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Extended versions exist of all but structures (a) and (j). Structures (m) and (n) also exist in alternate forms shifted with respect to the background.

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Smallest solutions for various sequences of integer (or so-called Diophantine) equations. indicates that it can be proved that no solution exists. A blank indicates that I know only that no solution exists below a billion.

The asymmetry between the left and right-hand sides is a direct consequence of asymmetry that exists in the particular underlying cellular automaton rule used.

From a practical point of view the fact that a short description may exist is presumably not too relevant if we can never find this description by any of the methods of perception and analysis that are available to us. … So one possibility is to define randomness so that something is considered random only if no short description whatsoever exists of it. … So to say that something is random only if no short description whatsoever exists of it turns out to be a highly restrictive definition of randomness.

And as a result, no finite process can in general be used to guarantee that there is no short description that exists of a particular thing.
… Indeed, what I suspect is that ultimately no useful definition of randomness can be based solely on the issue of what short descriptions of something may in principle exist. … But despite some confusion in the past it is certainly not true that just checking equality of frequencies of blocks of colors—even arbitrarily long ones—is sufficient to ensure that no regularities at all exist.

A cyclic tag system in general operates by removing the first element from the sequence that exists at each step, and then adding a new block of elements to the end of the sequence if this element is black. … Then when the stripe hits the first element in the sequence that exists at that step, it is allowed to pass only if the element is black. … When a dashed line hits the first element in the sequence that exists at a particular step, it effectively bounces back in the form of a line propagating to the left that carries the color of the first element.

For several decades it has been known that universal integer equations exist. … Yet from the discoveries in this book I am quite certain that there are vastly simpler examples that exist—so that in fact there are in the end rather simple integer equations for which the absence of solutions can never be proved from the normal axioms of arithmetic.
… And with equations that have higher powers or more variables questions of whether solutions exist quickly end up being unsolved problems of number theory.

Nevertheless, as the last stage demonstrates, a repetitive pattern does in fact exist. In case (c), the automatic procedure finds a fairly large and almost regular pattern that satisfies the constraints, but in this case it turns out that no infinite pattern exists.

Yet in the previous section we discovered the rather remarkable fact that there exist rules with the property that essentially regardless of how they are applied, the same causal network—and thus the same perceived history for the universe—will always emerge.
… And in general every path through a picture like this corresponds to a possible history that exists in the evolution of the multiway system.

Sarkovskii's theorem
For any iterated map based on a continuous function such as a polynomial it was shown in 1962 that if an initial condition exists that gives period 3, then other initial conditions must exist that give any other period.