Unproven conjectures

[galoomba]

A significant question is still life finitization (I think it has been called that). I do not know if it is currently open. I have had some ideas and never made progress. It is hard but seems tractable (unless it's actually easy and I'm missing something).

The question is: given a still life without finite boundaries (e.g. covering the plane; it need not be periodic or have any other properties), can any MxN finite window of it be preserved within a finite still life by adding appropriate unchanging cells around it (no particular bounds on this, though the question could place size limits in terms of M and N).

A natural conjecture based on experience and the prevalence of still lifes is that there is some fairly small constant k such that any MxN window that exists within an infinite still life can be stabilized within an (M+k)x(N+k) still life.

For instance, take a still life consisting of alternating stripes. That is only stable when infinite, but it is not hard to select a rectangle inside it and add a boundary to stabilize it. This appears to be true for less regular still lifes as well.

The first idea I had was backtracking from a row of empty cells to a populated row. The first row facing empty space must not have more than two live cells adjacent, but it is unclear what conditions are on its successor. My guess is that after several iterations, the conditions are identical to a row within an infinite still life, but I don't see how to prove it (I thought about treating these rows as regular languages, which they are, but that hasn't helped).

The second idea, which seems more tractable, is to pick a specific window size (say 10x10). For any still life of this size including with unstable boundaries, if we remove all cells in a centered subwindow (say 6x6) can we restabilize the bordering cells leaving an empty subwindow (I think 2x2 is big enough). If so, we have an algorithm for "punching holes" in any infinite still life and should be able to separate finite windows from the infinite stabilizer given a succession of hole punches.

I believe the second approach is suitable to an automated proof, but it might not result in a very readable explanation of how it works.

Or, again has someone already figured this out (Noam Elkies?) using more elegant mathematics and am I just way behind the state of the art.

I may have found a counterexample:

Code: Select all

x = 90, y = 3, rule = B3/S23
b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b
2o2b2o3b2o$2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o
2b2o3b2o2b2o3b2o2b2o$3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o
3b2o2b2o3b2o2b2o3b2o2b2o3b2o2b2o!

According to JLS, the only way to stabilize it is with more copies of itself, bounded by a line of slope -1/3.

Tried stabilising it by hand, it's probably far from optimal

Code: Select all

x = 102, y = 123, rule = LifeHistory
7.2A$6.A2.A$.2A3.3A2.2A$.A2.2A3.2A.A$2.3A2.2A$6.2A2.2A3.A$4.2A3.2A.A
2.3A$2.3A2.2A3.3A3.A$.A4.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A.A$2.3A2.2A
3.2A2.2A$6.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A
3.3A3.A$.A4.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A
2.2A3.2A2.2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A
3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A2.2A
3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.
3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
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2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.
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2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A2.2A3.2A
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3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.
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3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A
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2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
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2.2A3.3A3.A.A$6.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.A2.A.A$4.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A.A.2A$4.A2.2C3.2C2.
2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.
2C2.A$.2A.A.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.
2C2.2C3.2C2.2C3.2C2.2C3.2A$A.A2.A3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C
2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C$A.A3.3A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$.A6.
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3.2A2.2A3.2A2.A$9.A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$11.A3.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$12.3A2.A.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$14.A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$21.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$
17.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.A$17.2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A4.A$20.A3.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.3A$21.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A$23.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A$30.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.3A$26.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.A$26.2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A4.A$29.A3.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.3A$30.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A$32.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A$39.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
3A$35.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.A$35.
2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$38.A3.3A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$39.3A2.A.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$41.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A$48.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$
44.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.A$44.2A2.3A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$47.A3.3A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.3A$48.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$
50.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$57.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.3A$53.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.A$53.
2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$56.A3.3A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.3A$57.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$59.A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A$66.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$62.A.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.A$62.2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A4.A$65.A
3.3A3.2A2.2A3.2A2.2A3.2A2.3A$66.3A2.A.2A3.2A2.2A3.2A2.2A3.2A$68.A3.2A
2.2A3.2A2.2A3.2A2.2A$75.2A2.2A3.2A2.2A3.2A2.3A$71.A.2A3.2A2.2A3.2A2.
2A3.2A2.A$71.2A2.3A3.2A2.2A3.2A2.2A4.A$74.A3.3A3.2A2.2A3.2A2.3A$75.3A
2.A.2A3.2A2.2A3.2A$77.A3.2A2.2A3.2A2.2A$84.2A2.2A3.2A2.3A$80.A.2A3.2A
2.2A3.2A2.A$80.2A2.3A3.2A2.2A4.A$83.A3.3A3.2A2.3A$84.3A2.A.2A3.2A$86.
A3.2A2.2A$93.2A2.3A$89.A.2A3.2A2.A$89.2A2.3A3.2A$92.A2.A$93.2A!

[dvgrn]

Tried stabilising it by hand, it's probably far from optimal

Code: Select all

x = 102, y = 123, rule = LifeHistory
7.2A$6.A2.A$.2A3.3A2.2A$.A2.2A3.2A.A$2.3A2.2A$6.2A2.2A3.A$4.2A3.2A.A
2.3A$2.3A2.2A3.3A3.A$.A4.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A.A$2.3A2.2A
3.2A2.2A$6.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A
3.3A3.A$.A4.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A
2.2A3.2A2.2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A
3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A2.2A
3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.
3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A2.2A$.A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.A$4.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A
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2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$2.3A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$6.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.A$4.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A.A2.3A$2.3A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.3A3.A$.A4.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.3A2.A$.A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A6.A$2.3A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.3A3.A.A$6.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.A2.A.A$4.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A.A.2A$4.A2.2C3.2C2.
2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.
2C2.A$.2A.A.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.
2C2.2C3.2C2.2C3.2C2.2C3.2A$A.A2.A3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C
2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C$A.A3.3A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$.A6.
A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.A$9.A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$11.A3.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$12.3A2.A.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$14.A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$21.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$
17.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.A$17.2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A4.A$20.A3.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.3A$21.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A$23.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A$30.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.3A$26.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.A$26.2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A4.A$29.A3.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.3A$30.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A$32.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A$39.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
3A$35.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.A$35.
2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$38.A3.3A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$39.3A2.A.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$41.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A$48.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$
44.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.A$44.2A2.3A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$47.A3.3A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.3A$48.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$
50.A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$57.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.3A$53.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.A$53.
2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$56.A3.3A3.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.3A$57.3A2.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A$59.A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A$66.2A2.2A3.2A2.2A3.2A2.2A3.2A2.3A$62.A.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.A$62.2A2.3A3.2A2.2A3.2A2.2A3.2A2.2A4.A$65.A
3.3A3.2A2.2A3.2A2.2A3.2A2.3A$66.3A2.A.2A3.2A2.2A3.2A2.2A3.2A$68.A3.2A
2.2A3.2A2.2A3.2A2.2A$75.2A2.2A3.2A2.2A3.2A2.3A$71.A.2A3.2A2.2A3.2A2.
2A3.2A2.A$71.2A2.3A3.2A2.2A3.2A2.2A4.A$74.A3.3A3.2A2.2A3.2A2.3A$75.3A
2.A.2A3.2A2.2A3.2A$77.A3.2A2.2A3.2A2.2A$84.2A2.2A3.2A2.3A$80.A.2A3.2A
2.2A3.2A2.A$80.2A2.3A3.2A2.2A4.A$83.A3.3A3.2A2.3A$84.3A2.A.2A3.2A$86.
A3.2A2.2A$93.2A2.3A$89.A.2A3.2A2.A$89.2A2.3A3.2A$92.A2.A$93.2A!

