Abstract
Starting with the well-known Towers of Hanoi, we create a new sequence of puzzles which can essentially be solved in the same way. Since graphs and puzzles are intimately connected, we define a sequence of graphs, the iterated complete graphs, for our puzzles. To create puzzles for all these graphs, we need to generalize another puzzle, Spin-Out, and cross the generalized Towers puzzles with the the generalized Spin-Out puzzles. We show how to solve these combined puzzles. We also show how to compute distances between puzzle configurations. We show that our graphs have Hamiltonian paths and perfect one-error-correcting codes. (Properties that are \(\mathcal{NP}\)-complete for general graphs.) We also discuss computational complexity and show that many properties of our graphs and puzzles can be calculated by finite state machines.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Burkhardt, C., Pitts, T.: Hamiltonian paths and perfect one-error-correcting codes on iterated complete graphs. Oregon State REU Proceedings (2012)
Cull, P., Ecklund Jr., E.F.: Towers of Hanoi and Analysis of Algorithms. American Mathematical Monthly 92(6), 407–420 (1985)
Cull, P., Merrill, L., Van, T.: A Tale of Two Puzzles: Towers of Hanoi and Spin-Out. Journal of Information Processing 21(3), 1–15 (2013)
Cull, P., Flahive, M., Robson, R.: Difference Equations. Springer, New York (2005)
Cull, P., Nelson, I.: Error-correcting codes on the Towers of Hanoi graphs. Discrete Math. 208(209), 157–175 (1999)
Cull, P., Nelson, I.: Perfect Codes, NP-Completeness, and Towers of Hanoi Graphs. Bull. Inst. Combin. Appl. 26, 13–38 (1999)
Doran, R.W.: The Gray code. Journal of Universal Computer Science 13(11), 1573–1597 (2007)
Gardner, M.: Curious properties of the Gray code and how it can be used to solve puzzles. Scientific American 227(2), 106–109 (1972)
Jaap. Jaap’s puzzle page, http://www.jaapsch.net/puzzles/spinout.htm
Klažar, S., Milutinović, U., Petr, C.: 1-perfect codes in Sierpinski graphs. Bull. Austral. Math. Soc. 66, 369–384 (2002)
Kleven, S.: Perfect Codes on Odd Dimension Serpinski Graphs. Oregon State REU Proceedings (2003)
Li, C.-K., Nelson, I.: Perfect codes on the Towers of Hanoi graph. Bull. Austral. Math. Soc. 57, 367–376 (1998)
Pruhs, K.: The SPIN-OUT puzzle. ACM SIGCSE Bulletin 25, 36–38 (1993)
Savage, C.: A survey of combinatorial Gray codes. SIAM Review 39, 605–629 (1996)
Spin-Out. Amazon, http://www.amazon.com/Think-Fun-5401-Thinkfun-Spinout/dp/B000EGI4IA
Weaver, E.: Gray codes and puzzles on iterated complete graphs. Oregon State REU Proceedings (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cull, P., Merrill, L., Van, T., Burkhardt, C., Pitts, T. (2013). Solving Towers of Hanoi and Related Puzzles. In: Moreno-DÃaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory - EUROCAST 2013. EUROCAST 2013. Lecture Notes in Computer Science, vol 8111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-53856-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-53856-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-53855-1
Online ISBN: 978-3-642-53856-8
eBook Packages: Computer ScienceComputer Science (R0)