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Problem definition The 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.

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

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Some applications

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Some applications (Cont.) It is a 'smart' way of archiving an effective number of backups as well as the ability to go back over time The 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.

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

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Recursive solution

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Correctness & optimality

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Complexity

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

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Graphical representation

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Graphical representation (Cont.)

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Graphical representation (cont.)

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Sierpinski triangle

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

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Ants solve the TOH (2010)

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The results

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

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Relaxed definition

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Running example of Poole’s algorithm Disk n Small(n-1)

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Some definition

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Poole’s algorithm (1992)

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Complexity

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A different strategy

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A different strategy (Cont.)

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

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Further research

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Further 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?

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References Optimality of an Algorithm Solving the Tower of Hanoi Problem – Yefim Dinitz, Shay Solomon Optimization in a natural system: Argentine ants solve the Towers of Hanoi – Chris R.Raid, David J. T. Sumpter and Madeleine Beekman The Tower of Hanoi: A Bibliography - Paul K. Stockmeyer

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1 Section 6.1 Recurrence Relations. 2 Recursive definition of a sequence Specify one or more initial terms Specify rule for obtaining subsequent terms.

1 Section 6.1 Recurrence Relations. 2 Recursive definition of a sequence Specify one or more initial terms Specify rule for obtaining subsequent terms.

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