Seems like it was much easier with the diamond-shaped patch leaning the other way. Or I guess I mean, at a slightly different angle.

[galoomba]

Seems like it was much easier with the diamond-shaped patch leaning the other way. Or I guess I mean, at a slightly different angle.

Oh, that does make it a lot easier.

Code: Select all

x = 94, y = 63, rule = LifeHistory
2A$A2.2A$2.2A2.2A$5.2A2.2A$.A.2A3.2A2.2A$A.2A2.2A3.2A2.2A$A4.2A2.2A3.
2A2.2A$.A.2A3.2A2.2A3.2A2.2A$2.2A2.2A3.2A2.2A3.2A2.2A$5.2A2.2A3.2A2.
2A3.2A2.2A$.A.2A3.2A2.2A3.2A2.2A3.2A2.2A$A.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A$A4.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$.A.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A$2.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$5.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A$.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$
A.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$A4.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$.A.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A$2.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A$5.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$.A.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$A.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$A4.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$.A.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$2.2A2.2A3.2A2.2A3.
2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$5.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$.A.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$
A.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A$A4.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A$.A.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C
2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C.A$2.2C2.2C3.2C2.2C3.2C2.2C3.
2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C4.A$5.2C2.2C3.
2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.2C3.2C2.
2C.A$8.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A.A$11.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A$14.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A$17.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$20.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$23.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A.A$26.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A.A$29.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A
3.2A2.2A3.2A2.2A$32.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.
2A2.2A$35.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$38.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$41.2A2.2A3.2A2.2A3.2A
2.2A3.2A2.2A3.2A2.2A3.2A2.2A.A$44.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A
2.2A3.2A.A$47.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A$50.2A2.2A3.2A2.
2A3.2A2.2A3.2A2.2A3.2A2.2A$53.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$
56.2A2.2A3.2A2.2A3.2A2.2A3.2A2.2A4.A$59.2A2.2A3.2A2.2A3.2A2.2A3.2A2.
2A.A$62.2A2.2A3.2A2.2A3.2A2.2A3.2A.A$65.2A2.2A3.2A2.2A3.2A2.2A$68.2A
2.2A3.2A2.2A3.2A2.2A$71.2A2.2A3.2A2.2A3.2A.A$74.2A2.2A3.2A2.2A4.A$77.
2A2.2A3.2A2.2A.A$80.2A2.2A3.2A.A$83.2A2.2A$86.2A2.2A$89.2A2.A$92.2A!

[dingxutong]

Conjecture: Every pattern has a glider destruction. This might be easy to prove, but I'm not sure if a pattern can survive all kinds of collisions with a glider, and with anything else.

[AlbertArmStain]

Conjecture: Every pattern has a glider destruction. This might be easy to prove, but I'm not sure if a pattern can survive all kinds of collisions with a glider, and with anything else.

There may be a self constructing spaceship that, during at least one point of its evolution, would be impossible to destroy

[dvgrn]

Conjecture: Every pattern has a glider destruction. This might be easy to prove, but I'm not sure if a pattern can survive all kinds of collisions with a glider, and with anything else.

There may be a self constructing spaceship that, during at least one point of its evolution, would be impossible to destroy

Eh? If it's not destructible at one point of its evolution, it can't be destructible at any point of its evolution. Suppose it's indestructible at time T+N but destructible at time T. Take the collection of gliders that destroys the spaceship at time T, rewind them N ticks, and you have a counterexample destruction that works at time T+N.

There doesn't seem to be any indication at all that any pattern exists with no glider destruction. People can design awesomely clever glider-resistant structures all they want, but it's going to be awfully tricky to shut down all the weak points when an engineered attack is allowed.

Even in the absence of clever destruction-engineering, just "try gliders at random until you get a big explosion, wait for it to die down, then clean up the ash" will work quite well in every case that we know about.

[AlbertArmStain]

Eh? If it's not destructible at one point of its evolution, it can't be destructible at any point of its evolution. Suppose it's indestructible at time T+N but destructible at time T. Take the collection of gliders that destroys the spaceship at time T, rewind them N ticks, and you have a counterexample destruction that works at time T+N.

Oh yeah, I forgot about that.

There doesn't seem to be any indication at all that any pattern exists with no glider destruction. People can design awesomely clever glider-resistant structures all they want, but it's going to be awfully tricky to shut down all the weak points when an engineered attack is allowed.

Even in the absence of clever destruction-engineering, just "try gliders at random until you get a big explosion, wait for it to die down, then clean up the ash" will work quite well in every case that we know about.

There is probably an object that can absorb the hit from one glider, but probably not two gliders or xwsses

One thing someone could try is enclosing data behind an impenetrable wall of still lifes.

[dvgrn]

There is probably an object that can absorb the hit from one glider, but probably not two gliders or xwsses

One thing someone could try is enclosing data behind an impenetrable wall of still lifes.

This is all ground that has been gone over many times, decades ago.

It's trivial to build a constellation that can absorb a hit from any one glider. Just scatter blocks the right distance apart, such that a pi explosion from any one block won't touch any adjacent blocks -- but add enough layers of sparse blocks to ensure that there's a block in the way of every incoming glider lane.

A hollowed-out region of sparse blocks like this will prevent any single glider from reaching the central protected space, without causing any chaotic explosions... and it will stop most random pairs of gliders, and many random triplets of gliders... and so on. But as soon as you're allowed to carefully pick input glider lanes, of course you can get through such a wall very quickly.

An "impenetrable wall of still lifes" is one of those hopeful phrases like "a pair of wings that you can strap on and flap your arms and fly to the moon". It's not clear that these phrases refer to anything real, and in fact it seems pretty clear that they don't.

[galoomba]

Well, impenetrable walls/indestructible objects exist in some INT rules, so the idea isn't that ridiculous.

[hotdogPi]

Well, impenetrable walls/indestructible objects exist in some INT rules, so the idea isn't that ridiculous.

Those indestructible objects can be easily proven, usually provable within a 6×6 box or less.

[Haycat2009]

Conjecture: It is impossible for a phoenix oscillator to have an odd period.
I do not know how to prove this. Got any ideas?

[hotdogPi]

Conjecture: It is impossible for a phoenix oscillator to have an odd period.
I do not know how to prove this. Got any ideas?

This is a well-known unsolved problem, although we know that 3 is impossible.

[Haycat2009]

Conjecture: It is impossible for a phoenix oscillator to have an odd period.
I do not know how to prove this. Got any ideas?

This is a well-known unsolved problem, although we know that 3 is impossible.

We also know that 5 is impossible. What about 7, 19 and 41?

[AlbertArmStain]

Conjecture, there is a still life in every rule with 5 survival specifications

[squareroot12621]

Conjecture, there is a still life in every rule with 5 survival specifications

If you mean outer-totalistic rules with 5 survival transitions, B1/S45678 is a counterexample due to B1c.
If you mean isotropic rules with 5 survival transitions, B1c/S6ace78 is a counterexample.

