The Infinite Tower of Hanoi as a Numeration System

The paper derives and proves a non recursive, non iterative formula (simpler than the Gray algorithm and counting the bits of a binary representation) that computes the state of the Tower of Hanoi at an arbitrary moment in time. An infinite generalization of the Hanoi problem is considered. It is shown that the Tower of Hanoi is equivalent to representing a number in a peculiar, “Hanoi” positional numeral system. Attention is drawn to the importance of a number of related problems, whose solutions are only sketched. The results were verified in Mathcad 15. Some illustrations are provided in .zip files.Volodymyr Zhuravlov, https://orcid.org/0009-0002-3637-9000msc: 11A99, 03A99, udc 511 Volodymyr M. Zhuravlov