## The dynamics of a trick ### Some patterns II

For $$D$$ even, it seems there are more ways of building the cycles of $$N_{D}$$. For example, we can fix an element of a cycle of $$N_{D-2}$$ and place a zero in each extremity (such as in $$\{0090, 0810, 0630, 0270, 0450\}$$ when $$D=4$$); or choose an element of a cycle of $$N_{\frac{D}{2}}$$ and repeat it twice (as in $$\{0909, 8181, 6363, 2727, 4545\}$$, when $$D=4$$); or select cycles for smaller values of $$D$$ and join them after a suitable permutation (such as in the cycle $\{978021, 857142, 615384, 131868, 736263, 373626, 252747, 494505, 010989\}$ of $$D = 6 = 2+4$$, whose first element comes from joining in this way the cycle $$\{09, 81, 63, 27, 45\}$$ of $$N_{2}$$ and the cycle $$\{2178, 6534\}$$ of $$N_{4}$$).

The list of procedures detected for $$1 \leq D \leq 12$$ is a large one, with some of them generating cycles with periods which are new in relation to those already obtained for smaller values of $$D$$. The algorithm elaborated by Atractor to compute the cycles of $$f_{D}$$ took hundredths of a second to give the complete list of such special orbits when $$2 \leq D \leq 4$$; it took some few seconds for $$D = 6$$ and about two minutes for $$D = 8$$. However, for $$D = 12$$, the expected time rose to about one year (although, using random samples of elements in the image of $$f_{12}$$, the search for cycles became faster). This leaves the challenge of proving, for $$D =12$$ and without the help of a computer, some of the dynamical properties previously detected.