[May13]

Conjecture, there is a still life in every rule with 5 survival specifications

If you mean outer-totalistic rules with 5 survival transitions, B1/S45678 is a counterexample due to B1c.
If you mean isotropic rules with 5 survival transitions, B1c/S6ace78 is a counterexample.

What about a less strict conjecture?
For any combination of 5 survival transitions, there is a set of birth conditions under which at least one still life exists.

If a still life exists in a rule with certain birth conditions, then it will exist without these conditions, because not a single cell becomes alive from the dead. Therefore, it suffices to consider rules without birth transitions.

In OT case, it's true, according to oscillators.db viewed with new-glider.py:

Code: Select all

>>>;c1
             B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B
                                             4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
                             5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
                     6 6 6 6 6 6 6 6                 6 6 6 6 6 6 6 6
                 7 7 7 7         7 7 7 7         7 7 7 7         7 7 7 7
               8 8     8 8     8 8     8 8     8 8     8 8     8 8     8 8
            ----------------------------------------------------------------
S          : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S        8 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S       78 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S       7  : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S      67  : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S      678 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S      6 8 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S      6   : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     56   : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     56 8 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     5678 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     567  : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     5 7  : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     5 78 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     5  8 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S     5    : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S    45    : 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 . . . . 1 1 1 1 2 2 2 2
S    45  8 : 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 . . . . 1 1 1 1 2 2 2 2
S    45 78 : 4 4 4 4 3 3 3 3 2 2 2 2 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3
S    45 7  : 4 4 4 4 3 3 3 3 2 2 2 2 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3
S    4567  : 6 5 4 5 4 3 4 5 4 3 2 3 4 3 4 5 4 3 2 3 2 1 2 3 4 3 2 3 4 3 4 5
S    45678 : 5 5 4 4 3 3 4 4 3 3 2 2 3 3 4 4 3 3 2 2 1 1 2 2 3 3 2 2 3 3 4 4
S    456 8 : 4 4 3 3 2 2 3 3 2 2 1 1 2 2 3 3 2 2 1 1 . . 1 1 2 2 1 1 2 2 3 3
S    456   : 5 4 3 4 3 2 3 4 3 2 1 2 3 2 3 4 3 2 1 2 1 . 1 2 3 2 1 2 3 2 3 4
S    4 6   : 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1
S    4 6 8 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S    4 678 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S    4 67  : 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
S    4  7  : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S    4  78 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S    4   8 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S    4     : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S   34     : 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3
S   34   8 : 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3
S   34  78 : 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4
S   34  7  : 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4
S   34 67  : 7 6 6 7 6 5 5 6 6 5 5 6 6 5 5 6 4 3 3 4 4 3 3 4 4 3 3 4 5 4 4 5
S   34 678 : 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4
S   34 6 8 : 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3
S   34 6   : 6 5 5 6 5 4 4 5 5 4 4 5 5 4 4 5 3 2 2 3 3 2 2 3 3 2 2 3 4 3 3 4
S   3456   : c b a b 9 8 9 a 9 8 7 8 9 8 9 a 7 6 5 6 5 4 5 6 7 6 5 6 8 7 8 9
S   3456 8 : c c b b 9 9 a a 9 9 8 8 9 9 a a 6 6 5 5 4 4 5 5 6 6 5 5 7 7 8 8
S   345678 : d d c c a a b b a a 9 9 a a b b 7 7 6 6 5 5 6 6 7 7 6 6 8 8 9 9
S   34567  : d c b c a 9 a b a 9 8 9 a 9 a b 8 7 6 7 6 5 6 7 8 7 6 7 9 8 9 a
S   345 7  : a a a a 8 8 8 8 7 7 7 7 8 8 8 8 6 6 6 6 5 5 5 5 6 6 6 6 8 8 8 8
S   345 78 : a a a a 8 8 8 8 7 7 7 7 8 8 8 8 6 6 6 6 5 5 5 5 6 6 6 6 8 8 8 8
S   345  8 : 9 9 9 9 7 7 7 7 6 6 6 6 7 7 7 7 5 5 5 5 4 4 4 4 5 5 5 5 7 7 7 7
S   345    : 9 9 9 9 7 7 7 7 6 6 6 6 7 7 7 7 5 5 5 5 4 4 4 4 5 5 5 5 7 7 7 7
S   3 5    : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 5  8 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 5 78 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 5 7  : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 567  : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 5678 : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 56 8 : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3 56   : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S   3  6   : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3  6 8 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3  678 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3  67  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3   7  : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3   78 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3    8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S   3      : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S  23      : c c c c b b b b 7 7 7 7 8 8 8 8 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23    8 : c c c c b b b b 7 7 7 7 8 8 8 8 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23   78 : c c c c b b b b 7 7 7 7 8 8 8 8 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23   7  : c c c c b b b b 7 7 7 7 8 8 8 8 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23  67  : d d d d c c c c 8 8 8 8 9 9 9 9 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23  678 : e e e e d d d d 9 9 9 9 a a a a 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23  6 8 : e e e e d d d d 9 9 9 9 a a a a 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23  6   : d d d d c c c c 8 8 8 8 9 9 9 9 5 5 5 5 4 4 4 4 7 7 7 7 8 8 8 8
S  23 56   : f f f f e e e e a a a a b b b b 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 56 8 : g g g g f f f f b b b b c c c c 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 5678 : g g g g f f f f b b b b c c c c 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 567  : f f f f e e e e a a a a b b b b 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 5 7  : e e e e d d d d 9 9 9 9 a a a a 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 5 78 : e e e e d d d d 9 9 9 9 a a a a 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 5  8 : e e e e d d d d 9 9 9 9 a a a a 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  23 5    : e e e e d d d d 9 9 9 9 a a a a 7 7 7 7 6 6 6 6 9 9 9 9 a a a a
S  2345    : o o n n k k k k d d d d f f f f 9 9 9 9 7 7 7 7 d d d d g g h h
S  2345  8 : o o n n k k k k d d d d f f f f 9 9 9 9 7 7 7 7 d d d d g g h h
S  2345 78 : p p o o l l l l e e e e g g g g a a a a 8 8 8 8 e e e e h h i i
S  2345 7  : p p o o l l l l e e e e g g g g a a a a 8 8 8 8 e e e e h h i i
S  234567  : s r p q n m n o h g f g i h i j c b a b 9 8 9 a g f e f i h j k
S  2345678 : s s q q n n o o h h g g i i j j b b a a 8 8 9 9 f f e e h h j j
S  23456 8 : r r p p m m n n g g f f h h i i a a 9 9 7 7 8 8 e e d d g g i i
