I think the original counting argument was Moore's, and it easily applied directly to Life. However, there's an easy example of a GOE from the counting argument: The argument shows that there's some GOE pattern, of size < a computable number N. To make the argument constructive, just make up a supersize pattern, with all possible patterns of size N. Wrt showing Louiville numbers are transcendental: Use the example L = sum 10 ^ (- N!). In this special case, it's not too hard to check that L defeats any prospective polynomial. Rich --- Quoting Allan Wechsler <acwacw@gmail.com>:
This is the one I've been trying to remember. Thank you, Alon.
This also reminds me of the proof that the Game of Life has a finite Garden of Eden pattern. (Since the original proof, many explicit predecessorless patterns have been found.) It depends on counting possible predecessors and shows that for a certain class of patterns, the number of candidate predecessors is smaller than the number of patterns, so one of that class must lack a predecessor. The class is astronomically large, though, and the counting argument gives no hints about how to find an example. <...>