### 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 \(N_{5}\)).