Selenite Towers Move Faster Than Hanoï Towers, But Still Require Exponential Time

The Hanoï Tower problem is a classic exercise in recursive programming: the solution has a simple recursive definition, and its complexity and the matching lower bound correspond to the solution of a simple recursive function (the solution is so simple that most students memorize it and regurgitate it at exams without truly understanding it). We describe how some minor change in the rules of the Hanoï Tower yields various increases of difficulty in the solution, so that to require a deeper mastery of recursion than the classical Hanoï Tower problem. In particular, we analyze the Selenite Tower problem, where just changing the insertion and extraction positions from the top to the middle of the tower results in a surprising increase in the intricacy of the solution: such a tower of n disks can be optimally moved in a √ 3 moves for n even (i.e. less than a Hanoï Tower of same height), via five recursive functions following three distinct patterns. 1998 ACM Subject Classification F.2.2 [Analysis of Algorithms and Problem Complexity] Nonnumerical Algorithms and Problems, F.2.m [Analysis of Algorithms and Problem Complexity] Miscellaneous