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Solving Mathematical Problems
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Professor Joseph Vaisman
Department of Computer Science and Engineering, NYU-Poly
[email protected]
Table of Contents
Problems, Solutions, and Techniques
Game Theory
Towers of Hanoi
Magic Squares
Fifteen-Puzzle
Fibonacci Numbers
Pascal’s Triangle
The Quest for PI
The Science of Sticky Spheres
Problems, Solutions, and Techniques ======
Chapter 5 How to solve problems
http://dx.doi.org/10.1017/CBO9780511808258.006
How to Think Like a Mathematician:
A Companion to Undergraduate Mathematics
Kevin Houston
Cambridge University Press, 2009, ISBN 9780511808258
http://dx.doi.org/10.1017/CBO9780511808258
How to Solve It: Modern Heuristics;
Second, Revised and Extended Edition
Zbigniew Michalewicz and David B. Fogel
Springer, 2004, ISBN 978-3-662-07807-5
**** book on AI – relevant chapters ********
http://dx.doi.org/10.1007/978-3-662-07807-5
Mathematics as Problem Solving, Second Edition
Alexander Soifer
Springer, 2009, ISBN 978-0-387-74647-0
http://dx.doi.org/10.1007/978-0-387-74647-0
Problem-Solving Strategies
Arthur Engel
Springer, 1998, ISBN 978-0-387-22641-5
http://dx.doi.org/10.1007/b97682
Problem-Solving Through Problems
Loren C. Larson
Springer, 1983, ISBN 978-1-4612-5498-0
http://dx.doi.org/10.1007/978-1-4612-5498-0
The Beauty of Everyday Mathematics
Norbert Hermann
Springer, 2012, ISBN 978-3-642-22104-0
http://dx.doi.org/10.1007/978-3-642-22104-0
Problems and Proofs in Numbers and Algebra
Richard S. Millman, Peter J. Shiue, and Eric Brendan Kahn
Springer, 2015, ISBN 978-3-319-14427-6
http://dx.doi.org/10.1007/978-3-319-14427-6
Complex Numbers From A to … Z, Second Edition
Titu Andreescu and Dorin Andrica
Springer, 2014, ISBN 978-0-8176-8415-0
http://dx.doi.org/10.1007/978-0-8176-8415-0
Mathematical Problems and Proofs:
Combinatorics, Number Theory, and Geometry
Branislav Kisacanin
Springer, 2002, ISBN 978-0-306-46963-3
http://dx.doi.org/10.1007/b115295
Inequalities: A Mathematical Olympiad Approach
Radmila Bulajich Manfrino, Jose Antonio Gomez Ortega,
and Rogelio Valdez Delgado
Springer, 2009, ISBN 978-3-0346-0050-7
http://dx.doi.org/10.1007/978-3-0346-0050-7
Equations and Inequalities: Elementary Problems and
Theorems in Algebra and Number Theory
Jiri Herman, Radan Kucera, and Jaromir Simsa
Springer, 2000, ISBN 978-1-4612-1270-6
http://dx.doi.org/10.1007/978-1-4612-1270-6
Topics in Algebra and Analysis:
Preparing for the Mathematical Olympiad
Radmila Bulajich Manfrino, Jose Antonio Gomez Ortega,
and Rogelio Valdez Delgado
Springer, 2015, ISBN 978-3-319-11946-5
http://dx.doi.org/10.1007/978-3-319-11946-5
Problem-Solving Methods in Combinatorics:
An Approach to Olympiad Problems
Pablo Soberon
Springer, 2013, ISBN 978-3-0348-0597-1
http://dx.doi.org/10.1007/978-3-0348-0597-1
Problem-Solving and Selected Topics in Number Theory:
In the Spirit of Mathematical Olympiads
Michael Th. Rassias
Springer, 2011, ISBN 978-1-4419-0495-9
http://dx.doi.org/10.1007/978-1-4419-0495-9
104 Number Theory Problems:
From the Training of the USA IMO Team
Titu Andreescu, Dorin Andrica, and Zuming Feng
Springer, 2007, ISBN 978-0-8176-4561-8
http://dx.doi.org/10.1007/978-0-8176-4561-8
Mathematical Olympiad Treasures, Second Edition
Titu Andreescu and Bogdan Enescu
Springer, 2011, ISBN 978-0-8176-8253-8
http://dx.doi.org/10.1007/978-0-8176-8253-8
Mathematical Olympiad Challenges, Second Edition
Titu Andreescu and Razvan Gelca
Springer, 2009, ISBN 978-0-8176-4611-0
http://dx.doi.org.databases/10.1007/978-0-8176-4611-0
The IMO Compendium - A Collection of Problems Suggested
for
The International Mathematical Olympiads: 1959-2009,
Second Edition
Dusan Djukic, Vladimir Jankovic, Ivan Matic,
and Nikola Petrovic
Springer, 2011, ISBN 978-1-4419-9854-5
http://dx.doi.org/10.1007/978-1-4419-9854-5
Geometric Problems on Maxima and Minima
Titu Andreescu, Oleg Mushkarov, and Luchezar Stayanov
Springer, 2006, ISBN 978-0-8176-4473-3
http://dx.doi.org/10.1007/0-8176-4473-3
103 Trigonometry Problems:
From the Training of the USA IMO Team
Titu Andreescu and Zuming Feng
Springer, 2005, ISBN 978-0-8176-4432-1
http://dx.doi.org/10.1007/b139082
An Introduction to Diophantine Equations:
A Problem-Based Approach
Titu Andreescu, Dorin Andrica, and Ion Cucurezeanu
Springer, 2010, ISBN 978-0-8176-4549-6
http://dx.doi.org/10.1007/978-0-8176-4549-6
Putnam and Beyond
Razvan Gelca and Titu Andreescu
Springer, 2007, ISBN 978-0-387-68445-1
http://dx.doi.org/10.1007/978-0-387-68445-1
An Invitation to Mathematics: From Competition to Research
Dierk Schleicher and Malte Lackmann (Editors)
Springer, 2011, ISBN 978-3-642-19533-4
http://dx.doi.org/10.1007/978-3-642-19533-4
The Art of Mathematics – Coffee Time in Memphis
Bela Bollobas
Cambridge University Press, 2006, ISBN 9780511816574
http://dx.doi.org/10.1017/CBO9780511816574
Thinking in Problems: How Mathematicians Find Creative
Solutions
Alexander A. Roytvarf
Springer, 2013, ISBN 978-0-8176-8406-8
http://dx.doi.org/10.1007/978-0-8176-8406-8
Problem Solving for Engineers and Scientists:
A Creative Approach
Raymond Friedman
Springer, 1991, ISBN 978-1-4615-3906-3
http://dx.doi.org/10.1007/978-1-4615-3906-3
Arnold’s Problems
Vladimir I. Arnold
Springer, 2005, ISBN 978-3-540-28666-6
http://dx.doi.org/10.1007/b138219
Exercises in Analysis, Part I
Leszek Gazinski and Nikolaos S. Papageorgiou
Springer, 2014, ISBN 978-3-319-06176-4
http://dx.doi.org/10.1007/978-3-319-06176-4
People, Problems, and Proofs –
Essays from Godel’s Lost Letter: 2010
Richard J. Lipton and Kenneth W. Regan
Springer, 2013, ISBN 978-3-642-41422-0
http://dx.doi.org/10.1007/978-3-642-41422-0
Inside Interesting Integrals
Paul J. Nahin
Springer, 2015, ISBN 978-1-4939-1277-3
http://dx.doi.org/10.1007/978-1-4939-1277-3
Problems from the Discrete to the Continuous:
Probability, Number Theory, Graph Theory,
and Combinatorics
Ross G. Pinsky
Springer, 2014, ISBN 978-3-319-07965-3
http://dx.doi.org/10.1007/978-3-319-07965-3
Back to the Table of Contents
=======================================
Game Theory ==========================
Also see:
Game Theory shelf in the
Economics and Finance section.
