22A.A$23.2A$22A.A
$A2.A2.A2.A2.A2.A2.A2.A.A$23.2A$A2.A2.A2.A2.A2.A2.A2.A.A$22A.A$23.2A$
22A.A$A2.A2.A2.A2.A2.A2.A2.A.A$23.2A$A2.A2.A2.A2.A2.A2.A2.A.A$22A.A$
23.2A$22A$A2.A2.A2.A2.A2.A2.A2.A.2A$23.A$A2.A2.A2.A2.A2.A2.A2.A.A$22A
.2A$23.A$22A.A$A2.A2.A2.A2.A2.A2.A2.A.2A$23.A$A2.A2.A2.A2.A2.A2.A2.A.
A$22A.2A$23.A$22A.A$A2.A2.A2.A2.A2.A2.A2.A.2A$23.A$A2.A2.A2.A2.A2.A2.
A2.A.A$22A.2A$23.A$22A.A$A2.A2.A2.A2.A2.A2.A2.A.2A$23.A$A2.A2.A2.A2.A
2.A2.A2.A.A$22A.2A2$22A.2A$A2.A2.A2.A2.A2.A2.A2.A.A$23.A$A2.A2.A2.A2.
A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A2.A2.A2.A2.A2.A.A$23.A$A2.A2.A
2.A2.A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A2.A2.A2.A2.A2.A.A$23.A$A
2.A2.A2.A2.A2.A2.A2.A.2A$22A.A$23.A$22A.2A$A2.A2.A2.A2.A2.A2.A2.A.A$
23.A$23.2A!

EDIT2: ...But I don't think that gets us infinitely many though. Hmmm...

But this does!:

Code: Select all

x = 67, y = 74, rule = LifeHistory
44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A3.A.A3.A.A3.A.
A$44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A3.A.A3.A.A3.
A.A$44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A3.A.A3.A.A
3.A.A$44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A3.A.A3.A
.A3.A.A$44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A3.A.A
3.A.A3.A.A$44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A3.A
.A3.A.A3.A.A$44.2A.2A.2A.2A.2A.2A.2A.2A$45.A.A3.A.A3.A.A3.A.A$45.A.A
3.A.A3.A.A3.A.A$44.2A.2A.2A.2A.2A.2A.2A.2A2$22.A2.A2.A2.A2.A2.A2.A2.A
.22A$22.22A.A2.A2.A2.A2.A2.A2.A2.A2$22.2A.2A.2A.2A.2A.2A.2A.2A$23.A.A
3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.2A.2A.2A.2A.2A$23.A
.A3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.2A.2A.2A.2A.2A$
23.A.A3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.2A.2A.2A.2A.
2A$23.A.A3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.2A.2A.2A.
2A.2A$23.A.A3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.2A.2A.
2A.2A.2A$23.A.A3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.2A.
2A.2A.2A.2A$23.A.A3.A.A3.A.A3.A.A$23.A.A3.A.A3.A.A3.A.A$22.2A.2A.2A.
2A.2A.2A.2A.2A2$A2.A2.A2.A2.A2.A2.A2.A.22A$22A.A2.A2.A2.A2.A2.A2.A2.A
2$2A.2A.2A.2A.2A.2A.2A.2A$.A.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.
2A.2A.2A.2A.2A.2A.2A$.A.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.2A.2A
.2A.2A.2A.2A.2A$.A.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.2A.2A.2A.
2A.2A.2A.2A$.A.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.2A.2A.2A.2A.2A
.2A.2A$.A.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.2A.2A.2A.2A.2A.2A.
2A$.A.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.2A.2A.2A.2A.2A.2A.2A$.A
.A3.A.A3.A.A3.A.A$.A.A3.A.A3.A.A3.A.A$2A.2A.2A.2A.2A.2A.2A.2A!

