Details of trading Cynics might suggest that much of the randomness in practical markets is associated with details of trading.

And the picture shows that they too exhibit the kind of complexity and apparent randomness that is evident at the level of digits.

Once again, these sequences seem for practical purposes random.

In the case of both simple and completely random images, many features are recognizable even with fairly few basic forms—implying that a highly compressed representation can be given.

If one samples these rules at random, one finds that more than 99% of them just yield simple repetitive behavior.

Note that in a related situation almost any fairly random overlaid pattern containing many local features can successfully be used to mask the contents of a paper envelope.

Detection methods [for SETI] Ways to identify computational origins include looking for repeatability in apparently random signals and comparing with output from large collections of possible simple programs.

(The average number of faces for each object depends on the details of the random process used to pack them, but is typically around 14. Note that for a 3D Voronoi diagram with randomly placed points, the average number of faces for each region is 2 + 48 π 2 /35 ≃ 15.5 .)

To get some idea about the origin of this behavior, one can assume that successive values of n are randomly even and odd with equal probability. … Indeed, averaging over many initial values of n , there is good quantitative agreement between the predictions of the randomness approximation and the actual 3n+1 problem. But since there is no fundamental basis for the randomness approximation, it is still conceivable that a particular value of n exists that does not follow its predictions.

Frequency of behavior [in multiway systems] Among multiway systems with randomly chosen rules, one finds about equal numbers that grow rapidly and die out completely.