complicated—so that the approach probably ends up essentially being no better than just enumerating possible initial conditions.

The conclusion therefore is that at least with standard methods of cryptanalysis—as well as a few others—there appears to be no easy way to deduce the key for rule 30 from any suitably chosen encrypting sequence. But how can one be sure that there really is absolutely no easy way to do this? In Chapter 12 I will discuss some fundamental approaches to such a question. But as a practical matter one can say that not only have direct attempts to find easy ways to deduce the key in rule 30 failed, but also—despite some considerable effort—little progress has been made in solving any of various problems that turn out to be equivalent to this one.

Traditional Mathematics and Mathematical Formulas

Traditional mathematics has for a long time been the primary method of analysis used throughout the theoretical sciences. Its goal can usually be thought of as trying to find a mathematical formula that summarizes the behavior of a system. So in a simple case if one has an array of black and white squares, what one would typically look for is a formula that takes the numbers which specify the position of a particular square and from these tells one whether the square is black or white.

An example of how the color of any square in a repetitive pattern can be found from its coordinates by a simple mathematical procedure. The procedure takes the x and y coordinates of the square, and computes their remainders after division by 3 and 2 respectively. Using these remainders—which are shown inside each square—the color of a particular square can be determined by a simple lookup in the repeating block shown on the bottom left. The whole procedure can be represented using a mathematical formula that involves either functions like Mod or more traditional functions like Sin.