The Towers of Hanoi Problem in Graph Form

1. The Towers of Hanoi Problem in Graph Form Finding bounds for the number of moves • Presented by: • Shibo Fang • 9 July 2010 • Mentor: • Dr. Ernst Leiss

2. Background • Towers of Hanoi: a classic problem • Three poles to move disks on • Goal is to move all disks to third pole • Certain rules apply in movement of disks • Can be represented as a direct graph • S = start node, A = auxiliary node (between S and D nodes), D = destination node • Two edges between every two nodes

3. Solvable and Finite Graphs • A graph is solvable when: • There exist vertices S, D, and A • There exist paths from S to A, from A to D, and from D to S • Solvable means any number of disks can be moved from the S to the D node • Any graph that is not solvable is a finite graph

4. Hanoi Graphs • Original Hanoi problem with three nodes requires 2d – 1 moves (d = number of disks) • Modified Hanoi problem (there are no edges between S and D) requires 3d – 1 moves

5. Special Hanoi Graphs • First special Hanoi graph (k graph) • Has k+3 nodes, S0 through Sk, A, D • Requires k*d + 3d – 1 moves

6. Special Hanoi Graphs (Cont.) • Second special Hanoi graph (cycle graph) • An edge between each two nodes • First edge from S node to first A node • Last edge from D node to S node

7. Findings • Needed to find algorithm and formula for number of moves for cycle graph • Found for d ≤ n – 1 (n = number of nodes) • Need to find for d ≥ n • K graph thought to be upper bound • Requires most moves out of all Hanoi graphs • Cycle graph worse than K graph • When d ≤ n – 1, small d and large n

8. Questions Thank you. Any questions?