Bouncing Towers move faster than Hanoi Towers, but still require exponential time

TL;DR

This paper explores how minor rule changes in the Hanoi Tower problem, specifically the Bouncing Tower variant, significantly increase complexity, requiring more intricate recursive solutions and resulting in faster solutions than classical Hanoi Towers.

Contribution

It introduces the Bouncing Tower problem, demonstrating how small rule modifications lead to higher complexity and new recursive solution structures.

Findings

Bouncing Tower can be solved in ab3^n moves for even n.

The problem requires 5 recursive functions or states.

Bouncing Tower solutions are faster than classical Hanoi Towers.

Abstract

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 so easy that most students memorize it and regurgitate it at exams without truly understanding it). We describe how some very minor changes in the rules of the Hanoi Tower yield various increases of complexity in the solution, so that they require a deeper analysis than the classical Hanoi Tower problem while still yielding exponential solutions. In particular, we analyze the problem fo the Bouncing Tower, where just changing the insertion and extraction position from the top to the middle of the tower results in a surprising increase of complexity in the solution: such a tower of $n$ disks can be optimally moved in $\sqrt{3}^n$ moves for $n$ even (i.e. less than a Hanoi Tower of same height), via $5$ recursive functions (or, equivalently, one recursion function with $5$ states).