With 6 states, a machine is known that takes about 3.002 × 10 1730 steps to halt, and leaves about 1.29 ×10 865 black cells.

As in other class 4 cellular automata, there are structures in Life which take a very long time to settle down.

With the rule illustrated above, however, those clusters that do successfully grow exhibit complicated and irregular shapes, but nevertheless eventually seem to take on a roughly circular shape, as in the pictures below.

In flow past a cylinder it is conventional to take L to be the diameter of the cylinder.

But then, particularly as I began to think about doing explicit computer simulations, I decided to take a different tack and instead to look for the most idealized possible models.

The predecessors of a given state can be found from Cases[Map[Fold[Prepend[#1, If[#2  1 ⊻ , Take[#1, 2]  {0, 0}], 0, 1]] &, #, Reverse[list]] &, {{0, 0}, {0, 1}, {1, 0}, {1, 1}}], {a_, b_, c___, a_, b_}  {b, c, a}]

In the most direct approach, one takes a string and at each step just applies the underlying rules or axioms of the multiway system.

The following will update triples of cells in the specified order by using the function f : OrderedUpdate[f_, a_, order_]:= Fold[ReplacePart[ #1, f[Take[#1, {#2 - 1, #2 + 1}]], #2] &, a, order] A random ordering of n cells corresponds to a random permutation of the form Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Range[n]]

The original suggestion made by Derrick Lehmer in 1948 was to take a number n and at each step to replace it by Mod[a n, m] . … But the presence of the Mod takes the points off this line whenever a n[i] ≥ m . … In fact, the first known generator for digital computers was John von Neumann 's "middle square method" n  FromDigits[Take[IntegerDigits[n 2 , 10, 20], {5, 15}], 10] In practice this generator has too short a repetition period to be useful.

DES takes 64-bit blocks of data and a 56-bit key, and applies 16 rounds of substitutions and permutations.