[Layz Boi]

n=8, found manually:

Code: Select all

x = 27, y = 24, rule = LifeHistory
3.2A2.2A8.2A2.2A$4.4A2.A4.A2.4A$2.A6.2A4.2A6.A$2.6A.A6.A.6A$7.A.A6.A.
A$6A3.2A4.2A3.6A$A3.A4.A6.A4.A3.A$2.A3.A4.A2.A4.A3.A$.6A3.6A3.6A2$.6C
3D6C3D6C2D$.C3DC4DC3DC4DC3DC3D$.2DC3DC4DC3DC4DC3DCD$.D6C3D6C3D6CD$.26D
$2.6A3.6A3.6A$3.A3.A4.A2.A4.A3.A$.A3.A4.A6.A4.A3.A$.6A3.2A4.2A3.6A$8.
A.A6.A.A$3.6A.A6.A.6A$3.A6.2A4.2A6.A$5.4A2.A4.A2.4A$4.2A2.2A8.2A2.2A!

[HotWheels9232]

Here is 6n+4:

Code: Select all

x = 41, y = 31, rule = LifeSuper
36.2A$19.2A16.A$20.A16.A.2C$20.A.2C12.2A.C$2M17.2A.C14.A.C$.M18.A.C
14.A.2C$.M.2C15.A.2C12.2A.C$2M.C15.2A.C14.A.C$3.C18.C14.A.2C$2G.2C14.
2E.2C12.2A.C$.G18.E18.C$.G.2K15.E.2G12.2E.2C$2G.K15.2E.G14.E$3.K16.E.
G14.E.2G$2U.2K15.E.2G12.2E.G$.U17.2E.G14.E.G$.U.2A17.G14.E.2G$2U.A15.
2I.2G12.2E.G$3.A16.I16.E.G$3.2A15.I.2K13.E.2G$19.2I.K13.2E.G$20.I.K
16.G$20.I.2K12.2I.2G$19.2I.K14.I$22.K14.I.2K$22.2K12.2I.K$37.I.K$37.I
.2K$36.2I.K$39.K$39.2K!

[dani]

Here is 6n+4:

Code: Select all

x = 41, y = 31, rule = LifeSuper
36.2A$19.2A16.A$20.A16.A.2C$20.A.2C12.2A.C$2M17.2A.C14.A.C$.M18.A.C
14.A.2C$.M.2C15.A.2C12.2A.C$2M.C15.2A.C14.A.C$3.C18.C14.A.2C$2G.2C14.
2E.2C12.2A.C$.G18.E18.C$.G.2K15.E.2G12.2E.2C$2G.K15.2E.G14.E$3.K16.E.
G14.E.2G$2U.2K15.E.2G12.2E.G$.U17.2E.G14.E.G$.U.2A17.G14.E.2G$2U.A15.
2I.2G12.2E.G$3.A16.I16.E.G$3.2A15.I.2K13.E.2G$19.2I.K13.2E.G$20.I.K
16.G$20.I.2K12.2I.2G$19.2I.K14.I$22.K14.I.2K$22.2K12.2I.K$37.I.K$37.I
.2K$36.2I.K$39.K$39.2K!

Those aren't strict, as they can be split into separate still lifes consisting of two tables each. The last example in my last post cannot be split into smaller still lifes

EDIT: Improved 6+4m further (n=30 shown):

Code: Select all

x = 118, y = 11, rule = LifeHistory
12.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.
A2.A2.A2.A2.A2.A2.A2.A2.A2.A$12.22A2.22A2.22A2.22A2$A2.A2.A2.A2.10A2.
22A2.22A2.22A2.10A2.A2.A2.A2.A$13A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.
A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.13A2$13A2.A
2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A
2.A2.A2.A2.A2.A2.A2.13A$A2.A2.A2.A2.10A2.22A2.22A2.22A2.10A2.A2.A2.A
2.A2$12.22A2.22A2.22A2.22A$12.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.
A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A2.A!

[HotWheels9232]

Here is 6n+4:

Code: Select all

x = 41, y = 31, rule = LifeSuper
36.2A$19.2A16.A$20.A16.A.2C$20.A.2C12.2A.C$2M17.2A.C14.A.C$.M18.A.C
14.A.2C$.M.2C15.A.2C12.2A.C$2M.C15.2A.C14.A.C$3.C18.C14.A.2C$2G.2C14.
2E.2C12.2A.C$.G18.E18.C$.G.2K15.E.2G12.2E.2C$2G.K15.2E.G14.E$3.K16.E.
G14.E.2G$2U.2K15.E.2G12.2E.G$.U17.2E.G14.E.G$.U.2A17.G14.E.2G$2U.A15.
2I.2G12.2E.G$3.A16.I16.E.G$3.2A15.I.2K13.E.2G$19.2I.K13.2E.G$20.I.K
16.G$20.I.2K12.2I.2G$19.2I.K14.I$22.K14.I.2K$22.2K12.2I.K$37.I.K$37.I
.2K$36.2I.K$39.K$39.2K!

