# Vanishing line on Conway's game of life

If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$.

Question: Is such a line non-vanishing for any $n \in [25,\infty)$?

Edit

Definition: a finite pattern $p$ has a weak period $wp$ if for any cell $c$ in the grid, there is $k>0$ such that the set of cells which are neighbours of neighbours of neighbours... ($k$ times) of $c$, is periodic of period $wp$ after sufficiently many generations, from the initial state $p$.

The sequence A061342 gives the weak period $wp_n$ of a line of $n$ alive cells. By combining the checking above with the fact that $wp_n \ge 2$ for $n \in [84,1000]$, we deduce that the pattern is non-vanishing for $n \in [25,1000]$. We observe that for $n=500$, four gliders are produced on the boundaries after $435$ steps, but $435<500$, so this must happens $\forall n \ge 500$. Assuming that these gliders (or others) are perpetual (as stated implicitly by Nathaniel Johnston in A061342, although without reference, while the proof could be non-trivial, as pointed out by Will Sawin in the comments), the answer to the above question would be yes.

Definition: a finite pattern $p$ is weakly-vanishing if any cell $c$ in the grid becomes perpetually dead after sufficiently many generations (depending on $c$), from the initial state $p$.

Improved question: Is there a weakly-vanishing line of $n$ alive cells with $n \in [25,\infty)$?

Stronger question: Is $wp_n \ge 2$ for any $[84,\infty)$ ?

Tobias Fritz pointed out in the comments that there is a one-cell thick pattern with infinite growth (see this page), but it is disconnected. Bonus question: Can that happen in the connected case?

• Is your question about all $n>24$ or just $n\in[25,100]$? Also, you state "iff", but are the non-vanishing instances actually known? (e.g. eventually periodic) – Joel David Hamkins Dec 13 '17 at 14:28
• @JoelDavidHamkins: for $n \in [0,100]$ I have checked that the vanishing happens iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. The question is about $n>24$ (and so $n>100$ after my checking). – Sebastien Palcoux Dec 13 '17 at 14:45
• @JoelDavidHamkins: $n=10$ produces a pattern of period $15$, but that's not the point of this question. – Sebastien Palcoux Dec 13 '17 at 15:14
• What do you mean by stationary component? Do you know that the pattern can be decomposed for long times into gliders/spaceships plus something that happens in a bounded region? This is not always the case, e.g. there even are patterns that grow quadratically (pentadecathlon.com/lifeNews/2011/05/…). – Tobias Fritz Dec 13 '17 at 18:05
• How do you know the gliders are perpetual? Couldn't something else catch up and stop them? Or can you rule that out? – Will Sawin Dec 13 '17 at 19:08

Just to help develop intuition - here is a fragment of a typical evolution for large $n$ (made with Golly)
• I presume that this is large (size) $n$ and large (time) $t$. But does this screen shot correspond to $n$ and $t$ of comparable size? Or are we looking here at a case of $0<<n<<t$? – André Henriques Dec 15 '17 at 9:27
• @AndréHenriques To my shame I did not record exact numbers, but your last suggestion is correct: $n$ is around $23000$ and $t$ is around 2 million. – მამუკა ჯიბლაძე Dec 15 '17 at 10:00
• The first glider produced (at $t=435$, visible here at extreme left) did not exceed the half high of the pyramid, so here $t$ must be less than $23000$. – Sebastien Palcoux Dec 15 '17 at 13:31