Towers of Hanoi

The Towers of Hanoi problem with a relaxed placement rule

Problem definitionThe Tower of Hanoi is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:Only one disk may be moved at a time.Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.No disk may be placed on top of a smaller disk.

Outline Some Applications Classical version Graphical representation Ants solve the problem Relaxed version Further research & References

Some applicationsSome applications (Cont.)It is a 'smart' way of archiving an effective number of backups as well as the ability to go back over timeThe Tower of Hanoi is also used in psychological research on problem solving.The Tower of Hanoi is popular for teaching recursive algorithms to beginning programming students.Outline Some Applications Classical version Graphical representation Ants solve the problem Relaxed version Further research & ReferencesRecursive solutionCorrectness & optimalityComplexityOutline Some Applications Classical version Graphical representation Ants solve the problem Relaxed version Further research & ReferencesGraphical representation

Graphical representation (Cont.)

Graphical representation (cont.)Sierpinski triangle

Outline Some Applications Classical version Graphical representation Ants solve the problem Relaxed version Further research & References

Ants solve the TOH (2010)

The resultsOutline Some Applications Classical version Graphical representation Ants solve the problem Relaxed version Further research & ReferencesRelaxed definitionRunning example of Pooles algorithmDisk nSmall(n-1)Some definition

Pooles algorithm (1992)ComplexityA different strategyA different strategy (Cont.)Outline Some Applications Classical version Graphical representation Ants solve the problem Relaxed version Further research & References

Further researchFurther research (Cont.)While the shortest perfect-to-perfect sequence of moves had been found, we do not know what is such a sequence for transforming one given (legal) configuration to another given (legal) one, and what is its length?In particular, what is the length of the longest one among all shortest sequences of moves, over all pairs of initial and final configuration?

ReferencesOptimality of an Algorithm Solving the Tower of Hanoi Problem Yefim Dinitz, Shay SolomonOptimization in a natural system: Argentine ants solve the Towers of Hanoi Chris R.Raid, David J. T. Sumpter and Madeleine BeekmanThe Tower of Hanoi: A Bibliography - Paul K. Stockmeyer