Those aren't strict, as they can be split into separate still lifes consisting of two tables each. The last example in my last post cannot be split into smaller still lifes

I am not good at determining what is strict and what is psuedo, but I think you can do things better along the lines of that.

[dani]

1+4m general solution (13 shown):

Code: Select all

x = 23, y = 9, rule = LifeHistory
5.2A6.2A.2A$6.A7.A.A$A2.A2.A.A2.A2.A.A2.A2.A$7A.7A.7A2$7A.7A.7A$A2.A
2.A.A2.A2.A.A2.A2.A$6.A7.A.A$5.2A6.2A.2A!

[Layz Boi]

Edit:
Known. (couldn't delete post)

[henryz]

Code: Select all

x = 83, y = 71, rule = B3/S23
3$60b2ob2o$61bobo$61bobo$60b2ob2o$61bobo$61bobo$60b2ob2o$61bobo$61bob
o$60b2ob2o$55b2obo2bobo2bob2o$56bob2o5b2obo$56bobo7bobo$55b2obo7bob2o
$56bob2o5b2obo$56bobo7bobo$55b2obo7bob2o$56bob2o5b2obo$56bobo7bobo$55b
2obo7bob2o$56bob2o5b2obo$16b2ob2o24bo2bo2bo2bo15bo2bo2bo2bo$17bobo25b
11o13b11o$17bobo$16b2ob2o24b11o13b11o$17bobo25bo2bo2bo2bo2bob2o3b2obo
2bo2bo2bo2bo$17bobo36b2obo5bob2o$16b2ob2o36bobo5bobo$17bobo37bob2o3b2o
bo$9bo2bo2bo5bo2bo2bo28b2obo5bob2o$9b8o3b8o29bobo5bobo$57bob2o3b2obo$
9b8o3b8o28b2obo5bob2o$bo2bo2bo2bo2bo2bo3bo2bo2bo30bobo5bobo$b8o19bob2o
25bob2o3b2obo$27b2obo25b2obo5bob2o$b8o19bobo30bobo$bo2bo2bo2bo2bo2bo11b
ob2o28b2ob2o$9b8o10b2obo30bobo$28bobo30bobo$9b8o11bob2o28b2ob2o$9bo2b
o2bo2bob2o5b2obo2bob2o24bobo$17b2obo11b2obo25bobo$18bobo12bobo24b2ob2o
$18bob2o11bob2o24bobo$17b2obo11b2obo25bobo$18bobo12bobo24b2ob2o$18bob
2o11bob2o$17b2obo11b2obo$22bo2bo2bo8bo2bo2bo$21b8o7b8o2$21b8o5b8o$21b
o2bo2bo2bobo2bo2bo2bo$29b2ob2o$30bobo$30bobo$29b2ob2o$30bobo$30bobo$29b
2ob2o!

Any n = 3+4m is possible using the same construction from the n=7

[ihatecorderships]

Regarding n=4, the problem reduces to finding an induction coil with a three-cell leading edge that is MOIP (made of isolated polyominoes). However, I believe experimentally that this is impossible.

[Wyirm]

is block not a polymino?

[Book]

is block not a polymino?

indeed it is a tetromino

[Wyirm]

Unless I am mistaken, n=4

Code: Select all

x = 24, y = 5, rule = B3/S23
2o6b2ob2o6b2ob2o$2o6b2ob2o6b2ob2o2$19b2ob2o$19b2ob2o!

[Chris857]

Unless I am mistaken, n=4

For any n, can an infinite amount of strict still lifes be made of n-ominoes that do not directly touch eachother? If not, can only one be made?

I think the problem with just a grid of blocks is that they are pseudo-still lifes instead of strict i.e. you can remove any block and it's still a still life. I guess to the posted question a single block does satisfy, for n=4, a single strict still life made of 4-ominos, but are there others?