S  23456   : r q o p m l m n g f e f h g h i b a 9 a 8 7 8 9 f e d e h g i j
S  234 6   : k j i j h g g h c b b c d c c d 7 6 6 7 6 5 5 6 a 9 9 a c b c d
S  234 6 8 : k k j j h h h h c c c c d d d d 6 6 6 6 5 5 5 5 9 9 9 9 b b c c
S  234 678 : l l k k i i i i d d d d e e e e 7 7 7 7 6 6 6 6 a a a a c c d d
S  234 67  : l k j k i h h i d c c d e d d e 8 7 7 8 7 6 6 7 b a a b d c d e
S  234  7  : j j i i g g g g b b b b c c c c 7 7 7 7 6 6 6 6 a a a a c c d d
S  234  78 : j j i i g g g g b b b b c c c c 7 7 7 7 6 6 6 6 a a a a c c d d
S  234   8 : i i h h f f f f a a a a b b b b 6 6 6 6 5 5 5 5 9 9 9 9 b b c c
S  234     : i i h h f f f f a a a a b b b b 6 6 6 6 5 5 5 5 9 9 9 9 b b c c
S  2 4     : 8 8 7 7 7 7 7 7 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 3 3 3 3 3 3 4 4
S  2 4   8 : 8 8 7 7 7 7 7 7 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 3 3 3 3 3 3 4 4
S  2 4  78 : 9 9 8 8 8 8 8 8 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5
S  2 4  7  : 9 9 8 8 8 8 8 8 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5
S  2 4 67  : a 9 8 9 9 8 8 9 7 6 6 7 7 6 6 7 3 2 2 3 3 2 2 3 5 4 4 5 5 4 5 6
S  2 4 678 : 9 9 8 8 8 8 8 8 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5
S  2 4 6 8 : 8 8 7 7 7 7 7 7 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 3 3 3 3 3 3 4 4
S  2 4 6   : 9 8 7 8 8 7 7 8 6 5 5 6 6 5 5 6 2 1 1 2 2 1 1 2 4 3 3 4 4 3 4 5
S  2 456   : e d b c b a b c 8 7 6 7 8 7 8 9 4 3 2 3 2 1 2 3 7 6 5 6 7 6 8 9
S  2 456 8 : d d b b a a b b 7 7 6 6 7 7 8 8 3 3 2 2 1 1 2 2 6 6 5 5 6 6 8 8
S  2 45678 : e e c c b b c c 8 8 7 7 8 8 9 9 4 4 3 3 2 2 3 3 7 7 6 6 7 7 9 9
S  2 4567  : f e c d c b c d 9 8 7 8 9 8 9 a 5 4 3 4 3 2 3 4 8 7 6 7 8 7 9 a
S  2 45 7  : d d c c b b b b 7 7 7 7 8 8 8 8 3 3 3 3 2 2 2 2 6 6 6 6 7 7 8 8
S  2 45 78 : d d c c b b b b 7 7 7 7 8 8 8 8 3 3 3 3 2 2 2 2 6 6 6 6 7 7 8 8
S  2 45  8 : c c b b a a a a 6 6 6 6 7 7 7 7 2 2 2 2 1 1 1 1 5 5 5 5 6 6 7 7
S  2 45    : c c b b a a a a 6 6 6 6 7 7 7 7 2 2 2 2 1 1 1 1 5 5 5 5 6 6 7 7
S  2  5    : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  5  8 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  5 78 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  5 7  : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  567  : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  5678 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  56 8 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2  56   : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2   6   : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2   6 8 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2   678 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2   67  : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2    7  : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2    78 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2     8 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S  2       : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
S 12       : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12     8 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12    78 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12    7  : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12   67  : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12   678 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12   6 8 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12   6   : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  56   : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  56 8 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  5678 : 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  567  : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  5 7  : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  5 78 : 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  5  8 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12  5    : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 12 45    : i i h h g g g g b b b b c c c c 6 6 6 6 5 5 5 5 a a a a b b c c
S 12 45  8 : i i h h g g g g b b b b c c c c 6 6 6 6 5 5 5 5 a a a a b b c c
S 12 45 78 : k k j j i i i i d d d d e e e e 7 7 7 7 6 6 6 6 b b b b c c d d
S 12 45 7  : j j i i h h h h c c c c d d d d 7 7 7 7 6 6 6 6 b b b b c c d d
S 12 4567  : m l j k j i j k f e d e f e f g 9 8 7 8 7 6 7 8 d c b c d c e f
S 12 45678 : m m k k j j k k f f e e f f g g 8 8 7 7 6 6 7 7 c c b b c c e e
S 12 456 8 : k k i i h h i i d d c c d d e e 7 7 6 6 5 5 6 6 b b a a b b d d
S 12 456   : l k i j i h i j e d c d e d e f 8 7 6 7 6 5 6 7 c b a b c b d e
S 12 4 6   : g f e f f e e f c b b c c b b c 6 5 5 6 6 5 5 6 9 8 8 9 9 8 9 a
S 12 4 6 8 : f f e e e e e e b b b b b b b b 5 5 5 5 5 5 5 5 8 8 8 8 8 8 9 9
S 12 4 678 : g g f f f f f f c c c c c c c c 6 6 6 6 6 6 6 6 9 9 9 9 9 9 a a
S 12 4 67  : h g f g g f f g d c c d d c c d 7 6 6 7 7 6 6 7 a 9 9 a a 9 a b
S 12 4  7  : f f e e e e e e b b b b b b b b 6 6 6 6 6 6 6 6 9 9 9 9 9 9 a a
S 12 4  78 : f f e e e e e e b b b b b b b b 6 6 6 6 6 6 6 6 9 9 9 9 9 9 a a
S 12 4   8 : e e d d d d d d a a a a a a a a 5 5 5 5 5 5 5 5 8 8 8 8 8 8 9 9
S 12 4     : e e d d d d d d a a a a a a a a 5 5 5 5 5 5 5 5 8 8 8 8 8 8 9 9
S 1234     : q q p p n n n n h h h h i i i i a a a a 9 9 9 9 e e e e g g h h
S 1234   8 : q q p p n n n n h h h h i i i i a a a a 9 9 9 9 e e e e g g h h
S 1234  78 : r r q q o o o o i i i i j j j j b b b b a a a a f f f f h h i i
S 1234  7  : r r q q o o o o i i i i j j j j b b b b a a a a f f f f h h i i
S 1234 67  : u t s t r q q r l k k l m l l m c b b c b a a b g f f g i h i j
S 1234 678 : u u t t r r r r l l l l m m m m b b b b a a a a f f f f h h i i
S 1234 6 8 : t t s s q q q q k k k k l l l l a a a a 9 9 9 9 e e e e g g h h
S 1234 6   : t s r s q p p q k j j k l k k l b a a b a 9 9 a f e e f h g h i
S 123456   : X z x y v u v w o n m n p o p q f e d e c b c d k j i j m l n o
S 123456 8 : X X y y v v w w o o n n p p q q e e d d b b c c j j i i l l n n
S 12345678 : X X X X x x y y q q p p r r s s f f e e c c d d k k j j m m o o
S 1234567  : X X y z w v w x p o n o q p q r g f e f d c d e l k j k n m o p
S 12345 7  : x x w w t t t t l l l l n n n n e e e e c c c c j j j j m m n n
S 12345 78 : y y x x u u u u m m m m o o o o e e e e c c c c j j j j m m n n
S 12345  8 : w w v v s s s s k k k k m m m m d d d d b b b b i i i i l l m m
S 12345    : w w v v s s s s k k k k m m m m d d d d b b b b i i i i l l m m
S 123 5    : j j j j i i i i e e e e f f f f a a a a 9 9 9 9 c c c c d d d d
S 123 5  8 : j j j j i i i i e e e e f f f f a a a a 9 9 9 9 c c c c d d d d
S 123 5 78 : k k k k j j j j f f f f g g g g a a a a 9 9 9 9 c c c c d d d d
S 123 5 7  : j j j j i i i i e e e e f f f f a a a a 9 9 9 9 c c c c d d d d
S 123 567  : k k k k j j j j f f f f g g g g a a a a 9 9 9 9 c c c c d d d d
S 123 5678 : m m m m l l l l h h h h i i i i a a a a 9 9 9 9 c c c c d d d d
S 123 56 8 : l l l l k k k k g g g g h h h h a a a a 9 9 9 9 c c c c d d d d
S 123 56   : k k k k j j j j f f f f g g g g a a a a 9 9 9 9 c c c c d d d d
S 123  6   : i i i i h h h h d d d d e e e e 8 8 8 8 7 7 7 7 a a a a b b b b
S 123  6 8 : j j j j i i i i e e e e f f f f 8 8 8 8 7 7 7 7 a a a a b b b b
S 123  678 : j j j j i i i i e e e e f f f f 8 8 8 8 7 7 7 7 a a a a b b b b
S 123  67  : i i i i h h h h d d d d e e e e 8 8 8 8 7 7 7 7 a a a a b b b b
S 123   7  : h h h h g g g g c c c c d d d d 8 8 8 8 7 7 7 7 a a a a b b b b
S 123   78 : h h h h g g g g c c c c d d d d 8 8 8 8 7 7 7 7 a a a a b b b b
S 123    8 : h h h h g g g g c c c c d d d d 8 8 8 8 7 7 7 7 a a a a b b b b
S 123      : h h h h g g g g c c c c d d d d 8 8 8 8 7 7 7 7 a a a a b b b b
