Pascal's Triangle and Its Relatives

Exercises

Pascal's triangles and CA rules

1. Here is Pascal's triangle for (1 + x + x²)ⁿ, shading the boxes containing odd terms. A similar, but symmetric pattern is produced by the CA on the right.

We see this CA gives a live cell for all nbhd configs with 1 or 3 live cells.

Knowing that live cells on the left side of the nbhd give patterns growig to the right, we use all configs of 1 or 3 live cells in the center and two left cells. For each of these, all four combinations of the right two nhbd cells are used.

For example, all these configurations give a live cell if the left-most cell is alive, regardlass of the states of the two right cells.

Here are the corresponding rule, and pattern.

Return to CA rule Exercises.