Is there a sequence $\:\langle S_0,S_1,S_2,...,S_{n-1},S_n\rangle\:$ of generations for Conway's Game of Life such that
[$\hspace{.01 in}S_0 = S_n$ $\:$ and $\:$ $S_0$ has at least one live cell $\:$ and $\:$ [for all members $i$ of $\:\{0,1,2,...,n-2,n-1\}\:$,
for all grid states $S$, if $S$ differs from $S_i$ on only one cell, then $S$ eventually evolves into $S_0$]]$\hspace{.04 in}$?
(I thought of the above question while I was reading
an article about self-organization and stability.)
I would guess the answer is no, unless there happens to be a very simple (and therefore boring) solution. Something simple and unchanging like a $2\times2$ square, or with a very short period like a $3\times1$ "blinker"; however it turns out these are not examples. The reason for my guess is that any interesting repetitive pattern (with period${}>1$) will involve some region where interesting changes are going on, which means creation of "fertile" cells in some generations (a cell that vanishes improductively after creation does not contribute to reproduction of the population). Each cell creation will be undone if any one of the $8$ neighbour cells is toggled, which gives a large number of possible perturbances (and the number increases as the configuration gets larger), each of which must somehow magically be brought back into the right track. If small configurations do not provide examples, larger ones will have have an even harder time to do so.
There is of course nothing rigorous about this, but I feel that as one proceeds in exploring without finding examples, the probability that an example remains to be found will rapidly tend to $0$. It is like conjecturing that in the sequence of decimals of $\pi$, there is no initial subsequence that is immediately repeated a second time. An easy computation shows that the probability of such a repetition becomes vanishingly small after the first candidates have been checked (and ever smaller as more decimals are computed), even though I think it is also extremely unlikely that this conjecture can ever be proved (one could imagine some result about rational approximation of $\pi$ being applied, but for one thing I don't think any remarkable such properties are known, and on the other hands such results are usually asymptotic only, therefore independent of any particular isolated approximation).
