### The case with four or more digits

Which properties does this dynamical system have when we consider numbers with four or more digits? Let \(N_{D}\) be the set of natural numbers with \(D\) digits from \(\{0,1,...,9\}\), where we allow zeros at the left; and let \(i_{D}:N_{D}\rightarrow N_{D}\) be the function defined as follows: \(0\) maps to \(0\) and each nonzero \(x\) in \(N_{D}\), written in base \(10\) and represented by \(D\) digits \[x=x_{D-1}...x_{m}x_{m-1}...x_{1}x_{0},\] with \(m=\mbox{maximum}\{i: 0\leq i\leq D-1 \land x_{i}\neq 0\}\), is mapped to the natural \[i_{D}(x)=x_{0}x_{1} ...x_{m-1}x_{m}...x_{D-1}\] obtained by reversing the order of the digits of \(x\).

If \(f_{D}\) designates the function \(N_{D}\rightarrow N_{D}\) defined by \(f_{D}(x)=\left|x-i_{D}(x)\right|\), then all numbers in the image of \(f_{D}\) are multiples of \(9\) (and, when \(D\) is odd, they are simultaneously multiples of \(9\) and \(11\)).

In fact, that image is reduced to \(\frac{19^{\frac{D}{2}} +1}{2}\) elements if \(D\) is even, and to \(\frac{19^{\frac{D-1}{2}} +1}{2}\) elements if \(D\)is odd (see proof). Moreover, since \(N_{D}\) is finite, each orbit of \(f_{D}\) must end in a cycle whose elements are in the image of \(f_{D}\).

The table below contains some information about the dynamics of \(f_{D}\) for \(1\leq D\leq 11\): the numbers of cycles, the corresponding periods and, for \(1\leq D\leq 7\), the maximum preperiods.

\(D\) | Number of cycles | Periods | Maximum preperiod | [Period; Number of cycles for each period] |
---|---|---|---|---|

\(1\) | \(1\) | \(1\) | \(1\) | \([1; 1]\) |

\(2\) | \(2\) | \(1,\) \(5\) | \(2\) | \([1; 1],\) \([5; 1]\) |

\(3\) | \(2\) | \(1,\) \(5\) | \(2\) | \([1; 1],\) \([5; 1]\) |

\(4\) | \(5\) | \(1,\) \(2,\) \(5\) | \(12\) | \([1; 1],\) \([2;1],\) \([5; 3]\) |

\(5\) | \(5\) | \(1,\) \(2,\) \(5\) | \(12\) | \([1; 1],\) \([2;1],\) \([5; 3]\) |

\(6\) | \(12\) | \(1,\) \(2,\) \(5,\) \(9,\) \(18\) | \(47\) | \([ 1; 1 ],\) \([ 2; 2 ],\) \([ 5; 7 ],\) \([ 9; 1 ],\) \([ 18; 1 ]\) |

\(7\) | \(12\) | \(1,\) \(2,\) \(5,\) \(9,\) \(18\) | \(47\) | \([ 1; 1 ],\) \([ 2; 2 ],\) \([ 5; 7 ],\) \([ 9; 1 ],\) \([ 18; 1 ]\) |

\(8\) | \(26\) | \(1,\) \(2,\) \(5,\) \(9,\) \( 10,\) \(14,\) \(18\) | \([ 1; 1 ],\) \([ 2; 4 ],\) \([ 5; 15 ],\) \([ 9; 2 ],\) \([ 10; 1 ],\) \([ 14;1 ],\) \([ 18; 2 ]\) | |

\(9\) | \(26\) | \(1,\) \(2,\) \(5,\) \(9,\) \( 10,\) \(14,\) \(18\) | \([ 1; 1 ],\) \([ 2; 4 ],\) \([ 5; 15 ],\) \([ 9; 2 ],\) \([ 10; 1 ],\) \([ 14;1 ],\) \([ 18; 2 ]\) | |

\(10\) | \(49\) | \(1,\) \(2,\) \(5,\) \(9,\) \( 10,\) \(14,\) \(18\) | \([ 1; 1 ],\) \([ 2; 7 ],\) \([ 5; 31 ],\) \([ 9; 3 ],\) \([ 10; 2 ],\) \([ 14; 2 ],\) \([ 18; 3 ]\) | |

\(11\) | \(49\) | \(1,\) \(2,\) \(5,\) \(9,\) \( 10,\) \(14,\) \(18\) | \([ 1; 1 ],\) \([ 2; 7 ],\) \([ 5; 31 ],\) \([ 9; 3 ],\) \([ 10; 2 ],\) \([ 14; 2 ],\) \([ 18; 3 ]\) |