Discrete Mathematics

Andreas M. Hinz

University of Munich & University of Maribor

This is a complete list of my published papers on the Tower of Hanoi and related topics.

An Iterative Algorithm for the Tower of Hanoi with Four Pegs,
Computing 42(1989), 133-140.

The Tower of Hanoi, Enseign. Math. (2) 35(1989), 289-321.

(with A. Schief) The average distance on the Sierpiński gasket,
Probab. Theory Related Fields 87(1990), 129-138.

Solution to Problem 1350 (Math. Mag. 63(1990), 189),
Math. Mag. 64(1991), 203.

Shortest Paths between Regular States of the Tower of Hanoi,
Inform. Sci. 63(1992), 173-181.

Pascal's Triangle and the Tower of Hanoi, Amer. Math. Monthly
99(1992), 538-544.

Square-free Tower of Hanoi sequences, Enseign. Math. (2) 42(1996), 257-264.

The Tower of Hanoi, in: Shum, K.-P., Taft, E. J., Wan, Z.-X. (Eds.),
Algebras and Combinatorics, An International Congress, ICAC '97, Hong Kong,
Springer, Singapore, 1999, 277-289.

(with J.-P. Bode ) Results and open problems on the Tower of Hanoi,
Congr. Numer. 139(1999), 113-122.

(with D. Parisse) On the Planarity of Hanoi Graphs, Exposition. Math. 20(2002), 263-268.

(with S. Klavžar, U. Milutinović, D. Parisse, C. Petr) Metric properties
of the Tower of Hanoi graphs and Stern's diatomic sequence,
European J. Combin. 26(2005), 693-708.

(with A. Kostov, F. Kneißl, F. Sürer, A. Danek) A mathematical model
and a computer tool for the Tower of Hanoi and Tower of London puzzles,
Inform. Sci. 179(2009), 2934-2947.

(with A. H. Faber, N. Kühnpast, F. Sürer, A. Danek) The iso-effect:
Is there specific learning of Tower of London iso-problems?,
Thinking & Reasoning 15(2009), 237-249.

Graph theory of tower tasks, Behavioural Neurology 25(2012), 13-22.

(with D. Parisse) The average eccentricity of Hanoi and Sierpiński graphs, Graphs Combin. 28(2012), 671-686.

(with D. Parisse) Coloring Hanoi and Sierpiński Graphs, Discrete Math. 312(2012), 1521-1535.

(with S. Klavžar and S. S. Zemljič) Sierpiński graphs as spanning subgraphs of Hanoi graphs, Cent. Eur. J. Math. 11(2013), 1153-1157.

(with C. Holz auf der Heide) An efficient algorithm to determine all shortest paths in Sierpiński graphs, Discrete Appl. Math. 177(2014), 111-120.

(with M. Freiberger) The Tower of Hanoi: Where mathematics meets psychology, Chapter 22 in: S. Parc (ed.), 50 Visions of Mathematics, Oxford University Press, Oxford, 2014.

(with S. Aumann, K. A. M. Götz and C. Petr) The number of moves of the largest disc in shortest paths on Hanoi graphs, Electron. J. Combin. 21(2014), P4.38.

(with S. Döll) Kyu-renkan - the Arima sequence, Adv. Stud. Pure Math. 79(2018), 321-335.

(with C. Holz auf der Heide and S. S. Zemljič) Metric properties of Sierpiński triangle graphs, Discrete Appl. Math. 319(2022), 439-453.

(with C. Petr) Computational Solution of An Old Tower of Hanoi Problem, Electron. Notes Discrete Math. 53(2016), 445-458.

(with S. Varghese and A. Vijayakumar) Power domination in Knödel graphs and Hanoi graphs, Discuss. Math. Graph Theory 38(2018), 63-74.

(with S. Klavžar and S. S. Zemljič) A survey and classification of Sierpiński-type graphs, Discrete Appl. Math. 217(2017), 565-600.

The Lichtenberg Sequence, Fibonacci Quart. 55(2017), 2-12.

(with A. Heeffer) "A difficult case": Pacioli and Cardano on the Chinese Rings, Recreat. Math. Mag. 8(2017), 5-23.

Problems 3 (History of the Chinese rings, p. 214f), 5 (Jeux Scientifiques par Edouard Lucas (1842-1891), p. 215), and 12 (Hanoi sequences, p. 218f), in: C. Kimberling, Problem proposals, Fibonacci Quart. 55 (5)(2017), 213-221.

Open Problems for Hanoi and Sierpiński Graphs, Electron. Notes Discrete Math. 63(2017), 23-31.

(with N. Movarraei) The Hanoi Graph $H_4^3$, Discuss. Math. Graph Theory, 40(2020), 1095-1109.

(with B. Lužar and C. Petr) The Dudeney-Stockmeyer Conjecture, Discrete Appl. Math. 319(2022), 19-26.

(with P. K. Stockmeyer) Discovering Fibonacci numbers, Fibonacci words, and a Fibonacci fractal in the Tower of Hanoi, Fibonacci Quart. 57 (5)(2019), 72-83.

Are the Chinese Rings Chinese?, in: J. N. Silva (Ed.), Proceedings of the Recreational Mathematics Colloquium VI---G4G Europe, Ludus, Lisboa, 2019, 83--101.

(with T. Bousch, S. Klavžar, D. Parisse, C. Petr, and P. K. Stockmeyer) A note on the Frame-Stewart conjecture, Discrete Math. Algorithms Appl. 11 (4) (2019), 1950049.

The Tower of Brahma, Bull. Mathematical Consortium 1 (2)(2019), 13-19.

(with K. Balakrishnan, M. Changat, and D. S. Lekha) The Median of Sierpiński Graphs, Discrete Appl. Math. 319(2022), 159-170.

(with P. K. Stockmeyer) Precious Metal Sequences and Sierpiński-Type Graphs, J. Integer Sequences 25(2022), Article 22.4.8.

I once gave a seminar (in German) for high-school students in connection with
the program TU München Mathematik Stipendium on the Turm von Hanoi.
(This is a slightly extended version of an article in mathe-lmu.de 4(2001), 20-25.)

In collaboration with neuropsychologists and computer scientists we develop a computer tool for tower puzzles.
The corresponding poster has been awarded a Second Prize in the Poster competition of the International Congress of Mathematicians 2006 in Madrid.

I would be happy to hear from anyone interested in these topics.

In particular I am interested in any new development about the
Tower of Hanoi with more than three pegs. However, only serious
attempts to prove minimality of the presumed minimal solution
will be considered.

I am also looking for material about the biography of the inventor
of the Tower of Hanoi, the French mathematician Édouard Lucas,
who lived 1842-1891 (cf. my article in the Monthly ).

Some people have taken up my challenge to hold a conference on the Tower of Hanoi, albeit not in Hanoi!
It was organized by S. Klavžar, U. Milutinović, and C. Petr in the Slovenian town of Maribor (Marburg an der Drau);
see Workshop on the Tower of Hanoi and Related Problems for details.

With my Slovenian colleagues I wrote a comprehensive book The Tower of Hanoi - Myths and Maths ,
which has appeared at Birkhäuser/Springer Basel (2013) and now in its Second Edition, Cham (2018).

My further research interests cover analysis , mathematical modelling , and other topics .

You may want to know more about my courses and lectures .

My coordinates are on my personal homepage .

A. M. Hinz, hinz@math.lmu.de, 2023-04-14