S 1 3      : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3    8 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3   78 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3   7  : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3  67  : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3  678 : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3  6 8 : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3  6   : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S 1 3 56   : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 56 8 : 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 5678 : 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 567  : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 5 7  : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 5 78 : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 5  8 : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 3 5    : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
S 1 345    : f f f f d d d d b b b b c c c c 8 8 8 8 7 7 7 7 9 9 9 9 b b b b
S 1 345  8 : f f f f d d d d b b b b c c c c 8 8 8 8 7 7 7 7 9 9 9 9 b b b b
S 1 345 78 : h h h h f f f f d d d d e e e e 9 9 9 9 8 8 8 8 a a a a c c c c
S 1 345 7  : g g g g e e e e c c c c d d d d 9 9 9 9 8 8 8 8 a a a a c c c c
S 1 34567  : k j i j h g h i g f e f g f g h b a 9 a 9 8 9 a c b a b d c d e
S 1 345678 : l l k k i i j j h h g g h h i i a a 9 9 8 8 9 9 b b a a c c d d
S 1 3456 8 : j j i i g g h h f f e e f f g g 9 9 8 8 7 7 8 8 a a 9 9 b b c c
S 1 3456   : j i h i g f g h f e d e f e f g a 9 8 9 8 7 8 9 b a 9 a c b c d
S 1 34 6   : d c c d c b b c b a a b b a a b 6 5 5 6 6 5 5 6 7 6 6 7 8 7 7 8
S 1 34 6 8 : d d d d c c c c b b b b b b b b 5 5 5 5 5 5 5 5 6 6 6 6 7 7 7 7
S 1 34 678 : e e e e d d d d c c c c c c c c 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8
S 1 34 67  : e d d e d c c d c b b c c b b c 7 6 6 7 7 6 6 7 8 7 7 8 9 8 8 9
S 1 34  7  : b b b b a a a a 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8
S 1 34  78 : b b b b a a a a 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8
S 1 34   8 : a a a a 9 9 9 9 8 8 8 8 8 8 8 8 5 5 5 5 5 5 5 5 6 6 6 6 7 7 7 7
S 1 34     : a a a a 9 9 9 9 8 8 8 8 8 8 8 8 5 5 5 5 5 5 5 5 6 6 6 6 7 7 7 7
S 1  4     : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
S 1  4   8 : 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
S 1  4  78 : 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 1  4  7  : 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 1  4 67  : 8 7 7 8 8 7 7 8 7 6 6 7 7 6 6 7 5 4 4 5 5 4 4 5 6 5 5 6 6 5 5 6
S 1  4 678 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S 1  4 6 8 : 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
S 1  4 6   : 7 6 6 7 7 6 6 7 6 5 5 6 6 5 5 6 4 3 3 4 4 3 3 4 5 4 4 5 5 4 4 5
S 1  456   : b a 9 a 9 8 9 a 8 7 6 7 8 7 8 9 6 5 4 5 4 3 4 5 7 6 5 6 7 6 7 8
S 1  456 8 : a a 9 9 8 8 9 9 7 7 6 6 7 7 8 8 5 5 4 4 3 3 4 4 6 6 5 5 6 6 7 7
S 1  45678 : c c b b a a b b 9 9 8 8 9 9 a a 6 6 5 5 4 4 5 5 7 7 6 6 7 7 8 8
S 1  4567  : c b a b a 9 a b 9 8 7 8 9 8 9 a 7 6 5 6 5 4 5 6 8 7 6 7 8 7 8 9
S 1  45 7  : 9 9 9 9 8 8 8 8 6 6 6 6 7 7 7 7 5 5 5 5 4 4 4 4 6 6 6 6 7 7 7 7
S 1  45 78 : a a a a 9 9 9 9 7 7 7 7 8 8 8 8 5 5 5 5 4 4 4 4 6 6 6 6 7 7 7 7
S 1  45  8 : 8 8 8 8 7 7 7 7 5 5 5 5 6 6 6 6 4 4 4 4 3 3 3 3 5 5 5 5 6 6 6 6
S 1  45    : 8 8 8 8 7 7 7 7 5 5 5 5 6 6 6 6 4 4 4 4 3 3 3 3 5 5 5 5 6 6 6 6
S 1   5    : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   5  8 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   5 78 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   5 7  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   567  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   5678 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   56 8 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1   56   : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1    6   : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1    6 8 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1    678 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1    67  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1     7  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1     78 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1      8 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S 1        : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S01        : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01      8 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01     78 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01     7  : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01    67  : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01    678 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01    6 8 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01    6   : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   56   : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   56 8 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   5678 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   567  : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   5 7  : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   5 78 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   5  8 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01   5    : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
S01  45    : 9 9 9 9 8 8 8 8 6 6 6 6 7 7 7 7 5 5 5 5 4 4 4 4 6 6 6 6 7 7 7 7
S01  45  8 : 9 9 9 9 8 8 8 8 6 6 6 6 7 7 7 7 5 5 5 5 4 4 4 4 6 6 6 6 7 7 7 7
S01  45 78 : b b b b a a a a 8 8 8 8 9 9 9 9 6 6 6 6 5 5 5 5 7 7 7 7 8 8 8 8
S01  45 7  : a a a a 9 9 9 9 7 7 7 7 8 8 8 8 6 6 6 6 5 5 5 5 7 7 7 7 8 8 8 8
S01  4567  : d c b c b a b c a 9 8 9 a 9 a b 8 7 6 7 6 5 6 7 9 8 7 8 9 8 9 a
S01  45678 : d d c c b b c c a a 9 9 a a b b 7 7 6 6 5 5 6 6 8 8 7 7 8 8 9 9
S01  456 8 : b b a a 9 9 a a 8 8 7 7 8 8 9 9 6 6 5 5 4 4 5 5 7 7 6 6 7 7 8 8
S01  456   : c b a b a 9 a b 9 8 7 8 9 8 9 a 7 6 5 6 5 4 5 6 8 7 6 7 8 7 8 9
S01  4 6   : 8 7 7 8 8 7 7 8 7 6 6 7 7 6 6 7 5 4 4 5 5 4 4 5 6 5 5 6 6 5 5 6
S01  4 6 8 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S01  4 678 : 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S01  4 67  : 9 8 8 9 9 8 8 9 8 7 7 8 8 7 7 8 6 5 5 6 6 5 5 6 7 6 6 7 7 6 6 7
S01  4  7  : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S01  4  78 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S01  4   8 : 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S01  4     : 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
S01 34     : b b b b a a a a 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8
S01 34   8 : b b b b a a a a 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8
S01 34  78 : c c c c b b b b a a a a a a a a 7 7 7 7 7 7 7 7 8 8 8 8 9 9 9 9
S01 34  7  : c c c c b b b b a a a a a a a a 7 7 7 7 7 7 7 7 8 8 8 8 9 9 9 9
S01 34 67  : f e e f e d d e d c c d d c c d 8 7 7 8 8 7 7 8 9 8 8 9 a 9 9 a
S01 34 678 : f f f f e e e e d d d d d d d d 7 7 7 7 7 7 7 7 8 8 8 8 9 9 9 9
S01 34 6 8 : e e e e d d d d c c c c c c c c 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8
S01 34 6   : e d d e d c c d c b b c c b b c 7 6 6 7 7 6 6 7 8 7 7 8 9 8 8 9
S01 3456   : k j i j h g h i g f e f g f g h b a 9 a 9 8 9 a c b a b d c d e
S01 3456 8 : k k j j h h i i g g f f g g h h a a 9 9 8 8 9 9 b b a a c c d d
S01 345678 : m m l l j j k k i i h h i i j j b b a a 9 9 a a c c b b d d e e
S01 34567  : l k j k i h i j h g f g h g h i c b a b a 9 a b d c b c e d e f
S01 345 7  : h h h h f f f f d d d d e e e e a a a a 9 9 9 9 b b b b d d d d
S01 345 78 : i i i i g g g g e e e e f f f f a a a a 9 9 9 9 b b b b d d d d
S01 345  8 : g g g g e e e e c c c c d d d d 9 9 9 9 8 8 8 8 a a a a c c c c
S01 345    : g g g g e e e e c c c c d d d d 9 9 9 9 8 8 8 8 a a a a c c c c
S01 3 5    : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 5  8 : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 5 78 : 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 5 7  : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 567  : 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 5678 : a a a a a a a a a a a a a a a a 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 56 8 : 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3 56   : 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
S01 3  6   : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3  6 8 : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3  678 : 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3  67  : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3   7  : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3   78 : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3    8 : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S01 3      : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0123      : j j j j i i i i e e e e f f f f 9 9 9 9 8 8 8 8 b b b b c c c c
S0123    8 : j j j j i i i i e e e e f f f f 9 9 9 9 8 8 8 8 b b b b c c c c
S0123   78 : j j j j i i i i e e e e f f f f 9 9 9 9 8 8 8 8 b b b b c c c c
S0123   7  : j j j j i i i i e e e e f f f f 9 9 9 9 8 8 8 8 b b b b c c c c
S0123  67  : k k k k j j j j f f f f g g g g 9 9 9 9 8 8 8 8 b b b b c c c c
S0123  678 : l l l l k k k k g g g g h h h h 9 9 9 9 8 8 8 8 b b b b c c c c
S0123  6 8 : l l l l k k k k g g g g h h h h 9 9 9 9 8 8 8 8 b b b b c c c c
S0123  6   : k k k k j j j j f f f f g g g g 9 9 9 9 8 8 8 8 b b b b c c c c
S0123 56   : m m m m l l l l h h h h i i i i b b b b a a a a d d d d e e e e
S0123 56 8 : n n n n m m m m i i i i j j j j b b b b a a a a d d d d e e e e
S0123 5678 : o o o o n n n n j j j j k k k k b b b b a a a a d d d d e e e e
S0123 567  : m m m m l l l l h h h h i i i i b b b b a a a a d d d d e e e e
S0123 5 7  : l l l l k k k k g g g g h h h h b b b b a a a a d d d d e e e e
S0123 5 78 : m m m m l l l l h h h h i i i i b b b b a a a a d d d d e e e e
S0123 5  8 : l l l l k k k k g g g g h h h h b b b b a a a a d d d d e e e e
S0123 5    : l l l l k k k k g g g g h h h h b b b b a a a a d d d d e e e e
S012345    : y y x x u u u u m m m m o o o o e e e e c c c c j j j j m m n n
S012345  8 : y y x x u u u u m m m m o o o o e e e e c c c c j j j j m m n n
S012345 78 : X X z z w w w w o o o o q q q q f f f f d d d d k k k k n n o o
S012345 7  : z z y y v v v v n n n n p p p p f f f f d d d d k k k k n n o o
S01234567  : X X X X y x y z r q p q s r s t h g f g e d e f m l k l o n p q
S012345678 : X X X X z z X X s s r r t t u u g g f f d d e e l l k k n n p p
S0123456 8 : X X X X x x y y q q p p r r s s f f e e c c d d k k j j m m o o
S0123456   : X X z X x w x y q p o p r q r s g f e f d c d e l k j k n m o p
S01234 6   : v u t u s r r s m l l m n m m n c b b c b a a b g f f g i h i j
S01234 6 8 : v v u u s s s s m m m m n n n n b b b b a a a a f f f f h h i i
S01234 678 : w w v v t t t t n n n n o o o o c c c c b b b b g g g g i i j j
S01234 67  : w v u v t s s t n m m n o n n o d c c d c b b c h g g h j i j k
S01234  7  : t t s s q q q q k k k k l l l l c c c c b b b b g g g g i i j j
S01234  78 : t t s s q q q q k k k k l l l l c c c c b b b b g g g g i i j j
S01234   8 : s s r r p p p p j j j j k k k k b b b b a a a a f f f f h h i i
S01234     : s s r r p p p p j j j j k k k k b b b b a a a a f f f f h h i i
S012 4     : g g f f f f f f c c c c c c c c 6 6 6 6 6 6 6 6 9 9 9 9 9 9 a a
S012 4   8 : g g f f f f f f c c c c c c c c 6 6 6 6 6 6 6 6 9 9 9 9 9 9 a a
S012 4  78 : h h g g g g g g d d d d d d d d 7 7 7 7 7 7 7 7 a a a a a a b b
S012 4  7  : h h g g g g g g d d d d d d d d 7 7 7 7 7 7 7 7 a a a a a a b b
S012 4 67  : j i h i i h h i f e e f f e e f 8 7 7 8 8 7 7 8 b a a b b a b c
S012 4 678 : i i h h h h h h e e e e e e e e 7 7 7 7 7 7 7 7 a a a a a a b b
S012 4 6 8 : h h g g g g g g d d d d d d d d 6 6 6 6 6 6 6 6 9 9 9 9 9 9 a a
S012 4 6   : i h g h h g g h e d d e e d d e 7 6 6 7 7 6 6 7 a 9 9 a a 9 a b
S012 456   : n m k l k j k l g f e f g f g h 9 8 7 8 7 6 7 8 d c b c d c e f
S012 456 8 : m m k k j j k k f f e e f f g g 8 8 7 7 6 6 7 7 c c b b c c e e
S012 45678 : o o m m l l m m h h g g h h i i 9 9 8 8 7 7 8 8 d d c c d d f f
S012 4567  : o n l m l k l m h g f g h g h i a 9 8 9 8 7 8 9 e d c d e d f g
S012 45 7  : l l k k j j j j e e e e f f f f 8 8 8 8 7 7 7 7 c c c c d d e e
S012 45 78 : m m l l k k k k f f f f g g g g 8 8 8 8 7 7 7 7 c c c c d d e e
S012 45  8 : k k j j i i i i d d d d e e e e 7 7 7 7 6 6 6 6 b b b b c c d d
S012 45    : k k j j i i i i d d d d e e e e 7 7 7 7 6 6 6 6 b b b b c c d d
S012  5    : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  5  8 : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  5 78 : b b b b b b b b a a a a a a a a 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  5 7  : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  567  : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  5678 : b b b b b b b b a a a a a a a a 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  56 8 : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012  56   : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012   6   : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012   6 8 : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012   678 : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012   67  : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012    7  : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012    78 : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012     8 : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S012       : a a a a a a a a 9 9 9 9 9 9 9 9 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6
S0 2       : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2     8 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2    78 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2    7  : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2   67  : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2   678 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2   6 8 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2   6   : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  56   : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  56 8 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  5678 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  567  : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  5 7  : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  5 78 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  5  8 : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2  5    : 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3
S0 2 45    : e e d d c c c c 8 8 8 8 9 9 9 9 3 3 3 3 2 2 2 2 6 6 6 6 7 7 8 8
S0 2 45  8 : e e d d c c c c 8 8 8 8 9 9 9 9 3 3 3 3 2 2 2 2 6 6 6 6 7 7 8 8
S0 2 45 78 : f f e e d d d d 9 9 9 9 a a a a 4 4 4 4 3 3 3 3 7 7 7 7 8 8 9 9
S0 2 45 7  : f f e e d d d d 9 9 9 9 a a a a 4 4 4 4 3 3 3 3 7 7 7 7 8 8 9 9
S0 2 4567  : h g e f e d e f b a 9 a b a b c 6 5 4 5 4 3 4 5 9 8 7 8 9 8 a b
S0 2 45678 : g g e e d d e e a a 9 9 a a b b 5 5 4 4 3 3 4 4 8 8 7 7 8 8 a a
S0 2 456 8 : f f d d c c d d 9 9 8 8 9 9 a a 4 4 3 3 2 2 3 3 7 7 6 6 7 7 9 9
S0 2 456   : g f d e d c d e a 9 8 9 a 9 a b 5 4 3 4 3 2 3 4 8 7 6 7 8 7 9 a
S0 2 4 6   : b a 9 a a 9 9 a 8 7 7 8 8 7 7 8 3 2 2 3 3 2 2 3 5 4 4 5 5 4 5 6
S0 2 4 6 8 : a a 9 9 9 9 9 9 7 7 7 7 7 7 7 7 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5