New Dilemmas for the Prisoner
Brian Hayes
American Scientist,
Volume 101, Number 6 (November-December 2013)
http://dx.doi.org/10.1511/2013.105.422
Insights into Game Theory:
An Alternative Mathematical Experience
Ein-Ya Gura and Michael B. Maschler
Cambridge University Press, 2008, ISBN 9780511754326
http://dx.doi.org/10.1017/CBO9780511754326
Game Theory: An Introduction, Second Edition
E.N. Barron
Wiley, 2013, ISBN 9781118547168
http://dx.doi.org/10.1002/9781118547168
Back to the Table of Contents
=======================================
Towers of Hanoi ========================
The Tower of Hanoi – Myths and Maths
Andreas M. Hinz, Sandy Klavzar, Uros Milutinovic, and Ciril Petr
Springer, 2013, ISBN 978-3-0348-0237-6
http://dx.doi.org/10.1007/978-3-0348-0237-6
About the remarkable similarity between the Icosian
Game and the Tower of Hanoi
Martin Gardner
Scientific American, Volume 196, Number 5 (May 1957) Pages 150-157
http://www.nature.com/scientificamerican/journal/v196/n5/pdf/scien
tificamerican0557-150.pdf
The curious properties of Gray code and how it can be
used to solve puzzles
Martin Gardner
Scientific American, Volume 227, Number 2 (September 1972)
http://www.nature.com/scientificamerican/journal/v227/n2/pdf/scien
tificamerican0872-106.pdf
Yin and yang: recursion and iteration, the Tower of Hanoi and the
Chinese rings
A. K. Dewdney
Scientific American, Volume 251, Number 5 (November 1984)
http://www.nature.com/scientificamerican/journal/v251/n5/pdf/scien
tificamerican1184-19.pdf
Sierpinski’s Ubiquitous Gasket
Ian Stewart
Scientific American, Volume 281, Number 2 (August 1999)
http://www.nature.com/scientificamerican/journal/v281/n2/pdf/scien
tificamerican0899-90.pdf
Chapter 1 “The Lion, the Llama, and the Lettuce”
Another Fine Math You’ve Got Me Into…
Ian Stewart
http://books.google.com/books?id=u5GPE97ZhsC&printsec=frontcover&dq=intitle:another+intitle:fine+intitle:math
&hl=en&ei=PJygTr2GLKna0QGKx7T6BA&sa=X&oi=book_result&ct=resu
lt&resnum=1&ved=0CD0Q6AEwAA#v=onepage&q&f=false
The Generalized Towers of Hanoi for Space-Deficient Computers and
Forgetful Humans
Timothy R. Walsh
The Mathematical Intelligencer, Volume 20, Number 1 (March 1998)
http://dx.doi.org/10.1007/BF03024398
Back to the Table of Contents
=======================================
Magic Squares =========================
In general, a magic square is an arrangement of the integers from 1 to
n*n, in cells of an n-by-n square, such that the numbers in each row,
column, and diagonal give the same sum, the magic sum.
You have access to a thorough explanation at the link below –
disregard that it is intended for a very young audience.
http://www.dr-mikes-math-games-for-kids.com/3x3-magicsquare.html
Anything but square: from magic squares to Sudoku
Hardeep Aiden
http://plus.maths.org/content/anything-square-magic-squares-sudoku
Magic Squares
Pages 16-24
http://dx.doi.org/10.1007/978-1-4612-1088-7_2
Ramanujan’s Notebooks, Part I
Bruce C. Berndt
Springer, 1985, ISBN 978-1-4612-1088-7
http://dx.doi.org/10.1007/978-1-4612-1088-7
Magic Squares of Order Three
T. S. K. V. Iyer
Pages 76-78
http://dx.doi.org/10.1007/BF02834335
Basis Properties of Third Order Magic Squares
Shailesh A Shirali
Pages 79-89
http://dx.doi.org/10.1007/BF02834336
Resonance, Volume 11, Number 9 (September 2006)
Magic “Squares” Indeed!
