alan2here
Registered: Oct 2007
Posts: 12 
How to solve a CA with algebra
I'm not sure if this is my method or an obvious algebraic method or even if it definitely works but here goes.
It's based on something similar I did for Minesweeper that did work, inspired by something I read online about suprisingly long distance behaviour in certain Minesweeper configuations.
Start with a simple example from WireWorld.
http://mathworld.wolfram.com/WireWorld.html
An amazing CA, LifeLike and Generations CA's should work in a simerlar way though.
Lets say for simplicity:
000
133
000
According to the rules of WireWorld this will progress as so.
000
213
000
000
321
000
000
332
000
000
333
000
As the signal travels down the wire
First create (a map?) for each state that changes.
All
000
133
000
State 1
000
100
000
State 2
000
000
000
And assign letters to each cell in each map.
All
abc
def
ghi
State 1
rst
uvw
xyz
State 2
ABC
DEF
GHI
Construct the formuli
(I'm using square brackets here to represent subscript as I don't know how to do that formatting in the forum)
a[0] = 0
...
I[0] = 0
Skip a to i for this next part.
r[n] = (0 < s[n1] + u[n1] + v[n1] < 3)
...
I...
Now you can rearange the equation to ask questions like for example, does this cell EVER become state 1 or what are all the possiblilites for the previous generation that could lead to this one.
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