Strings of length n take n steps to produce. • (g) The same strings as in (f) are produced, but now a string of length n with m black elements takes n + m - 1 steps. • (h) All strings appear in which the first run of black elements is of length 1; a string of length n with m black elements appears after n + m - 1 steps. • (i) All strings containing an odd number of black elements are produced; a string of length n with m black cells occurs at step n + m - 1 . • (j) All strings that end with a black element are produced. • (k) Above length 1, the strings produced are exactly those starting with a white element. … takes 14 steps. • (q) All strings are produced, with a string of length n with m white elements taking n + 2m steps. • (r) All strings are ultimately produced—which is inevitable after the lemmas  and  appear at steps 12 and 13.

Affine transformations Any set of so-called affine transformations that take the vector for each point, multiply it by a fixed matrix and then add a fixed vector, will yield nested patterns similar to those shown in the main text.

An example of a system defined by the following rule: at each step, take the number obtained at that step and write its base 2 digits in reverse order, then add the resulting number to the original one.

But if one takes into account the actual frequencies of different blocks, as well as their ranking, then it turns out that there are better ways to assign codewords.

The plots show how many steps this takes for successive inputs with lengths up to 9.

And while some such questions may be answered by fairly straightforward computational or mathematical means, there will be no upper bound on the amount of effort that it could take to answer any particular question.

But with n variables a DNF-type canonical form can be of size 2 n —and can take up to at least 2 n proof steps to reach. … The longest of these are respectively {57, 94, 42, 57, 55, 53, 179, 157} and occur for theorems {(((a ⊼ a) ⊼ b) ⊼ b)  (((a ⊼ b) ⊼ a) ⊼ a), (a ⊼ (a ⊼ (a ⊼ a)))  (a ⊼ ((a ⊼ b) ⊼ b)), (((a ⊼ a) ⊼ a) ⊼ a)  (((a ⊼ a) ⊼ b) ⊼ a), (((a ⊼ a) ⊼ b) ⊼ b)  (((a ⊼ b) ⊼ a) ⊼ a), (a ⊼ ((b ⊼ b) ⊼ a))  (b ⊼ ((a ⊼ a) ⊼ b)), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b)} Note that for systems that do not already have it as an axiom, most theorems use the lemma (a ⊼ b)  (b ⊼ a) which takes respectively {6, 1, 8, 49, 8, 1, 119, 118} steps to prove.

This has k piles of objects, and on alternate steps each of two players takes as many objects as they want from any one of the piles. The winner is the player who manages to take the very last object.

Realistically this seems to take several months even for the most talented and open-minded people. … For someone to assimilate all of the new kind of science I describe in this book will take a very significant time. … How long it will take a given individual to get to the point of being able to do something specific with the new kind of science in this book will depend greatly on their background and particular goals.

Note that in applying the rule to a particular square, one must take account of the orientation of that square.