I believe that such a 3x3 square does not necessarily exist.

A counterexample would take the form of an infinite [still life][1] pattern in the [life-like cellular automaton][2] rule B123678/S34 (these rules are chosen so that the only patterns that remain stable are the ones in which the number of live cells in each 3x3 box is 4 or 5). Additionally, both the live and dead cells of the pattern should be connected.

But as the following partial double spiral shows (copy and paste it into [Golly][3] to view and test) it's possible to form partial double-spiral patterns that, at least in the center of the pattern, have the desired properties. I don't see any good reason why it shouldn't be possible to continue the spiral infinitely.

<pre>x = 16, y = 19, rule = B123678/S34
8b4o$6b3o2b3o$4b3o6b3o$3b2o3b4o3bo$3bo2b3o2b3o$2b2ob2o6b2o$2b2obo2b4o
2bo$2bo2bob2o2b2obo$2bob2o2bo2bo2b2o$b2ob2ob2ob2ob2o$bo2bo2bo2bo2bo$2o
b2ob2ob2o2bo$o2bo2b2o2bob2o$2ob2o4b2obo$bo2b6o2bo$b2o3b2o3b2o$2b3o4b3o
$4b6o$6b2o!</pre>

Here's a screenshot:

![alt text][4]


  [1]: http://en.wikipedia.org/wiki/Still_life
  [2]: http://en.wikipedia.org/wiki/Life-like_cellular_automaton
  [3]: http://golly.sourceforge.net/
  [4]: http://www.ics.uci.edu/~eppstein/0xDE/b123678s34-spiral.png