The Parity-Constrained Four-Peg Tower of Hanoi Problem and Its Associated Graph

TL;DR

This paper introduces a new parity-constrained four-peg Tower of Hanoi variant, deriving exact move counts, recursive relations, and analyzing the associated graph's properties, revealing slower exponential growth compared to the classical case.

Contribution

It formulates the parity-constrained four-peg Tower of Hanoi problem, derives explicit formulas for optimal moves, and characterizes the properties of the related Hanoi graph.

Findings

Optimal move sequences grow exponentially but slower than classical case.

Derived explicit formulas and recursive relations for move counts.

Analyzed the properties of the associated Hanoi graph, including diameter and planarity.

Abstract

We introduce and study a new four-peg variant of the Tower of Hanoi problem under parity constraints. Two pegs are neutral and allow arbitrary disc placements, while the remaining two pegs are restricted to discs of a prescribed parity: one for even-labelled discs and the other for odd-labelled discs. Within this constrained setting, we investigate four canonical optimization objectives according to distinct target configurations and derive for each the exact number of moves required to optimally transfer the discs. We establish a coupled system of recursive relations governing the four optimal move functions and unfold them into higher-order recurrences and explicit closed forms. These formulas exhibit periodic growth patterns and reveal that all solutions grow exponentially, but at a significantly slower rate than the classical three-peg case. In particular, each optimal move sequence has an a half-exponential-like asymptotic order induced by the parity restriction. In addition, we define the associated parity-constrained Hanoi graph, whose vertices represent all feasible states and whose edges represent legal moves. We determine its order, degrees, connectivity, planarity, diameter, Hamiltonicity, clique number, and chromatic properties, and show that it lies strictly between the classical three- and four-peg Hanoi graphs via the inclusion relation.