1. 1 On a question of Leiss regarding the Towers of Hanoi problem

2. 2 Introduction The classic problem (3 pegs) The generalization: A Hanoi graph is a finite directed graph with two distinct vertices denoted by S,D. such that for each vertex there is a path from S to and a path from to D. The source initially contains m discs, no two of which are of equal size, such that smaller discs rest on top of larger ones.

3. 3 The task is to move the m discs from S to D. To this end we may use the other vertices of G. the transfer is subject to the rules of the classic problem, in addition we add the following rule: A disk may be moved from a peg v to another peg w only if there is an edge from v to w.

4. 4 Definitions and problem description The Hanoi Towers problem HAN(G,m) for m >= 0 is to transfer the m disks from S to D, subject to the above rules. HAN(G,m) is solvable if the task may be accomplished. G is solvable if HAN(G,m) is such for all m > 0.

5. 5 Leiss (1983) obtained the following characterization of solvable graphs.

6. 6 The main result An unsolvable graph G has a maximal m for which HAN(G,m) is solvable. Denote it by M(G). There is a maximal such m for graphs with n vertices. Denote this maximum by, i.e.

7. 7 We prove the following:

8. 8 The best unsolvable graphs We characterize a family of graphs which are (among) the “best” within the family of unsolvable graphs; more accurately: An unsolvable Hanoi graph G=(V,E) is a ladder graph if E(G) is maximal with respect to G being unsolvable (i.e. by adding an edge to G, one makes it solvable)

9. 9 The following lemma shows that if G is a ladder graph, then the set of edges, E, coincides with its transitive closure. Let G=(V,E) be a ladder graph. Define an equivalence relation on V by: Define an ordering on the set of equivalence classes:

10. 10 Note that the decomposition into equivalence classes has the property that there are no two consecutive of size 1. According to the last corrolary, if G is an unsolvable graph then it is possible to add to it some edges so that it will become a ladder graph. Hence, to prove the main result we may restrict ourselves to ladder graphs.

11. 11 The sequence The following lemma will be useful in estimating the numbers. It will be convenient to encode legal sequences of moves of disks by sequences of edges of G.

12. 12 In view of the last lemma, we may restrict ourselves to solutions of HAN(G,m) in which the largest disk moves but once.

13. 13 The following lemma gives us an upper bound for the sequence

14. 14 The upper bound of

15. 15 Clearly for all. Thus it is sufficient to prove: