Extending the 1089 Attractor to Any Number of Digits and Any Number of Steps

Le résumé fourni par la source

The well-known “1089 trick” reveals an amazing trait of the digit reversal process and is reminiscent of a limiting attractor in dynamical systems, even though it takes only two steps. R. Webster’s 1995 work generalized the above process to any number of digits. To generalize the 1089 trick to any number of steps, we propose a two-option iteration procedure named “iterative digital reversal” (IDR), which suitably interweaves additions and subtractions. The simplest limiting behavior of IDR is 2-member cycles. The smaller number in a 2-cycle consists of variations of the motif 10(9) L89 (L≥0), or symmetric concatenations of it. IDR also exhibits longer p-cycles (p = 10, 12, 71), as well as non-periodic trajectories of integers, which monotonously grow to infinity. In these diverging trajectories, the middle digits present an 8-step rhythm. We have also shown that the above findings are qualitatively shared in other bases, while the binary system represents a particular case. Overall, the generalization of the original 1089 trick in both “space” and “time” leads to many new patterns in integers and a new phenomenology in dynamics.

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