A Slightly Different Cellular Automaton


The New Rule:

Let's modify the last rule slightly - simply change the neighborhood-0 output from white to black. The new rule is shown below:

rule 125

What is the Wolfram code for this rule? With such a slight change in the rule, we ought to expect a sligh change in the output, right? Let's see.

The First Generation:



All of the neighborhoods to the left of the black cell are neighborhood 0, so all cells from the arrow leftward are black/on in the first generation. rule 125, gen 1a
neighborhood 1
rule 125 gen 1b
neighborhood 2
rule 125 gen 1c
neighborhood 4
rule 125 gen 1d
All neighborhoods from this one are neighborhood 0, so all remaining cells in this generation are black/on. This completes the first generation. rule 125 gen 1e

The Second Generation:

Action Result
All neighborhoods to the left of the neighborhood shown follow neighborhood 7, so all of those cells produce a white/empty cell. The neighborhood shown at right is neighborhood 6, so its successor is black/on. rule 125 gen2a
neighborhood 5
rule 125 gen 2b
neighborhood 3
rule 125 gen 2c
All of the remaining cells are neighborhood 7. This completes the second generation. rule 125 gen 2d

Moving On:

The third generation is rule 125 gen 3.

After eleven generations you should have rule 125 gen 11.

Switching to Mathematica (computer), the first 50 generations is: rule 125 gen 50

Making a tiny change in the rule certainly did not make a slight change in the output! This is a completely different pattern!


At this point, you should be able to answer the following questions. Check your answers here.

  1. Convert these binary (base-2) numbers to base-10:
    1. 10
    2. 101
    3. 10110010
    4. 11111111
  2. Convert theses base-10 numbers to base-2:
    1. 6
    2. 11
    3. 100
    4. 148
  3. Construct the first 5 generations of the following CAs, starting with a single "live" (black) cell: (Hint: Convert the rule number to binary, then convert the binary digits (bits) into a rule.)
    1. Rule 28
    2. Rule 150


last update November 24, 2009 by JL Stanbrough