Handbook of Cubik Math
This is a text project of the book Handbook of Cubik Math by Alexander H. Frey, Jr. and David Singmaster. If you want to make some translation contribution, please contact me [email protected]
Acknowledgments
The works of many people over the last decade have contributed to this book. Not only did Ernö Rubik devise the Magic Cube, but he also has contributed substantially to an understanding of the possible movements of cube pieces. Both John Conway and Tamas Varga are among the early pioneers in the study of cubik math, and to these and many others like them, the authors are particularly indebted.
Much of the material in this book is derived from David Singmaster's Notes on Rubik's Magic Cube. Related to this, there have been literally thousands of letters from many correspondents over the last several years that have contributed to both books.To facilitate the further exchange of new ideas, there is now a cubik circular.Persons may write to D.S. Ltd., 66 Mount View Rd., London N4 4JR, United kingdom for particulars.Interesting mathematical puzzles will also be distributed by this company.
The first U.S. importer of the Magic Cube was Logical Games Inc., Haymarket, Virginia. We are most grateful to its founder, Bela Szalai, whose idea for this book brought the authors together.
Many people have contributed to the preparation of the book. Lowell Smith and Alice H. Frey both read the entire manuscript, eliminating many errors. Cathy Schrott of Jantron Inc. did an outstanding job of page layout and typesetting. Special appreciation is extended to Helen Koehler Frey who not only read the entire manuscript several times, but also supplied the patient encouragement needed to finally complete this project.
Alexander H. Frey, Jr. and David Singmaster
Preface
Handbook of Cubik Math is a book about problem solving and some of the fundamental techniques used in problem solving throughout mathematics and science. Both the problems and the illustrations of concepts for solving them are drawn from Rubik’s Magic Cube.
Ernö Rubik invented the Magic Cube as an aid to developing three-dimensional skills in his students. Little did he realize the impact that this puzzle would come to have. In 1980 alone, approximately five million cubes were sold. Predictions for future years are that sales will continue at more than twice that rate. In almost every neighborhood children – and adults – are playing with the cube.
It certainly enhances three dimensional thinking. However, even greater educational value has been found by mathematicians. For them the cube gives a unique physical embodiment of many abstract concepts which otherwise must be presented with only trivial or theoretical examples. Cube processes are noncommutative – that is, changing the order of movements produces different results. Cube processes generate permutations of the pieces of the cube. Sometimes different processes generate the same permutation so that, by looking at the cube, you cannot tell which process was used. This defines an equivalence between processes. The concepts of an identity process, inverses, the cyclic order of a process, commutators, and conjugates all playa part in solving problems on the cube. By experimenting on the cube, a student learns about these concepts and their relation to problem defining and problem solving without having to rely solely on his faith in the teacher or the text.
Perhaps surprisingly, one of the most fundamental concepts which cubik math illustrates is the use of symbolic notation. It is extremely rare to find anyone who can master the complexities of the cube without writing down what movements he has made or is planning to make. Without a good symbolic notation this is cumbersome at best. For communicating about the cube with others a common notation is mandatory.
The Handbook of Cubik Math in the early technical chapters orients the reader to the basic problem of the cube. It introduces a standard notation – one which is internationally accepted. Then it describes a logical method for restoring any scrambled cube to its pristine state where every face is a solid color. No background of complex or sophisticated mathematical concepts is required in these first three chapters. Many good students in their early teens have mastered these ideas. At the end of Chapter 3, several games are introduced. Playing these will enhance the competitor’s understanding of the concepts inherent in controlled modification of the state of the cube.
One might think that after learning how to solve the cube – that is, how to restore it to its monochromatic-sided state – a person would lose interest in the cube. We thought so before we had taught many people how to solve it, only to find that with their increased understanding came increased curiosity. They wanted to understand more about how the cube worked, why processes produced the results they do, and what they could do to enhance their mastery of the cube.
Seldom does one realize at this point that the concepts which appeared so logical for solving the cube problem are, in fact, the concepts of identities, inverses, commutators, and conjugates. Chapter 4 defines these generalized concepts with many examples and exercises from the cube. These principles are applied to derive new techniques for manipulating the cube. Then in Chapter 5 these improvements are applied to obtain better ways to restore the cube.
It is in Chapter 6 that the mathematical concepts become more sophisticated. It is here that the concept of a group is introduced. The structure and the size of the cube group and its subgroups are explored in Chapters 6 and 7. This leads finally to a discussion of normal subgroups and the isomorphisms of subgroups and factor groups in Chapter 8.
It is expected that some students of the cube will only be ready to absorb material through Chapter 3. Others will be able to work through Chapter 5. The more advanced students will work all the way through to the end. At all stages it is necessary to have easy access to a cube. The cube is the best teacher and experimentation is the best learning technique.