Hanoi Three-Peg Shortest State Path

The Tower of Hanoi is a classic mathematical puzzle that perfectly illustrates recursive problem-solving and exponential growth. The objective is to move an entire stack of disks from one peg to another, subject to the rules that only one disk may be moved at a time, and a larger disk can never be placed on top of a smaller disk.

worked-example

Let us calculate the total number of shortest moves required for n=3 disks, according to the lower-bound recurrence T(n) = 2T(n-1) + 1, where T(0) = 0.

T(0) = 0
T(1) = 2*T(0) + 1 = 2(0) + 1 = 1
T(2) = 2*T(1) + 1 = 2(1) + 1 = 3
T(3) = 2*T(2) + 1 = 2(3) + 1 = 7

The total shortest moves for 3 disks is 7. If we inspect m = 3 moves into this path, the remaining shortest distance along this path is T(3) - 3 = 7 - 3 = 4.

experiment

You can interact with the puzzle parameters to trace the shortest path and count available moves from any valid state. Using the interactive controls in the Hanoi Three-Peg Shortest State Path module:

Set the Disks count to exactly 3.
Adjust the Inspected moves select.
Observe the visual graph and disk positions on pegs.
Inspect the resulting Legal Neighbors to count the possible moves from the current state.

limitations

This module is specifically focused on the classic three-peg Hanoi problem and its recursive shortest-path solution. It does not generalize to variations with four or more pegs (like the Reve's puzzle), nor does it explore non-optimal state sequences beyond what is necessary to highlight the neighbor transitions along the optimal path. The visualization focuses on the state prefix clamped to 2^n-1 rather than exhaustive state graph layouts.

references

The recursive lower bound and complexity details can be explored further in the following references:

Burch, C. "Tower of Hanoi Time complexity". https://www.cs.cmu.edu/~cburch/survey/recurse/hanoitime.html

Hanoi Three-Peg Shortest State Path