[Chris857]

For n=4, not a finite stabilization, but I have a wick and an agar.

Code: Select all

x = 33, y = 24, rule = B3/S23
17b3o$4bo9b2o2bo$4b2o6bo2b2o3bo$5bo6b2o5b2o7b3o$3bo8bo6bo5b2o2bo$3b2o
9bo6bobo2b2o3bo$4bo8b2o5b2ob2o5b2o$2bo10bo3b2o2bobo6bo$2b2o11bo2b2o5b
o6bo$3bo10b3o7b2o5b2o$bo22bo3b2o2bo$b2o11b3o9bo2b2o$2bo8b2o2bo9b3o$o8b
o2b2o3bo$2o7b2o5b2o7b3o$bo7bo6bo5b2o2bo$11bo6bobo2b2o3bo$10b2o5b2ob2o
5b2o$10bo3b2o2bobo6bo$12bo2b2o5bo6bo$11b3o7b2o5b2o$21bo3b2o2bo$23bo2b
2o$22b3o!

[squareroot12621]

More agars:

Code: Select all

x = 24, y = 24, rule = B3/S23:T24,24
b3o3b3o3b3o3b3o$2bo2bo2bo5bo2bo2bo$o3b3o3bobo3b3o3bo$2o7b2ob2o7b2o$o3b
3o3bobo3b3o3bo$2bo2bo2bo5bo2bo2bo$b3o3b3o3b3o3b3o2$b3o3b3o3b3o3b3o$2b
o2bo2bo5bo2bo2bo$o3b3o3bobo3b3o3bo$2o7b2ob2o7b2o$o3b3o3bobo3b3o3bo$2b
o2bo2bo5bo2bo2bo$b3o3b3o3b3o3b3o2$b3o3b3o3b3o3b3o$2bo2bo2bo5bo2bo2bo$
o3b3o3bobo3b3o3bo$2o7b2ob2o7b2o$o3b3o3bobo3b3o3bo$2bo2bo2bo5bo2bo2bo$
b3o3b3o3b3o3b3o$24b!

Code: Select all

x = 50, y = 50, rule = B3/S23:T50,50
b3o5bo2bobo3b2ob2o3bobo2bo5b3o$2bo2bobo3b2ob2o3bobo2bo5b3o11b3o$o3b2o
b2o3bobo2bo5b3o11b3o5bo2bo$2o3bobo2bo5b3o11b3o5bo2bobo3b2o$o2bo5b3o11b
3o5bo2bobo3b2ob2o3bo$2b3o11b3o5bo2bobo3b2ob2o3bobo2bo$9b3o5bo2bobo3b2o
b2o3bobo2bo5b3o$2b3o5bo2bobo3b2ob2o3bobo2bo5b3o$3bo2bobo3b2ob2o3bobo2b
o5b3o11b3o$bo3b2ob2o3bobo2bo5b3o11b3o5bo2bo$b2o3bobo2bo5b3o11b3o5bo2b
obo3b2o$bo2bo5b3o11b3o5bo2bobo3b2ob2o3bo$3b3o11b3o5bo2bobo3b2ob2o3bob
o2bo$10b3o5bo2bobo3b2ob2o3bobo2bo5b3o$3b3o5bo2bobo3b2ob2o3bobo2bo5b3o
$4bo2bobo3b2ob2o3bobo2bo5b3o11b3o$obo3b2ob2o3bobo2bo5b3o11b3o5bo$ob2o
3bobo2bo5b3o11b3o5bo2bobo3bo$obo2bo5b3o11b3o5bo2bobo3b2ob2o$4b3o11b3o
5bo2bobo3b2ob2o3bobo2bo$11b3o5bo2bobo3b2ob2o3bobo2bo5b3o$4b3o5bo2bobo
3b2ob2o3bobo2bo5b3o$5bo2bobo3b2ob2o3bobo2bo5b3o11b3o$bobo3b2ob2o3bobo
2bo5b3o11b3o5bo$2ob2o3bobo2bo5b3o11b3o5bo2bobo$bobo2bo5b3o11b3o5bo2bo
bo3b2ob2o$5b3o11b3o5bo2bobo3b2ob2o3bobo2bo$o11b3o5bo2bobo3b2ob2o3bobo
2bo5b2o$5b3o5bo2bobo3b2ob2o3bobo2bo5b3o$o5bo2bobo3b2ob2o3bobo2bo5b3o11b
2o$2bobo3b2ob2o3bobo2bo5b3o11b3o5bo$b2ob2o3bobo2bo5b3o11b3o5bo2bobo$2b
obo2bo5b3o11b3o5bo2bobo3b2ob2o$o5b3o11b3o5bo2bobo3b2ob2o3bobo$2o11b3o
5bo2bobo3b2ob2o3bobo2bo5bo$6b3o5bo2bobo3b2ob2o3bobo2bo5b3o$2o5bo2bobo
3b2ob2o3bobo2bo5b3o11bo$o2bobo3b2ob2o3bobo2bo5b3o11b3o$2b2ob2o3bobo2b
o5b3o11b3o5bo2bobo$3bobo2bo5b3o11b3o5bo2bobo3b2ob2o$bo5b3o11b3o5bo2bo
bo3b2ob2o3bobo$3o11b3o5bo2bobo3b2ob2o3bobo2bo$7b3o5bo2bobo3b2ob2o3bob
o2bo5b3o$3o5bo2bobo3b2ob2o3bobo2bo5b3o$bo2bobo3b2ob2o3bobo2bo5b3o11b3o
$3b2ob2o3bobo2bo5b3o11b3o5bo2bobo$o3bobo2bo5b3o11b3o5bo2bobo3b2obo$2b
o5b3o11b3o5bo2bobo3b2ob2o3bobo$b3o11b3o5bo2bobo3b2ob2o3bobo2bo$8b3o5b
o2bobo3b2ob2o3bobo2bo5b3o!

