Abstract
The classical towers of Hanoi have been generalized in several ways. In particular the second named author has studied the 3-peg Hanoi towers with all possible restrictions on the permitted moves between pegs. We prove that all these Hanoi puzzles give rise to infinite morphic sequences of moves, whose appropriate truncations describe the transfer of any given number of disks. Furthermore two of these infinite sequences are actually automatic sequences.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Allouche, J.-P.: Note on the cyclic towers of Hanoi, in Number theory, combinatorics and applications to computer science (Marseille, 1991). Theoret. Comput. Sci. 123, 3–7 (1994)
Allouche, J.-P., Astoorian, D., Randall, J., Shallit, J.: Morphisms, squarefree strings, and the Tower of Hanoi puzzle. Amer. Math. Monthly 101, 651–658 (1994)
Allouche, J.-P., Bacher, R.: Toeplitz sequences, paperfolding, Towers of Hanoi and progression-free sequences of integers. Enseign. Math. 38, 315–327 (1992)
Allouche, J.-P., Bétréma, J., Shallit, J.: Sur des points fixes de morphismes d’un monoïde libre, RAIRO Inform. Théor. Appl. 23, 235–249 (1989)
Allouche, J.-P., Dress, F.: Tours de Hanoï et automates, RAIRO Inform. Théor. Appl. 24, 1–15 (1990)
Allouche, J.-P., Shallit, J.: Automatic sequences. Theory, applications, generalizations, p. xvi+571. Cambridge University Press, Cambridge (2003)
Atkinson, M.D.: The cyclic towers of Hanoi. Inform. Process. Lett. 13, 118–119 (1981)
Berstel, J.: Sur la construction de mots sans carré, Séminaire de Théorie des Nombres de Bordeaux, 1978–1979. Exposé 18, 18-01–18-15
Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory 3, 186–192 (1969)
Cobham, A.: Uniform tag sequences. Math. Systems Theory 6, 164–192 (1972)
Durand, F.: A generalization of Cobham’s theorem. Theory Comput. Syst. 31, 169–185 (1998)
Durand, F.: A theorem of Cobham for non-primitive substitutions. Acta Arith. 104, 225–241 (2002)
Durand, F.: Combinatorial and dynamical study of substitutions around the theorem of Cobham. In: Maass, A., et al. (eds.) Dynamics and randomness, Universidad de Chile, Santiago, Chile, December 11-15. Lectures given at the conference. Kluwer, Boston (2000); Nonlinear Phenom. Complex Systems 7, 53–94 (2002)
Hinz, M.: The Tower of Hanoi. Enseign. Math. 35, 289–321 (1989)
Hinz, M.: Pascal’s triangle and the Tower of Hanoi. Amer. Math. Monthly 99, 538–544 (1992)
Hinz, M., Klavžar, S., Milutinović, U., Parisse, D., Petr, C.: Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence. European J. Combin. 26, 693–708 (2005)
Lucas, É.: Le calcul et les machines à calculer, Assoc. Française pour l’Avancement des Sciences. Comptes Rendus 13, 111–141 (1884)
Sapir, A.: the tower of Hanoi with forbidden moves. The Computer Journal 47, 20–24 (2004)
Scorer, R.S., Grundy, P.M., Smith, C.A.B.: Some binary games. Math. Gazette 280, 96–103 (1944)
Shallit, J.: A generalization of automatic sequences. Theoret. Comput. Sci. 61, 1–16 (1988)
Shallit, J., Swart, D.: An efficient algorithm for computing the i’th letter of ϕ n(a). In: Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 768–775 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Allouche, JP., Sapir, A. (2005). Restricted Towers of Hanoi and Morphisms. In: De Felice, C., Restivo, A. (eds) Developments in Language Theory. DLT 2005. Lecture Notes in Computer Science, vol 3572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11505877_1
Download citation
DOI: https://doi.org/10.1007/11505877_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26546-7
Online ISBN: 978-3-540-31682-4
eBook Packages: Computer ScienceComputer Science (R0)