An interesting feature of this task is that there are at most 256 different rule sets. You can see that you have to indicate what happens to the colour of a cell based on the three cells above it (directly above and the two diagonal ones). There are eight different ways the three cells are coloured black or white (www, wwb, wbw, wbb, bww, bwb, bbw, bbb). For each of these eight situations you have to decide on the colour of the cell below. This means you only have to make 8 choices of black and white. For 8 binary choices you have 2**8 possibilities, so 256. In the solution diagram we have added '1' and '0' below the choices. The table on the right shows all valid solutions. Some of these are easy to find (put a black square somewhere only if the square above it is white and either one or two of the diagonals are black), and some are very interesting to find.Having so many valid solutions means you can just play with this task to see the patterns emerge without having to spend too much time searching for the solution.