The Apprentices' Tower of Hanoi

65 Pages·2015·3.37 MB·English

East Tennessee State University
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5-2015
The Apprentices' Tower of Hanoi
Cory BH Ball
East Tennessee State University
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Ball, Cory BH, "The Apprentices' Tower of Hanoi" (2015).Electronic Theses and Dissertations.Paper 2512. https://dc.etsu.edu/etd/
2512
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The Apprentices’ Tower of Hanoi
A thesis
presented to
the faculty of the Department of Mathematics
East Tennessee State University
In partial fulfillment
of the requirements for the degree
Master of Science in Mathematical Sciences
by
Cory Braden Howell Ball
May 2015
Robert A. Beeler, Ph.D., Chair
Anant P. Godbole, Ph.D.
Teresa W. Haynes, Ph.D.
Jeff R. Knisley, Ph.D.
Keywords: graph theory, combinatorics, Tower of Hanoi, Hanoi variant, puzzle
ABSTRACT
The Apprentices’ Tower of Hanoi
by
Cory Braden Howell Ball
TheApprentices’TowerofHanoiisintroducedinthisthesis. Severalboundsarefound
in regards to optimal algorithms which solve the puzzle. Graph theoretic properties
of the associated state graphs are explored. A brief summary of other Tower of Hanoi
variants is also presented.
2
Copyright by Cory Braden Howell Ball 2015
3
DEDICATION
I dedicate this thesis to my late grandfather Brady Stevenson, who taught me
from a very young age how to read, how to write, and how to do arithmetic, and also
guided me in the development my faith, my character, and my morality.
4
ACKNOWLEDGMENTS
I would like to acknowledge Dr. Robert A. Beeler for his continued support as my
thesisadvisor. Withouthisknowledgeandexpertisecompletingthethesiswouldhave
been impossible. I would like to thank Dr. Anant Godbole, Dr. Jeff Knisley, and Dr.
Teresa Haynes for their valuable time and effort as my committee members. I would
like to express my gratitude to Dr. Robert Gardner without whom the opportunity to
further my education would not have been possible. Thank you to Dr. Rick Norwood
for sharing an intellectually stimulating teaching method with his students.
To my mother, Ronda, and my dad, Greg, thank you for providing for me for so
many years. I would like to thank my brother Adam for never staying angry with me,
despite the number of fights we’ve had. Thank you to my grandparents for always
helping me. I am eternally grateful to the Lord and Saviour Jesus Christ.
A special thanks is expressed to the good friends I have met during my time in
the ETSU Department of Mathematics and Statistics, Whitney Forbes, Tony “T-
Rod” Rodriguez, Ricky Blevins, Derek Bryant, and “President” Dustin Chandler.
Lastly, thank you to my cronies Joel Shelton, Derek Kiser, Brett Shields, and Samuel
Kakraba.
5
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 CLASSICAL TOWER OF HANOI . . . . . . . . . . . . . . . . . . . 10
1.1 The Legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 An Optimal Algorithm . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Hanoi Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 TOWER OF HANOI VARIANTS . . . . . . . . . . . . . . . . . . . . 18
2.1 The Reve’s Puzzle . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Frame-Stewart . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Bottle-Neck Tower . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Sinner’s Tower and Santa Claus’ Tower . . . . . . . . . . . . . 23
3 THE APPRENTICES’ TOWER OF HANOI . . . . . . . . . . . . . . 24
3.1 The Legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 The Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 QUEST FOR AN OPTIMAL ALGORITHM . . . . . . . . . . . . . . 27
5 AH GRAPH PROPERTIES . . . . . . . . . . . . . . . . . . . . . . 38
n
5.1 The Number of Vertices . . . . . . . . . . . . . . . . . . . . . 38
6
5.2 Figures of AH for n ≤ 3 . . . . . . . . . . . . . . . . . . . . . 46
n
5.3 Graph Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 GENERALIZING THE APPRENTICES’ TOWER OF HANOI . . . 52
6.1 s sins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
i
6.2 p pegs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 General Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 FURTHER QUESTIONS . . . . . . . . . . . . . . . . . . . . . . . . 59
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7
LIST OF TABLES
1 The Minimum Number of Moves for n ∈ [31] for the Reve’s Puzzle [15] 21
2 Allowed Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 2S +3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
n−2
4 2S (1,1,1)+2S (1,1,0)+1 . . . . . . . . . . . . . . . . . . . . 31
n−k k−1
5 n(AH ) for small n. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
n
8
LIST OF FIGURES
1 H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3
2 The Sierpin´ski Triangle[5] . . . . . . . . . . . . . . . . . . . . . . . . 17
3 The Reve’s Puzzle[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 16|34|52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
∼
5 AH = Trivial Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
0
∼
6 AH = K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1 3
∼
7 AH = SD(1) = Star of David Graph [20] . . . . . . . . . . . . . . . 47
2
8 The graph AH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3
9 The graph AH with selected vertices. . . . . . . . . . . . . . . . . . 50
3
9