This question spins off of Gil Kalai's recent question on Conway's game of life for a random initial configuration.
There are numerous configurations in the game of life that are known to be stable—such as blocks, beehives, blinkers and toads—in the sense that if they appear on an otherwise empty board or on part of the board that remains otherwise empty, then they will persevere (or at least reappear on some period) into the indefinite future. All of the common examples of such configurations, however, seem to disintegrate when placed into a hostile environment; when they are hit by a glider or other spaceship, for example, these common stable configurations can be completely ruined.
My question is whether there is any superstable configuration, which can survive even in any hostile environment.
Question 1. Is there any superstable configuration in the game of life?
Specifically, let us define that a finite configuration is superstable, if it can survive in any environment, no matter how hostile, meaning that if it should ever appear on the board, then it will definitely reappear later in exactly that same position, regardless of what else is happening on the board. Perhaps the position is somehow isolated, absorbing whatever is happening around it; or perhaps it is a strong source of some kind, spewing out gliders or other objects, regardless of what else is around it; or perhaps it is some core surrounded by encircling vacuum-cleaners, traveling patterns that sweep up whatever might interfere.
This question is related to Gil Kalai's, in that if there are such superstable configurations, then we will expect that the infinite random position will have them with some (albeit very small) density, which will enable us to prove lower bounds on the density of the expected living infinite random position.
One can also imagine a glider version of superstability, where the pattern survives, but with some nonzero displacement:
Question 2. Is there any superstable glider?
That is, is there a finite pattern that, regardless of the environment in which it is placed, will repeat itself at some future time with some displacement? A strong form of such a superstable glider would ask also that it be a vacuum cleaner, meaning that it glides around in any given environment while leaving only empty cells in its wake.
Question 2b. Is there a superstable glider vacuum?
I can imagine a small glider that erases everything in its path; or perhaps there is a kind of moving wall, which steadily pushes against whatever it faces, leaving emptiness behind. If there were such a superstable glider vacuum that also moved in a definite direction, then of course there could be no superstable stationary position, since otherwise we could vacuum it up.
Another alternative would seem to be that every finite configuration in the game of life is destructible, in the sense that one can design for it an especially hostile environment, leading to eventual death.
Question 3. Is every finite configuration destructible?
In other words, can every finite configuration in the game of life be extended to a larger configuration whose development leads in finite time to a position with no living cells? A weaker version of this would ask merely that the configuration be extended to a configuration such that eventually, the original configuration does not recur on any subportion of the board.
Best Answer
The existence of a "superstable configuration" is a long-standing open question in the Game of Life community. Years ago I saw Conway ask it as follows: Is there a finite $N$ and a configuration $C$ in an $N \times N$ square that contains some live cell $c$ and guarantees that $c$ will remain live for all time, regardless of what is placed outside the $N \times N$ square in the initial configuration? I think the expectation is that no such $C$ exists but a proof would be very difficult.