Appended below is an annotated bibliography of materials I have explored in bring fun, games, and puzzles into our “Explorations of Mathematics” course. At this point, it covers Dominoes, Connection Games, Go, Origami, as well as other games and puzzles.
A copy will be made available online under <http://www.wsc.ma.edu/ecke/>.
0. The Puzzle Instinct: The Meaning of Puzzles in Human Life, Marcel Danesi, Indiana University Press, 2004.
Why are humans fascinated by puzzles? Danesi not only has included many puzzles to, well, puzzle over, he also explores why we like to puzzle over them. What is the need puzzles meet? Why did they emerge at the same time in history as myth, magic, and the occult arts? And: why can't we put them down!?
Marcel Danesi is Professor of semiotics and linguistics at the University of Toronto. He is also cross-appointed as a professor of education, having established a continuing studies mathematics program for students with difficulty in the subject.
1. The Little Giant Book of Dominoes, Jennifer A. Kelley and Miguel Lugo, Sterling Publishing, 2003.
2. Great Book of Domino Games, Jennifer A. Kelley, Sterling Publishing, 1999.
3. How to play better Dominoes, Miguel Lugo, Sterling Publishing, 2002.
Jennifer Kelley and Miguel Lugo are the authors, separately and jointly, of three inexpensive Domino-related books from Sterling Publishers. It seems to me that their joint Little Giant Book is a re-worked combination of the earlier published separate books. I recommend the Little Giant Book as the place to start: it provides some historical background, a great variety of possible Domino-related games, as well as discussions of intermediate and advanced strategies.
4. Hex Strategy: Making the Right Connections, Cameron Browne. A.K. Peters, 2000.
5. Connection Games: Variations on a Theme, Cameron Browne, A.K. Peters, 2005.
In Hex, Cameron Browne lists a large variety of possible strategies for use in playing that connection game. We have also used it as a source for creating investigation worksheets for our course. Connection Games forms an encyclopedia of connection games of all shapes and forms. It formed the main source for our work with other connection games such as ConHex, Stymie, and Y.
In addition, Cameron has also provided us with a range of further game boards in order to explore with our students what kind of connection games we might be able to create on these boards. Cameron also has a nice web site about connection games, and his design work at <http://members.optusnet.com.au/cameronb/>.
6. Mathematical Go: Chilling Gets the Last Point, Elwyn R. Berlekamp, David Wolfe, A.K. Peters, 1997.
A natural progression from playing connection games like Hex is to explore the Japanese game of Go. The strategies in Go are more like strategies in Chess. They cannot be discerned, or described, quite as easily as connection game strategies. One advantage over chess is that the rules are a little easier to explain.
7. Modern Origami, James Minoru Sakoda, Dover, 1997.
Dover Publishers has a wide range of Origami books available. James Minoru Sakoda’s “Modern Origami” is an excellent first book. Sakoda starts with the observation that many people get easily frustrated with Origami because they jump too quickly to sophisticated figures which may have very terse instructions. Sakoda therefore places great emphasis on training a novice in developing a firm ability to comfortably read the pictorial and worded instructions, and to practice the fundamental Origami short-hand folds such as the “outside-reverse fold,” the “book-fold,” or the “accordion fold.” (This pictorial language of basic folds has been codified by Akira Yoshizawa.) It is only after about forty pages that the first “real” Origami figure, the bird base and crane, appear in the book.
Our little group of Origami explorers found ourselves is precisely the situation that Sakoda describes. After successfully folding the crane, we boldly skipped ahead to “something more interesting.” Our desire to create the seal from a one-page description (on p. 57) quickly hit a snag: “Start with the back-side of an eight-point star.” So we leaf back to the two-page description of an eight-point star (p.54-55) which starts with a “sunken bird-base” (p. 53) which leads us back to the bird-base (p. 40-41). So, after working through five pages of folding instructions, we are in a position to return to the very first step for the seal, only to find that the instructions rely on the folder’s mastery (or at least understanding) of basic moves like the “crimp fold” or the “inverse-reverse fold.” Fate sealed.
8. Origami Flowers, James Minoru Sakoda, Dover, 1998.
This book, also by Sakoda, focuses on Origami flowers. Interesting aspects include instructions on how to create and use paper with pentagonal and hexagonal symmetry, as present in most blossoms; how to work with free-form asymmetric paper (in order to create a more natural look to the blossoms); how to create paper vases; as well as some basic instructions in the Japanese art of Ikebana, or flower arranging. A beautiful source for more creative advanced student projects!
