Make the person in the middle of the top row of seats hold up a black card, and make everyone else in that row hold up a white card. Now each successive person in each successive row determines the color of the card they hold up by looking at the person directly above them, and above them immediately to their left and right, and then applying the simple rule on page 27 .

Card shuffling Another rather poor example of intrinsic randomness generation is perfect card shuffling.

Monocotyledons—of which palms and grasses are two examples—typically have only one primary site of growth, and thus do not exhibit repeated branching. (In grasses the growth site is at the bottom of the stem, and in bamboos there are multiple growth sites up the stem.)

The evolution of densities in the ensemble average is analogous to a traditional finite difference method with a real number at each site.

Note that in the later cases shown, the head often visits the same position on the grid many times.

In general, however, the maximum possible period for a system containing a certain number of cells can be achieved only if the evolution of the system from any initial condition eventually visits all the possible states of the system, as discussed on page 258 . … And instead, starting from any particular initial condition, the system will only ever visit a tiny fraction of all possible states. Yet since the total number of states is astronomically large—about 10 60 for size 100—the number of states visited by rule 37R, and therefore the repetition period, can still be extremely long.

Each cell corresponds to an entity that either buys or sells on each step.

So in practice what almost universally ends up being done is to consider not just an individual model, but rather a whole class of models, and then to try to identify which model from this class is the best one—as measured, say, by the criterion that its likelihood of generating the observed data is as large as possible. … Given a set of raw data the procedure for finding which model in this class is best—according, say, to the criterion of maximum likelihood—is extremely straightforward: all one does is to compute what fraction of squares in the data are black, and this value then immediately gives the value of p for the best model. … And in this case the best model is again straightforward to find: it simply takes the probabilities for different blocks to be equal to the frequencies with which these blocks occur in the data.

In a rectangular region, the position is given by Mod[a t, {w, h}] and every point will be visited if the parameters have irrational ratios. … For a system of balls in a region with cyclic boundaries, a complicated proof due to Yakov Sinai from the 1960s purports to show that every ball eventually visits every point in the region, and that certain simple statistical properties of trajectories are consistent with randomness.

Each cell then corresponds to a single trading entity, and the color of the cell at a particular step specifies whether that entity chooses to buy or sell at that step.