Arthur T. Benjamin and Kan Yasuda
American Mathematical Monthly,
Volume 106, Number 2 (February 1999) Pages 152-156
http://dx.doi.org/10.2307/2589051
Constructing All Magic Squares of Order Three
Guoce Xin
http://arxiv.org/abs/math/0409468
Quadramagicology
Dana Mackenzie
Pages 50-53
New Scientist, Volume 180, Issue 2426-2428 (December 20, 2003)
http://search.ebscohost.com/login.aspx?direct=true&db=n2h&AN=257
15053&site=ehost-live
Magic Square Patterns
D. B. Eperson
The Mathematical Gazette,
Volume 43, Number 346 (December 1959) Pages 273-275
http://www.jstor.org/stable/3610655
Magic Squares of Order 4
Dame Kathleen Ollerenshaw and Herman Bondi
Philosophical Transactions of the Royal Society of London, Series A,
Mathematical and Physical Sciences,
Volume 306, Number 1495 (October 15, 1982) Pages 443-532
http://www.jstor.org/stable/37143
On ‘Most Perfect’ or ‘Complete’ 8 X 8 Pandiagonal Magic Squares
Dame Kathleen Ollerenshaw
Proceedings of the Royal Society of London, Series A,
Mathematical and Physical Sciences,
Volume 407, Number 1833 (October 8, 1986) Pages 259-281
http://www.jstor.org/stable/2397989
On Karnaugh maps and magic squares
Dieter Schuett and Sebastian Meine
Pages 120-123
Informatik-Spektrum, Volume 28, Number 2 (April 2005)
http://dx.doi.org/10.1007/s00287-005-0468-3
Latin Squares
Bhaskar Bagchi
Pages 895-902
Resonance, Volume 17, Number 9 (September 2012)
http://dx.doi.org/10.1007/s12045-012-0098-4
Magic Graphs, Second Edition
Alison M. Marr and W.D. Wallis
Springer, 2013, ISBN 978-0-8176-8391-7
http://dx.doi.org/10.1007/978-0-8176-8391-7
Back to the Table of Contents
=======================================
Fifteen-Puzzle ==========================
How Safe Is Sam Lloyd’s Bet? The 15-Puzzle and Beyond
Jyoti Ramakrishnan
Pages 80-89
Resonance, Volume 5, Number 11 (November 2000)
http://dx.doi.org/10.1007/BF02868496
Back to the Table of Contents
=======================================
Fibonacci Numbers =====================
Fibonacci Numbers
Ron Knott
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/
Ron Knott’s web pages on Mathematics
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/
FIBONACCI – HIS RABBITS AND HIS NUMBERS and KEPLER
Keith Tognetti
http://www.austms.org.au/Modules/Fib/
Fibonacci Numbers
Nicolai N. Vorobiev
Springer, 2002, ISBN 978-3-0348-8107-4
http://dx.doi.org/10.1007/978-3-0348-8107-4
Fibonacci and Catalan Numbers: An Introduction
Ralph P. Grimaldi
Wiley, 2012, ISBN 9781118159743
http://dx.doi.org/10.1002/9781118159743
Catalan Numbers: with Applications
Thomas Koshy
Oxford University Press, 2009, ISBN 9780195334548
http://dx.doi.org/10.1093/acprof:oso/9780195334548.001.000
1
Fibonacci and Lucas Numbers with Applications
Thomas Koshy
Wiley, 2001, ISBN 9781118033067
http://dx.doi.org/10.1002/9781118033067
Pell and Pell-Lucas Numbers with Applications
Thomas Koshy
Springer, 2014, ISBN 978-1-4614-8489-9
http://dx.doi.org/10.1007/978-1-4614-8489-9
Applications of Fibonacci Numbers, Volume 9:
Proceedings of the Tenth International Research Conference
on Fibonacci Numbers and Their Applications
Frederic T. Howard (Editor)
Springer, 2004, ISBN 978-0-306-48517-6
http://dx.doi.org/10.1007/978-0-306-48517-6
Applications of Fibonacci Numbers, Volume 8:
Proceedings of ‘The Eighth International Research Conference
on Fibonacci Numbers and Their Applications’
Fredric T. Howard (Editor)
Springer, 1999, ISBN 978-94-011-4271-7
http://dx.doi.org/10.1007/978-94-011-4271-7
Applications of Fibonacci Numbers, Volume 7:
Proceedings of ‘The Seventh International Research
Conference on Fibonacci Numbers and Their Applications’
G. E. Bergum, A. N. Philippou, and A. F. Horadam (Editors)
Springer, 1998, ISBN 978-94-011-5020-0
http://dx.doi.org/10.1007/978-94-011-5020-0
Back to the Table of Contents
=======================================
Pascal’s Triangle ========================
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Pascal’s Triangle
Nathan Hoffman
The Arithmetic Teacher, Volume 21, Number 3 (March 1974)
http://www.jstor.org/stable/41188488
Pascal’s Triangle
Karl J. Smith
The Two-Year College Mathematics Journal,
Volume 4, Number 1 (Winter 1973)
http://www.jstor.org/stable/2698949
Differences and Pascal’s triangle
