Q11Data Structures

The Tower of Hanoi is a classic, fundamental mathematical puzzle that serves as the quintessential educational example for teaching the paradigm of recursive algorithmic design. The puzzle setup consists of three vertical pegs (or rods) and a specified number of perforated disks of differing, incrementally increasing radii. Initially, all disks are neatly stacked on the first originating peg in a perfectly conical shape, ordered strictly by size with the largest disk resting at the very bottom.

Operational Constraints and Rules

The overarching objective of the algorithm is to successfully transport the entire stacked tower from the initial source peg to a designated destination peg, utilizing the third remaining peg as an auxiliary workspace. This physical transfer must strictly adhere to three non-negotiable operational constraints: First, only one single disk may be manipulated and moved at any given time. Second, a valid move strictly consists of removing the topmost exposed disk from one peg and sliding it onto another peg. Third, and most crucially, a larger, heavier disk can never, under any circumstances, be placed on top of a smaller, lighter disk.

Recursive Algorithmic Logic

The problem can be elegantly deconstructed using a recursive mathematical methodology. To move a tower of height $n$ from the Source peg to the Destination peg, the algorithm implicitly assumes the capability to move a smaller tower of height $n-1$. The logical sequence is threefold: 1. Recursively move the top $n-1$ disks from the Source peg to the Auxiliary peg. 2. Physically move the largest, remaining $n$-th disk directly from the Source peg to the Destination peg. 3. Recursively move the $n-1$ disks from the Auxiliary peg to the final Destination peg, resting them on top of the largest disk. This recursive breakdown continues down to the trivial base case where $n=1$, requiring only a single direct move.

C++ Algorithmic Implementation