Key Concepts

• Success in “solving” Rubik’s Cube depends on discovering short se­quences of moves that accomplish limited goals.
• But the strategy is so successful that the authors yearned for puzzles whose solutions would require novel tactics.
• Basing their work on the mathematical theory of groups so well illustrated by Rubik’s Cube, the authors have devised three new games that challenge today’s generation of puzzle lovers with the complexities of “sporadic simple groups.”

Editor's Note: The online puzzles mentioned in the July magazine can be found here.

Millions of people have been perplexed at one time or another by Rubik’s Cube, a fascinating puzzle that took the world by storm in the 1980s. If you somehow missed the puzzle—or the 1980s—the cube is a plastic gizmo that appears to be made up of 27 small cubes, or “cubies,” stacked into a larger cube, three cubies to an edge. Each of the six square faces of the larger cube is colored in one of six eye-catching colors—typically blue, green, orange, red, yellow or white. We said the cube appears to be a stack of cubies, but appearances here are deceptive. An ingenious mechanism, invented in 1974 by a Hungarian teacher named Erno Rubik (and, independently, in 1976 by a Japanese engineer named Terutoshi Ishige), enables any of the six square faces of the large cube to be twisted about the center of that face. Twist the faces in some random sequence five or six times, and you have a cube so scrambled that only an expert—a cubemeister—can restore order. The object of the puzzle is to put an arbitrarily scrambled cube back into its original state, one solid color per face, thereby “solving” the cube.

Rubik’s Cube, Rubik’s polyhedra and all the many knockoffs that have appeared in the cube’s wake are known as permutation puzzles because they are based on moves that rearrange, or permute, the puzzle pieces (the cubies, in the case of Rubik’s Cube). The object in each case is to restore some scrambled arrangement of the pieces to some predetermined order, often their initial, “virgin” configuration. Permutation puzzles are closely related to a mathematical entity called a permutation group, the set of all the sequences of allowable moves that lead to distinct arrangements of the objects in the puzzle.

In mathematics, a group can be understood as a generalization of ordinary arithmetic. The positive and negative integers 0, ±1, ±2, and so on, together with the operation of addition for combining them, form a group. But groups can be made up of many other kinds of entities as well—the rotations and reflections of physical objects, the various kinds of permutations that can be applied to sets of letters or things, the groupings of numbers called square matrices, and so forth—as long as the group includes some operation for combining the entities in such a way that the combinations, too, are members of the group.

In addition to its interest within pure mathematics, the theory of groups also has powerful applications outside the discipline, in such fields as crystallography, elementary particle physics, string theory and even in telecommunications. So it can be challenging as well as scientifically important for students and working scientists to gain familiarity with the ways that groups behave. Puzzling out a solution to Rubik’s Cube has turned out to be a terrific way for people to get a feel for the ways that the elements of certain kinds of abstract groups combine.

But once people reach that level of mastery with the cube, they often find that their solution strategies are equally effective for solving virtually all the copycat permutation puzzles that it inspired. And, frankly, at that point this kind of permutation puzzle begins to lose its thrill. At least that was our experience with the cube. But we also knew there were good mathematical reasons for our disappointment. All the cubelike puzzles represent groups of a certain general kind, and so they all yield to the same general kinds of attack. Yet those groups by no means exhaust the mathematical diversity of the concept of a group.

What we wanted for educational purposes was an entertaining way to develop people’s intuitions for groups entirely unlike the ones represented by the cube. And what we wanted as puzzle fans was a new set of puzzles whose solutions require a substantially different strategy from that of Rubik’s Cube and its relatives. So we made the natural connection: we were able to develop three new puzzles based on groups known as sporadic simple groups, whose properties are both remarkable and not well known except to specialists. Happily, the experiences of our colleagues show that anyone who can learn to solve Rubik’s Cube can gain an equally substantial understanding of these sporadic simple groups by doing our puzzles. But more, these puzzles are challenging in the sense that they do not yield to the methods that work with Rubik’s Cube—and we think they are a lot of fun. Readers who want to get their hands on them right away can download them.