Chris Houghton
Mathematics in School,
Volume 20, Number 4 (September 1991)
http://www.jstor.org/stable/30214830
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Extended Pascal Triangles
Richard C. Bollinger
Mathematics Magazine, Volume 66, Number 2 (April 1993)
http://www.jstor.org/stable/2691114
Pascal + Fermat -> Gauss
Per Haggmark
The Mathematical Gazette
Volume 61, Number 418 (December 1977)
http://www.jstor.org/stable/3617404
Pascal’s Triangle and the Tower of Hanoi
Andreas M. Hinz
The American Mathematical Monthly,
Volume 99, Number 6 (June-July 1992)
http://www.jstor.org/stable/2324061
Pascal’s Triangle, Difference Tables and Arithmetic Sequences
of Order N
Calvin Long
The College Mathematics Journal,
Volume 15, Number 4 (September 1984)
http://www.jstor.org/stable/2686393
Pascal Triangles and Combinations Where Repetitions Are
Allowed
Kendell Hyde
The College Mathematics Journal,
Volume 19, Number 1 (January 1988)
http://www.jstor.org/stable/2686707
Paths and Pascal Numbers
John F. Lucas
The Two-Year College Mathematics Journal,
Volume 4, Number 1 (Winter 1973)
http://www.jstor.org/stable/3027286
Recurrent Sequences and Pascal’s Triangle
Thomas M. Green
Mathematics Magazine, Volume 41, Number 1 (January 1968)
http://www.jstor.org/stable/2687953
Relating Geometry and Algebra in the Pascal Triangle,
Hexagon, Tetrahedron, and Cuboctahedron
Part I: Binomial Coefficients, Extended Binomial Coefficients
and Preparation for Further Work
Peter Hilton and Jean Pedersen
The College Mathematics Journal,
Volume 30, Number 3 (May 1999)
http://www.jstor.org/stable/2687595
Relating Geometry and Algebra in the Pascal Triangle,
Hexagon, Tetrahedron, and Cuboctahedron
Part II: Geometry and Algebra in Higher Dimensions:
Identifying the Pascal Cuboctahedron
Peter Hilton and Jean Pedersen
The College Mathematics Journal,
Volume 30, Number 4 (September 1999)
http://www.jstor.org/stable/2687666
Restricted Occupancy Theory – A Generalization of Pascal’s
Triangle
J. E. Freund
The American Mathematical Monthly,
Volume 63, Number 1 (January 1956)
http://www.jstor.org/stable/2308048
The Towers and Triangles of Professor Claus
(or, Pascal Knows Hanoi)
David G. Poole
Mathematics Magazine,
Volume 67, Number 5 (December 1994)
http://www.jstor.org/stable/2690991
Zaphod Beeblebrox’s Brain and the Fifty-ninth Row of Pascal’s
Triangle
Andrew Granville
The American Mathematical Monthly,
Volume 99, Number 4 (April 1992)
http://www.jstor.org/stable/2324898
Correction to: Zaphod Beeblebrox’s Brain and the Fifty-ninth
Row of Pascal’s Triangle
Andrew Granville
The American Mathematical Monthly,
Volume 104, Number 9 (November 1997)
http://www.jstor.org/stable/2975291
Back to the Table of Contents
=======================================
The Quest for Pi ========================
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The Quest for Pi
D.H. Bailey, J.M. Borwein, P.B. Borwein, and S. Plouffe
Mathematical Intelligencer,
Volume 19, Number 1 (Winter 1997)
*** Item # 47 ****************************************
http://www.davidhbailey.com/dhbpapers/index.html#Technica
l-papers
http://dx.doi.org/10.1007/BF03024340
Home Page of David H. Bailey
http://www.davidhbailey.com/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Pi: A Source Book, Third Edition
Lennart Berggren, Jonathan Borwein, and Peter Borwein
Springer, 2004, ISBN 978-1-4757-4217-6
http://dx.doi.org/10.1007/978-1-4757-4217-6
Pi – Unleashed
Jorg Arndt and Christoph Haenel
Springer, 2001, ISBN 978-3-642-56735-3
http://dx.doi.org/10.1007/978-3-642-56735-3
Back to the Table of Contents
=======================================
The Science of Sticky Spheres =============
This shelf contains the original article, and will contain inks to
the article’s bibliography, and links to additional items I
consider relevant and useful.
The Science of Sticky Spheres
Brian Hayes
American Scientist,
Volume 100, Number 6 (November-December 2012)
http://dx.doi.org/10.1511/2012.99.442
Computing Science column and book reviews
http://www.americanscientist.org/issues/issue_byAuthor_list.a
spx?authorID=6660&pageID=1
Website
http://bit-player.org/
Back to the Table of Contents
=======================================