# Game of Life meets Chaos Theory

I was wondering if anyone had any examples of Chaos Theory in John Conway's Game of Life, i.e. a position which is stable, but change just one cell and the population becomes extinct.

• There is a big discussion on mathoverflow: mathoverflow.net/questions/132687/… – rhsquared Apr 29 '19 at 10:39
• (I'll note this is technically a list question - one which can never be exhaustively answered, and for which any relevant answer is equally valid. See in particular Are list questions off topic? for more reasons why these types of questions are generally off-topic everywhere on Stack Exchange. Having said that, I suspect that a strict reading of your questionâ€”changing any one cell, i.e. toggling any single cell in a candidate NxM grid, causing extinctionâ€”will have few if any solutions beyond degenerate ones or for uninterestingly small N and M) – Rubio May 4 '19 at 6:45

One of the simplest still-lifes is the "beehive":

. # # .
# . . #
. # # .

If you remove the cell at one end, it will eat itself over the next few generations and nothing will remain.

I suspect there is a position with the properties that (1) its population remains nonzero but bounded, (2) if you change one cell you can make it go extinct, and (3) if you change one cell you can make it grow without limit (via glider guns or the like), but that would be much more difficult to construct.

• As a fun little sidenote for this shape, if you fill in either one of the 2 interior cells, the construct will, over the course of a few generations, "explode" itself out until it becomes a cross-shaped layout of 4 "beehives" – Skidsdev Apr 29 '19 at 14:28
• I while ago, I constructed a self-destroying glider gun with an exponential delay fuse. I'm sure there are plenty of one-cell changes that will either stabilize it at a non-zero population (just break the gun somehow) or make it grow without bound (just break the fuse in a way that lets the glider stream escape). Of course, I'm sure there are far smaller and simpler patterns with the same property. – Ilmari Karonen May 5 '19 at 10:00
• @IlmariKaronen Nice! – Gareth McCaughan May 5 '19 at 11:03

There is this configuration which is stable:

.#.
#.#
.#.

If you take out any one of them, it will die.

# # . # #
# # . # #
. . . . .
# # . # #
# # . # #


On removing any one of the 4 innermost cells it will mutate and form another stable shape in 6 generations.

Remove any or all of the 4 corner cells and it will immediately self heal.

But if you remove any one of the other 8 outer cells it will mutate and die out in 33 generations.

I think this is actually a solution for your question as asked:

As in

No cells live at all. It's a stable formation (of zero cells), and changing any cell (toggling any single cell to Live anywhere on the grid) creates a population that will indeed become extinct, in one generation.

Another possibility:

.#.
#.#
.#.
specifically on a 3x3 grid. It's stable, but toggling any cell causes extinction.
(This pattern is the same as rhsquared's, but confining it to a 3x3 grid means it's a solution when removing any live cell or when adding any new live cell.)

Perhaps with some changes

to the question to clarify what is meant by an initial "position" an uninteresting answer like this one will be ruled out. :)