User blog:Wythagoras/Conway's game of life

Here are some functions I made up using Conway's Game of Life: I have the idea to make some sort of an competition.
 * Die(n) = the maximal number of steps that a configuration of at most n cells can make before dying out.
 * SS(n) = the maximal number of steps that a configuration of at most n cells can make when stabilizing. (when there is no configuration that stabilizes, this returns 0)
 * SC(n) = the maximal number of cells that a configuration of at most n cells can have when stabilizing. (when there is no configuration that stabilizes, this returns 0)
 * NGC(n) = the maximal number of cells that a configuration of at most n cells can have before stopping growing in number of cells (so only oscillators are left). Again, returns 0 if it doesn't exist.
 * NGS(n) = the maximal number of steps that a configuration of at most n cells can make before stopping growing in number of cells (so only oscillators are left). Again, returns 0 if it doesn't exist.
 * Osc(n) = the maximal period of an oscilliator with at most n cells. Again, returns 0 if no oscilliator exist.

Note that there can be no gliders for neither of these functions.

Here values are of all functions form n=1,2,3 to illustrate the idea.

Die(1), SS(1), SC(1), NGC(1), NGS(1) and Osc(1)
1 cell dies out after one step, so Die(1) = 1. No pattern with 1 cell stabilizes, so SS(1) = SC(1) = 0. NGC(1) is also 0, since a died out doesn't grow in number of cells. But a died out pattern stopped growing, so NGS(1) = 1. (just as Die(1)). Also, no oscilliator with 1 cell exists, so Osc(1) = 0.

Die(2), SS(2), SC(2), NGC(2), NGS(2) and Osc(2)
2 cells dies out after one step, so Die(2) = 1. No pattern with 2 cells stabilizes, so SS(2) = SC(2) = 0. NGC(2) is also 0, since a died out doesn't grow in number of cells, but has 0 cells. But a died out pattern stopped growing, so NGS(2) = 1. Also, no oscilliator with 1 cell exists, so Osc(2) = 0.

Die(3) and NGS(3)
So this proves NGS(3) ≥ Die(3) ≥ 2. By analysing all configurations it can be proven that NGS(3) = Die(3) = 2.

SS(3) and SC(3) and NGC(3)
SS(3) = 1 and NGC(3) = SC(3) = 4.

4 cells
Make some candidates!