Code: Select all

x = 101, y = 101, rule = B3/S23:T101,101
6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$bo2bo2bobo6b2ob2o
6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o$6b2ob2o6bobo7bobo6b2ob2o6bobo2b
o2bo5b6o14b6o5bo2bo2bobo$o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo
2bobo6b2obo$7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$6b
2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo$o6bobo2bo2bo5b6o
14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2obo$2bo2bo5b6o14b6o5bo2bo2bobo6b2o
b2o6bobo7bobo6b2ob2o6bobo$b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o
6bobo2bo2bo$11b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$b
6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$2bo2bo2bobo6b2ob
2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o$o6b2ob2o6bobo7bobo6b2ob2o6bob
o2bo2bo5b6o14b6o5bo2bo2bo$2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2b
o2bobo6b2o$o7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bo$o6b
2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bo$2o6bobo2bo2bo5b6o
14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2o$o2bo2bo5b6o14b6o5bo2bo2bobo6b2o
b2o6bobo7bobo6b2ob2o6bo$2b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6b
obo2bo2bo$12b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$2b6o
5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$3bo2bo2bobo6b2ob2o
6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o$bo6b2ob2o6bobo7bobo6b2ob2o6bobo
2bo2bo5b6o14b6o5bo2bo2bo$b2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2b
o2bobo6b2o$bo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bo$bo
6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bo$b2o6bobo2bo2bo5b
6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2o$bo2bo2bo5b6o14b6o5bo2bo2bobo6b
2ob2o6bobo7bobo6b2ob2o6bo$3b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o
6bobo2bo2bo$13b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$3b
6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$4bo2bo2bobo6b2ob
2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o$obo6b2ob2o6bobo7bobo6b2ob2o6b
obo2bo2bo5b6o14b6o5bo2bo$ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo
2bo2bobo6bo$obo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o$obo
6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$ob2o6bobo2bo2bo5b6o
14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6bo$obo2bo2bo5b6o14b6o5bo2bo2bobo6b2o
b2o6bobo7bobo6b2ob2o$4b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bob
o2bo2bo$14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$4b6o5b
o2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$5bo2bo2bobo6b2ob2o6b
obo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o$bobo6b2ob2o6bobo7bobo6b2ob2o6bobo
2bo2bo5b6o14b6o5bo2bo$2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2b
o2bobo$bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o$bobo6b2o
b2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$2ob2o6bobo2bo2bo5b6o14b
6o5bo2bo2bobo6b2ob2o6bobo7bobo$bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6b
obo7bobo6b2ob2o$5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2b
o$o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b5o$5b6o5bo2bo
2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$o5bo2bo2bobo6b2ob2o6bobo
7bobo6b2ob2o6bobo2bo2bo5b6o14b5o$2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2b
o2bo5b6o14b6o5bo2bo$b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo
2bobo$2bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o$2bobo6b
2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$b2ob2o6bobo2bo2bo5b6o
14b6o5bo2bo2bobo6b2ob2o6bobo7bobo$2bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2o
b2o6bobo7bobo6b2ob2o$o5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bo
bo2bo$2o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b4o$6b6o5b