9. Secrets of Origami: The Japanese Art of Paper Folding, Robert Harbin, Dover, 1997.
Robert Harbin’s “Secrets of Origami” is a collection of many hundred of Origami designs by different Origami artists, condensed into one-page instructions. It therefore necessitates a great familiarity with the symbols and language of paper folding to be able to fold these Origami figures. Secrets may serve to motivate students into exploring their own artistic folding aspirations, as well as a rich resource for more advanced student projects.
10. Build Your Own Polyhedra, Peter Hilton and Jean Pedersen, Dale Seymour Publications, 1994.
“Build Your Own Polyhedra” is a resource with explanations and patterns for cutting and folding paper into flexagons, polygons, polyhedra, and “collapsoids!” Build Your Own also includes some mathematical investigations into Euler’s formula, angle deficiencies, combinatorial properties of polyhedra, as well as symmetries of the cube. While the constructions and braidings are fascinating in their own right, they also provide a nice entry point into “modular” origami where a number of simple folded modules are combined to form larger and more complex creations.
11. Origami Design Secrets: Mathematical Methods for an Ancient Art, Robert J. Lang, A.K. Peters Publishing, 2003.
Origami Design Secrets is the bible of mathematical origami. It is a large, rich and in-depth compendium of the mathematical and geometric principles underlying origami. Going beyond providing folding instruction, Design Secrets exhibits and explains comprehensive methods that allow a dedicated reader to systematically design original origami figures.
12. The Encyclopedia of Origami: The complete, fully illustrated guided to the folded paper arts, Nick Robinson, Running Press.
The Encyclopedia of Origami, by contrast, is a misleading title for Nick Robinson’s book. The bulk of the material consists of very simple origami models that might be interesting for young children. Many of the very exciting designs on the cover appear merely in photographs in the last twenty pages of the book—without any instructions.
13. Inside Rubik's Cube and Beyond, Christoph Bandelow, Birkhäuser Boston, 1982.
This is the best cube book ever written. Besides the clever title, it also has a section on mathematical theory, a solution understandable to the average person, amusing cartoons, an exceptional collection of pretty patterns and maneuvers, and a flowchart for a cube-solving program. The German edition of the book, Einführung in die Cubologie is earlier, and not quite as complete as the English edition.
14. Notes on Rubik's Magic Cube, David Singmaster, Enslow, 1981.
The first book to appear about the cube and the most influential. This book popularized Singmaster's "FLUBRD" notation and other conventions which have become standard. It's not as polished as Bandelow's book, but it appeared much earlier. Appendices were added as new information was discovered. You can see cube history unfold before your eyes as you read them. It's still a classic.
15. Winning Ways (volume 2), by Elwyn Berlekamp, John Conway and Richard Guy, Academic Press, 2003.
It only has a brief section on Rubik's Cube, but it's quite worthwhile. Many other games are discussed, including Conway's Game of Life. Berlekamp has also written many other puzzle-related books.
16. Handbook of Cubik Math, Alexander Frey, Jr. and David Singmaster, Enslow, 2001.
This book supplements Singmaster's Notes on Rubik's Magic Cube, by taking a closer look at the group theory behind the cube.
17. Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, David Joyner, Johns Hopkins University Press, 2002.
Based on lecture notes by the author on the mathematics of the Rubik's cube, which are available in a somewhat more rough version at <http://web.usna.navy.mil/~wdj/rubik_nts.htm>.
Lots of nice ideas, some of which I borrow for the Explorations class. Especially helpful in illustrating abstract algebra ideas in the concrete setting of the cube. For instance, the abstract order of a particular group element reappears concretely with the cube: starting with a solved cube, how often do I have to repeat this sequence of moves before the cube is completely solved again?
18. Rubik's Cubic Compendium, Ernő Rubik, Tamas Varga, Gerzson Keri, Gyorgy Marx and Tamas Vekerdy, Oxford University Press, 1988.
English translation of A Buvös Kocka, with an after-word by David Singmaster.
19. Cubic Circular, David Singmaster.
A short-lived periodical in 8 issues. They are now online at Jaap's page <http://www.geocities.com/jaapsch/puzzles/cubic.htm>, but you are encouraged to purchase print versions from David Singmaster.
20. A lot more material is available on the web, such as Georges Helm's comprehensive collection of cube solutions (nearly 600 items) at <http://webplaza.pt.lu/public/geohelm/myweb/cubbib.htm>
21. The Official Book of Kakuro, Timothy E. Parker, Penguin Group.
22. The Dots-and-Boxes Game: Sophisticated Child's Play, Elwyn R. Berlekamp, A.K. Peters, 2000.
Many books with puzzles are available inexpensively at supermarkets.