S0 2 4 678 : b b a a a a a a 8 8 8 8 8 8 8 8 3 3 3 3 3 3 3 3 5 5 5 5 5 5 6 6
S0 2 4 67  : c b a b b a a b 9 8 8 9 9 8 8 9 4 3 3 4 4 3 3 4 6 5 5 6 6 5 6 7
S0 2 4  7  : b b a a a a a a 8 8 8 8 8 8 8 8 3 3 3 3 3 3 3 3 5 5 5 5 5 5 6 6
S0 2 4  78 : b b a a a a a a 8 8 8 8 8 8 8 8 3 3 3 3 3 3 3 3 5 5 5 5 5 5 6 6
S0 2 4   8 : a a 9 9 9 9 9 9 7 7 7 7 7 7 7 7 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5
S0 2 4     : a a 9 9 9 9 9 9 7 7 7 7 7 7 7 7 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5
S0 234     : k k j j h h h h c c c c d d d d 7 7 7 7 6 6 6 6 a a a a c c d d
S0 234   8 : k k j j h h h h c c c c d d d d 7 7 7 7 6 6 6 6 a a a a c c d d
S0 234  78 : l l k k i i i i d d d d e e e e 8 8 8 8 7 7 7 7 b b b b d d e e
S0 234  7  : l l k k i i i i d d d d e e e e 8 8 8 8 7 7 7 7 b b b b d d e e
S0 234 67  : n m l m k j j k f e e f g f f g 9 8 8 9 8 7 7 8 c b b c e d e f
S0 234 678 : n n m m k k k k f f f f g g g g 8 8 8 8 7 7 7 7 b b b b d d e e
S0 234 6 8 : m m l l j j j j e e e e f f f f 7 7 7 7 6 6 6 6 a a a a c c d d
S0 234 6   : m l k l j i i j e d d e f e e f 8 7 7 8 7 6 6 7 b a a b d c d e
S0 23456   : t s q r o n o p i h g h j i j k c b a b 9 8 9 a g f e f i h j k
S0 23456 8 : t t r r o o p p i i h h j j k k b b a a 8 8 9 9 f f e e h h j j
S0 2345678 : u u s s p p q q j j i i k k l l c c b b 9 9 a a g g f f i i k k
S0 234567  : u t r s p o p q j i h i k j k l d c b c a 9 a b h g f g j i k l
S0 2345 7  : r r q q n n n n g g g g i i i i b b b b 9 9 9 9 f f f f i i j j
S0 2345 78 : r r q q n n n n g g g g i i i i b b b b 9 9 9 9 f f f f i i j j
S0 2345  8 : q q p p m m m m f f f f h h h h a a a a 8 8 8 8 e e e e h h i i
S0 2345    : q q p p m m m m f f f f h h h h a a a a 8 8 8 8 e e e e h h i i
S0 23 5    : g g g g f f f f b b b b c c c c 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 5  8 : g g g g f f f f b b b b c c c c 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 5 78 : g g g g f f f f b b b b c c c c 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 5 7  : g g g g f f f f b b b b c c c c 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 567  : h h h h g g g g c c c c d d d d 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 5678 : i i i i h h h h d d d d e e e e 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 56 8 : i i i i h h h h d d d d e e e e 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23 56   : h h h h g g g g c c c c d d d d 8 8 8 8 7 7 7 7 a a a a b b b b
S0 23  6   : f f f f e e e e a a a a b b b b 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23  6 8 : g g g g f f f f b b b b c c c c 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23  678 : g g g g f f f f b b b b c c c c 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23  67  : f f f f e e e e a a a a b b b b 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23   7  : e e e e d d d d 9 9 9 9 a a a a 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23   78 : e e e e d d d d 9 9 9 9 a a a a 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23    8 : e e e e d d d d 9 9 9 9 a a a a 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0 23      : e e e e d d d d 9 9 9 9 a a a a 6 6 6 6 5 5 5 5 8 8 8 8 9 9 9 9
S0  3      : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3    8 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3   78 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3   7  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3  67  : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3  678 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3  6 8 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3  6   : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0  3 56   : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 56 8 : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 5678 : 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 567  : 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 5 7  : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 5 78 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 5  8 : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  3 5    : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
S0  345    : a a a a 8 8 8 8 7 7 7 7 8 8 8 8 6 6 6 6 5 5 5 5 6 6 6 6 8 8 8 8
S0  345  8 : a a a a 8 8 8 8 7 7 7 7 8 8 8 8 6 6 6 6 5 5 5 5 6 6 6 6 8 8 8 8
S0  345 78 : b b b b 9 9 9 9 8 8 8 8 9 9 9 9 7 7 7 7 6 6 6 6 7 7 7 7 9 9 9 9
S0  345 7  : b b b b 9 9 9 9 8 8 8 8 9 9 9 9 7 7 7 7 6 6 6 6 7 7 7 7 9 9 9 9
S0  34567  : e d c d b a b c b a 9 a b a b c 9 8 7 8 7 6 7 8 9 8 7 8 a 9 a b
S0  345678 : e e d d b b c c b b a a b b c c 8 8 7 7 6 6 7 7 8 8 7 7 9 9 a a
S0  3456 8 : d d c c a a b b a a 9 9 a a b b 7 7 6 6 5 5 6 6 7 7 6 6 8 8 9 9
S0  3456   : d c b c a 9 a b a 9 8 9 a 9 a b 8 7 6 7 6 5 6 7 8 7 6 7 9 8 9 a
S0  34 6   : 7 6 6 7 6 5 5 6 6 5 5 6 6 5 5 6 4 3 3 4 4 3 3 4 4 3 3 4 5 4 4 5
S0  34 6 8 : 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4
S0  34 678 : 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5
S0  34 67  : 8 7 7 8 7 6 6 7 7 6 6 7 7 6 6 7 5 4 4 5 5 4 4 5 5 4 4 5 6 5 5 6
S0  34  7  : 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5
S0  34  78 : 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5
S0  34   8 : 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4
S0  34     : 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4
S0   4     : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0   4   8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0   4  78 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0   4  7  : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0   4 67  : 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3
S0   4 678 : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
S0   4 6 8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0   4 6   : 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
S0   456   : 6 5 4 5 4 3 4 5 4 3 2 3 4 3 4 5 4 3 2 3 2 1 2 3 4 3 2 3 4 3 4 5
S0   456 8 : 5 5 4 4 3 3 4 4 3 3 2 2 3 3 4 4 3 3 2 2 1 1 2 2 3 3 2 2 3 3 4 4
S0   45678 : 6 6 5 5 4 4 5 5 4 4 3 3 4 4 5 5 4 4 3 3 2 2 3 3 4 4 3 3 4 4 5 5
S0   4567  : 7 6 5 6 5 4 5 6 5 4 3 4 5 4 5 6 5 4 3 4 3 2 3 4 5 4 3 4 5 4 5 6
S0   45 7  : 5 5 5 5 4 4 4 4 3 3 3 3 4 4 4 4 3 3 3 3 2 2 2 2 3 3 3 3 4 4 4 4
S0   45 78 : 5 5 5 5 4 4 4 4 3 3 3 3 4 4 4 4 3 3 3 3 2 2 2 2 3 3 3 3 4 4 4 4
S0   45  8 : 4 4 4 4 3 3 3 3 2 2 2 2 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3
S0   45    : 4 4 4 4 3 3 3 3 2 2 2 2 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3
S0    5    : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    5  8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    5 78 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    5 7  : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    567  : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    5678 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    56 8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0    56   : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0     6   : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0     6 8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0     678 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0     67  : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0      7  : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0      78 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0       8 : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
S0         : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Starting from line S34, all rules without birth conditions have at least one known still life. Above this line there is only one line with 5 survival conditions (S45678), but still lifes are known in B/S45678.

