Dear Remi,

Thanks for your answer.

But, that is not the problem. What I call "equilibration time" is the number of steps untill the system is not changing anymore. So it is a clear, integer number, there is no truncation/ loss of precision isue.

If this is not the case, then your problem has some kind of attractor to the equations. You might then want to look at the problem analytically.

Yes, that is the point.

Some numbers seem to be "special" and different configurations reach equilibrium in that number of steps (1109 and others, as can be seen in the graph).

So that was the question:  How to look at this problem analytically? Are these special numbers predictable? etc

Right, the question was cryptic...

Is just Conway's Came of Life, with the classic rules.

https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

I use lines as starting configurations and I check the equilibration time seeing when the system stops evolving. Like in this video.

Here Initial L = 138, and equilibration time is 1109 steps

Could you provide a table of length vs equilibration time?

Hi Andrew,

Enclosed are two figures where can be seen that 1109 is a frequent equilibration time for starting lengths in the range 150-250. (First time for L=138).

(A table will be uploaded soon)

A

 I would suggest perhaps that the number 1109 isn't special but rather a straight line of arbitrary length as a Game of Life starting configuration is special. I wasn't aware that the Game of Life ever reached equilibrium, by the way.

Hello Rhoderick,

Thanks for your comment.

The question "What is special in 1109?" was just a way of saying. We decided to study lines as a first step to understand some facts of Life, and lines, despite being "simple starting configurations (the simplest possible?) " also present very interesting behaviour.

Now, the fact is that different lines (i.e, different values of L) reach equilibrium exactly at 1109 steps, and this special number is not the only one. the same thing happens for 1388 for example. So we can conjecture that there are infinite special numbers like these (when L goes to infinite).