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The purpose of this paper is to prove the Frame-Stewart algorithm for the generalized Towers of Hanoi problem as well as finding the number of moves required to solve the problem and studying the multitude of optimal solutions. The main…
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…
In this paper, our aim is to prove that our recursive algorithm to solve the "Reve's puzzle" (four- peg Tower of Hanoi) is the optimal solution according to minimum number of moves. Here we used Frame's five step algorithm to solve the…
The Frame-Stewart conjecture states the least number of moves to solve a generalized Tower of Hanoi problem, of n disks and p pegs. In this paper, we prove a weaker version of the Frame-Stewart conjecture.
The Tower of Hanoi continues to provide a surprisingly rich meeting point for recursive reasoning, combinatorial geometry, and computational verification. Motivated by the editorial standards of the Bulletin of the Australian Mathematical…
We derive a unified closed-form expression for the Frame-Stewart algorithm in the multi-peg Tower of Hanoi: M(p,n) = 2^(i(p,n)+1)*n - sum_{k=0}^{i(p,n)} 2^k * C(p+k-2, k), where i(p,n) = min{ j >= 0 : n <= C(p-1+j, j+1) }. and prove it…
In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between…
The weighted Tower of Hanoi is a new generalization of the classical Tower of Hanoi problem, where a move of a disc between two pegs $i$ and $j$ is weighted by a positive real $w_{ij}\geq 0$. This new problem generalizes the concept of…
For the multi-peg Tower of Hanoi problem with $k \geqslant 4$ pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: $\mathrm{S}\_k(n)=\min\_{1 \leqslant t \leqslant n} \left\{2…
The generalized Tower of Hanoi problem with h \ge 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path_h variant, where…
We develop a unified algebraic theory of the weighted Tower of Hanoi with arbitrary nonnegative symmetric move costs depending on both disc index and pegs. Starting from a general optimality recurrence with two competing strategies -- one…
Considering the symmetries and self similarity properties of the corresponding labeled graphs, it is shown that the minimal number of moves in the Tower of Hanoi game with $p =4$ pegs and $n \geq p$ disks satisfies the recursive formula $…
We explore a method for computing admissible heuristic evaluation functions for search problems. It utilizes pattern databases, which are precomputed tables of the exact cost of solving various subproblems of an existing problem. Unlike…
The problem of the Hanoi Tower is a classic exercise in recursive programming: the solution has a simple recursive definition, and its complexity and the matching lower bound are the solution of a simple recursive function (the solution is…
Our theme bases on the classical Hanoi Towers Problem. In this paper we will define a new problem, permitting some positions, that were not legal in the classical problem. Our goal is to find an optimal (shortest possible) sequence of…
The Magnetic Tower of Hanoi puzzle - a modified "base 3" version of the classical Tower of Hanoi puzzle as described in earlier papers, is actually a small set of independent sister-puzzles, depending on the "pre-coloring" combination of…
Planning for sequential robotics tasks often requires integrated symbolic and geometric reasoning. TAMP algorithms typically solve these problems by performing a tree search over high-level task sequences while checking for kinematic and…
This work studies rearrangement problems involving the sorting of robots or objects in stack-like containers, which can be accessed only from one side. Two scenarios are considered: one where every robot or object needs to reach a…
We prove the exact formulae for the expected number of moves to solve several variants of the Tower of Hanoi puzzle with 3 pegs and n disks, when each move is chosen uniformly randomly from the set of all valid moves. We further present an…
We present efficient algorithms for constructing a shortest path between two states in the Tower of Hanoi graph, and for computing the length of the shortest path. The key element is a finite-state machine which decides, after examining on…