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

Let us analyze the data of the following table, which displays the cycles of $$f_{D}$$, for $$1 \leq D \leq 6$$.

$$D$$ Periods Cycles in $$N_{D}$$
$$1$$ $$1$$ $$\{0\}$$
$$2$$ $$1$$
$$5$$
$$\{00\}$$
$$\{09, 81, 63, 27, 45\}$$
$$3$$ $$1$$
$$5$$
$$\{000\}$$
$$\{099, 891, 693, 297, 495\}$$
$$4$$ $$1$$
$$2$$

$$5$$

$$\{0000\}$$
$$\{2178, 6534\}$$
$$\{0999, 8991, 6993, 2997, 4995\}$$
$$\{0090, 0810, 0630, 0270, 0450\}$$
$$\{0909, 8181, 6363, 2727, 4545\}$$
$$5$$ $$1$$
$$2$$

$$5$$

$$\{00000\}$$
$$\{21978, 65934\}$$
$$\{09999, 89991, 69993, 29997, 49995\}$$
$$\{09009, 81081, 63063, 27027, 45045\}$$
$$\{00990, 08910, 06930, 02970, 04950\}$$
$$6$$ $$1$$

$$2$$

$$5$$

$$9$$

$$18$$

$$\{000000\}$$
$$\{219978, 659934\}$$
$$\{021780, 065340\}$$
$$\{099999, 899991, 699993, 299997, 499995\}$$
$$\{090009, 810081, 630063, 270027, 450045\}$$
$$\{009990, 089910, 069930, 029970, 049950\}$$
$$\{000900, 008100, 006300, 002700, 004500\}$$
$$\{009090, 081810, 063630, 027270, 045450\}$$
$$\{090909, 818181, 636363, 272727, 454545\}$$
$$\{099099, 891891, 693693, 297297, 495495\}$$
$$\{978021, 857142, 615384, 131868, 736263, 373626,$$
$$252747, 494505, 010989\}$$
$$\{043659, 912681, 726462, 461835, 076329, 847341,$$
$$703593, 308286, 374517, 340956, 318087, 462726,$$
$$164538, 670923, 341847, 406296, 286308, 517374\}$$

We see here some patterns. For each natural $$D$$, the application $$f_{D}$$ has only one fixed point. If $$D > 1$$ is odd, then the cycles of $$N_{D}$$ originate from the cycles of $$N_{D-1}$$ by one of the following ways:

• fix an element of a cycle of $$N_{D-1}$$ and place a $$9$$ in between (consider, for example, the cycles $$\{2178, 6534\}$$ of $$N_{4}$$) and $$\{21978, 65934\}$$ of $$N_{5}$$);
• fix an element of a cycle of $$N_{D-1}$$ and add a $$0$$ at the central position (that is, for example, the relationship between the cycles $\{09009, 81081, 63063, 27027, 45045\}$ of $$N_{5}$$) and $\{0909, 8181, 6363, 2727, 4545\}$ of $$N_{4}$$).

These procedures do not alter the period of the cycle, but not all possibilities produce cycles (for example, $$09909$$ does not belong to any cycle of $$N_{5}$$).

See proof