o2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$2o5bo2bo2bobo6b2ob2o
6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b4o$o2bobo6b2ob2o6bobo7bobo6b2ob2o6b
obo2bo2bo5b6o14b6o5bo$2b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo
2bo2bobo$3bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o$3bob
o6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$2b2ob2o6bobo2bo2b
o5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo$3bobo2bo2bo5b6o14b6o5bo2bo2bob
o6b2ob2o6bobo7bobo6b2ob2o$bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2o
b2o6bobo2bo$3o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b3o
$7b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$3o5bo2bo2bobo
6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b3o$bo2bobo6b2ob2o6bobo7bobo
6b2ob2o6bobo2bo2bo5b6o14b6o5bo$3b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o
14b6o5bo2bo2bobo$4bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2o
b2o$4bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$3b2ob2o6b
obo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo$4bobo2bo2bo5b6o14b6o5b
o2bo2bobo6b2ob2o6bobo7bobo6b2ob2o$2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo
7bobo6b2ob2o6bobo2bo$4o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2b
o2bo5b2o$8b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$4o5bo
2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b2o$2bo2bobo6b2ob2o6b
obo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo$4b2ob2o6bobo7bobo6b2ob2o6bobo2b
o2bo5b6o14b6o5bo2bo2bobo$5bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2b
obo6b2ob2o$5bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo$4b
2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo$5bobo2bo2bo5b6o
14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o$o2bo5b6o14b6o5bo2bo2bobo6b2o
b2o6bobo7bobo6b2ob2o6bobo$5o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6b
obo2bo2bo5bo$9b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o$5o
5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14bo$o2bo2bobo6b2ob
2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o$5b2ob2o6bobo7bobo6b2ob2o6bobo
2bo2bo5b6o14b6o5bo2bo2bobo$6bobo7bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2b
o2bobo6b2ob2o$6bobo6b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo
$5b2ob2o6bobo2bo2bo5b6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo$6bobo2bo2bo5b
6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o$bo2bo5b6o14b6o5bo2bo2bobo6b
2ob2o6bobo7bobo6b2ob2o6bobo$6o14b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o
6bobo2bo2bo$10b6o5bo2bo2bobo6b2ob2o6bobo7bobo6b2ob2o6bobo2bo2bo5b6o!

[Wyirm]

I may be wrong, but I strongly believe that n=4 using any other polymino than the block, is impossible. Proof:
There are 5 tetrominoes, only 3 of which can survive isolated from other cells, and only one still life.

Code: Select all

x = 23, y = 6, rule = LifeSuper
4A2.3A2.3A2.2A3.2A$6.A5.A4.2A2.2A2$.2D4.D4.D$E2AE2.2AE2.3A2.2AD2.2A$.
2D3.A4.DAD2.D2A2.2A!

Imagine breaking a shape down into how many ways the shape can birth cells. The t has 3 birth "surfaces", the 3-long line and the 2 corners. The z has 2 birth surfaces, 2 corners. Normally, these surfaces are stabilized by other similar motifs, but in the case of the t and z, they have 2 and 3 respectively, and each needs at minimum 1 similar surface to stabilize it. every time the t is used in one of these n=4 constructs, an extra surface is needed , because 1 of the surfaces are covered up and 2 more are created, and z's are inconsequential to the number of surfaces to cover up. a block, a z and a t can be used to stabilize a 3 long surface like a snake:

Code: Select all

x = 5, y = 6, rule = LifeSuper
2D.D$2A.C$2A.2C$4.C$3E$.E!

but this costs more than it helps. It is impossible to reduce the number of cells being birthed using only tetrominoes. (unless i'm missing something)