In INT case, it's false. S6in78 is a counterexample, because it's impossible to keep alive any cell located on the edge of the bounding box.

[AlbertArmStain]

Conjecture: there is a wick that can only be supported by itself

[Moosey]

Well, GOL is omniperiodic now, thanks to the p41 that glider_rider found. That's one big unproven conjecture, PROVEN.

[wirehead]

Hooray on proving CGoL omniperiodic!!

Now I have a question.

Are there any 2-state rules that can be proven to be not omniperiodic (excluding trivial cases such as B0 rules and LwoD), but with an infinite number of "missing" periods? E.g. a rule where any period one less than a multiple of four is impossible?

In more mathy terms: Let the period density up to n be the number of possible periods less than or equal to n in some rule, divided by n. Are there any rules such that the period density approaches some number other than 0, 0.5, or 1 as n goes to infinity?

For example, because CGoL is omniperiodic, the period density is 1 no matter the n. In Wireworld it approaches 1 as the only impossible period is 2.


Separate question:

Are there also some set of rules where the period density approaches 1, but still, for any arbitrarily large n, there is a period greater than n that is impossible (i.e. the "holes" become less and less frequent)?

[toroidalet]

B2a/S and B2c/S only have even-period oscillators, and B3i/S2i probably has only a p2 and p4. Finally, I think B2aci/S can only have oscillators of period 4n.
However, it seems very hard to prove non-omniperiodicity for all but the simplest rules. How would you prove that B3/S has no p1000 oscillator?

[Haycat2009]

Can anyone think of (or engineer) a rule that allows for finite oscillators, but only of particular limited types -- no oscillators above period 2, for example? That would certainly make Turing completeness impossible. Maybe B3/S is such a rule? It doesn't seem possible to design phoenix-based circuitry capable of carrying signals of any kind.

Be careful that the circutry does not even have to be reusable - two examples would be the one-time/burnt-after-using Rule 110 unit cell in LWoD or DotLife. Basically you can have many options for presenting and interacting signals, so B3/S could actually have adequate signal manipulations.

Yes! The B35/S23 W110 circuitery is not reusable, but it still counts.

[confocaloid]

Yes! The B35/S23 W110 circuitery is not reusable, but it still counts.

I would say these are two different possibilities.
If a logic circuit is one-time by design, it can still count as a logic circuit.
But if someone finds a way to make reusable logic circuits in B35/S23 or another rule, that will be cool & notable, even if an one-time version already exists.

[C28]

the p280 U-turner hassler takes no more than 33 gliders to synthesize (i would try to prove this myself but i don't have the time to do so)

[confocaloid]

the p280 U-turner hassler takes no more than 33 gliders to synthesize (i would try to prove this myself but i don't have the time to do so)

If its glider constructible at all (which it is, as there is only one small isolated unstable object to insert), then it takes no more than 15 gliders to construct, via the universal constructor argument.

Code: Select all

x = 41, y = 44, rule = B3/S23
38bo$38bobo$38b2o$15bo2$13b3obo$12bob2o9bo$12b2obo8bobo$10bob3o10bo2$
12bo3$35b3o$35b3o$35b3o$38b3o$20bo17b3o$5bo13bobo16b3o$4bobo13b2o$5bo
27b2o$33bobo$34bo2$30bo$29bobo$3bo25b2o6bo$2b2o32bobo$bob2o32bo$3o2bo$
2bobobo$3bobobo$4bo2b3o$5b2obo$6b2o$6bo$30bo$29b3o$17bo10bob3o$16bobo
8bo3bo$17bo8bo3bo$25b3obo$26b3o$27bo!

[GUYTU6J]

Conjecture: there exists positive integers n, m ≤ 26 such that at least one maximum-density still life within bounding box n×m does not have a glider synthesis.

Conjecture: every strict or pseudo still life within a 6-cell-high bounding box has a glider synthesis.