Solving Towers of Hanoi and Related Puzzles

  • Paul Cull
  • Leanne Merrill
  • Tony Van
  • Celeste Burkhardt
  • Tommy Pitts
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8111)


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.


Puzzles Graphs Towers of Hanoi Spin-Out Algorithms Hamiltonian Paths Error-correcting Codes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Burkhardt, C., Pitts, T.: Hamiltonian paths and perfect one-error-correcting codes on iterated complete graphs. Oregon State REU Proceedings (2012)Google Scholar
  2. 2.
    Cull, P., Ecklund Jr., E.F.: Towers of Hanoi and Analysis of Algorithms. American Mathematical Monthly 92(6), 407–420 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Cull, P., Flahive, M., Robson, R.: Difference Equations. Springer, New York (2005)zbMATHGoogle Scholar
  5. 5.
    Cull, P., Nelson, I.: Error-correcting codes on the Towers of Hanoi graphs. Discrete Math. 208(209), 157–175 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cull, P., Nelson, I.: Perfect Codes, NP-Completeness, and Towers of Hanoi Graphs. Bull. Inst. Combin. Appl. 26, 13–38 (1999)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Doran, R.W.: The Gray code. Journal of Universal Computer Science 13(11), 1573–1597 (2007)MathSciNetGoogle Scholar
  8. 8.
    Gardner, M.: Curious properties of the Gray code and how it can be used to solve puzzles. Scientific American 227(2), 106–109 (1972)CrossRefGoogle Scholar
  9. 9.
    Jaap. Jaap’s puzzle page,
  10. 10.
    Klažar, S., Milutinović, U., Petr, C.: 1-perfect codes in Sierpinski graphs. Bull. Austral. Math. Soc. 66, 369–384 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kleven, S.: Perfect Codes on Odd Dimension Serpinski Graphs. Oregon State REU Proceedings (2003)Google Scholar
  12. 12.
    Li, C.-K., Nelson, I.: Perfect codes on the Towers of Hanoi graph. Bull. Austral. Math. Soc. 57, 367–376 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pruhs, K.: The SPIN-OUT puzzle. ACM SIGCSE Bulletin 25, 36–38 (1993)CrossRefGoogle Scholar
  14. 14.
    Savage, C.: A survey of combinatorial Gray codes. SIAM Review 39, 605–629 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
  16. 16.
    Weaver, E.: Gray codes and puzzles on iterated complete graphs. Oregon State REU Proceedings (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Paul Cull
    • 1
  • Leanne Merrill
    • 1
  • Tony Van
    • 1
  • Celeste Burkhardt
    • 1
  • Tommy Pitts
    • 1
  1. 1.Oregon State UniversityCorvallisUSA

Personalised recommendations