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.

Puzzles and Their Groups
To solve the new puzzles, it is useful to understand something about the sporadic simple groups from which they are constructed, as well as how they differ from the group represented by Rubik’s Cube: “Rubik’s group.” Groups can be infinite or finite in size. The additive group of integers we mentioned earlier obviously has infinitely many members. But the number of elements in Rubik’s group is finite, even though the set of all the allowable sequences of moves of Rubik’s Cube is infinite. The reason is that if two sequences of moves lead from the same starting arrangement of cubies to the same end point, the sequences are regarded as equivalent. In Rubik’s Cube the number of distinct con­figurations of cubies is astronomical—about 4 X 1019, or 43,252,­003,274,489,856,000, to be precise—so the number of elements, or distinct combinations of moves in the group represented by the cube, is gigantic but finite.

In spite of that vast “space” of moves, it is not hard to devise a solution to the cube by following a few general hints. You need a pencil, paper and one Rubik’s Cube, preferably unscrambled. Your aims are twofold: First, you want some convenient way of recording your moves. Second, you want to discover various short sequences of moves that you can write down for accomplishing specific tasks: exchanging certain pairs of corner cubies or edge cubies, for instance. The idea is to combine the sequences systematically to solve a scrambled cube.

It turns out that this systematic approach, begun by trial and error, almost invariably leads to useful sequences that give you enough flexibility to solve the cube. Roughly speaking, the reason is that the basic algebraic components of Rubik’s group are the so-called symmetric groups, which are groups of all possible permutations of a given number of objects, and their close relatives the alternating groups, each of which contains half the elements of the corresponding symmetric group. Thus, the symmetric group S3 contains all 3! (1 X 2 X 3), or six, possible permutations of three objects; its relative, the alternating group A3, has three elements. Among the symmetric groups related to Rubik’s group are the symmetric group S8 (all the 8!, or 40,320, ways the eight corner cubies can be rearranged) and the symmetric group S12 (all the 12!, or 479,001,600, ways the 12 edge cubies can be rearranged).

“Atoms” of Symmetry
Our puzzles, too, are permutation puzzles, but each of them is based on a so-called sporadic  simple group. To understand what a sporadic simple group is, begin with the concept of a subgroup. Suppose you are allowed to twist only the blue and yellow faces of Rubik’s Cube. Under that restriction you will never be able to move the side cubie colored green and white. Hence, the number of distinct sequences of restricted moves is smaller than the number of elements in Rubik’s group as a whole. Whenever all the combinations of some subset of the moves in a puzzle group are also moves within the subset, the subset is called a subgroup. Beyond that point the concept of a simple group is somewhat technical; suffice it to say that a simple group is a group that contains no “proper, normal” subgroups.

The term “simple” as it is applied in group theory may be one of the greatest misnomers in the history of mathematics. The simple groups have turned out to include some of the most complex entities known to mathematicians. Yet they are simple in the sense that they are the building blocks, or “atoms,” of group theory. In a way, the simple groups are also like the prime numbers, numbers divisible only by themselves and by 1 (2, 3, 5, 7, 11, and so on). Every finite group can be uniquely “decomposed” into simple groups, just as any whole number can be factored into primes.

All the finite simple groups have been identified and classified. They were discovered between the 1860s and 1980, and the classification was done mostly between the late 1940s and the early 1980s (with some more recent corrections), involving the work of hundreds of mathematicians. Reports of discoveries of simple groups and the proof that the final list was complete consumed more than 10,000 pages in professional mathematics journals, distributed across some 500 articles. Mathematicians are still working on a simpler version of that proof, which could clarify their understanding of the simple groups. But the proof already in hand shows that there are 18 families of finite simple groups—each family an infinite collection of specific kinds of groups—and 26 so-called sporadics that are oddballs, essentially mathematical entities unto themselves. There are no others.

We constructed puzzles based on the three sporadic simple groups known as M12, M24 and Co1. Those puzzles, like Rubik’s Cube, are permutation puzzles, but the permutations that represent sporadic simple groups are far more restrictive about the allowed permutations than the symmetric groups are. Thus, in our puzzles many arrangements of numbers are inaccessible, no matter how many moves one makes.

As we noted earlier, the strategy that works for solving the cube and other puzzles based on symmetric groups does not work for our new puzzles. But other strategies can be developed from only small hints about the groups.

The simplest of our three puzzles is M12, based on the sporadic simple group of the same name. The M12 group is one of the first five sporadic simple groups ever discovered; all five were found in the 1860s by French mathematician Émile Mathieu and are dubbed Mathieu groups. The would-be puzzle solver confronts a specially scrambled sequence of the numbers 1 through 12, arranged in a row. Only two moves are allowed, though they can be applied any number of times in any sequence. The object of the puzzle is to put the scrambled arrangement back into ordinary numerical order (1, 2, 3, ..., 12).

We will give just one hint to readers bold enough to take on our challenge. In the puzzle (and in the group M12), it is possible to move any five of the numbers to any five of the 12 positions in the row. Once that is done, all the remaining numbers fall into position; the puzzle is solved. The reason is that the group M12 has 12 X 11 X 10 X 9 X 8, or 95,040, permutations, which happens to be exactly the number of ways of selecting any five of the 12 numbers and placing each of them somewhere in the sequence. (The first number can take any one of the 12 positions, the second any one of the remaining 11 positions, and so on.) The fact that the entire permutation is specified by fixing the positions of five numbers implies that it is pointless to search for a sequence of moves that would shift only a few numbers. Except for the so-called dummy, or null, move, which leaves any arrangement just as it is, every move must leave fewer than five numbers fixed. In other words, every nontrivial sequence of moves must displace at least eight of the 12 numbers.

Puzzles Not for the Faint of Heart
Our second puzzle, M24, includes 23 numbers arrayed in a circle, as if on the face of a clock, and a 24th number placed just outside the circle at 12 o’clock. As in the M12 puzzle, just two moves are allowed. In principle, the M24 puzzle could be manufactured from real parts rather than just represented by a computer: the circle of 23 numbers could be moved by a rotating device, and a system of gears could swap pairs of numbers as dictated by the moves.

The group of permutations generated by the two moves of M24 is the Mathieu group M24. Like M12, M24 is “five-transitive”: with some combination of the two moves, it is possible to manipulate the arrangement until any five of the 24 numbers are positioned in any five of the 24 positions. Because of five-transitivity, our hint for solving the M12 puzzle helps in solving M24 as well: devise moves that return the numbers 1 through 5 to their proper positions without disturbing the numbers already in place. But this time the solver is not quite done. The group M24 has 24 X 23 X 22 X 21 X 20 X 48, or 244,823,040, elements; thus, even after the numbers 1 through 5 are returned to their proper places, the other 19 numbers can still be distributed around the circle in 48 different ways.

Dotto, our final puzzle, represents the Conway group Co0, published in 1968 by mathematician John H. Conway of Princeton University. Co0 contains the sporadic simple group Co1 and has exactly twice as many members as Co1. Conway is too modest to name Co0 after himself, so he denotes the group “.0” (hence the pronunciation “dotto”).

We will have to leave the details of the Dotto puzzle to our online discussion. But we can point out that both the puzzle and its underlying group have fascinating mathematical properties. The puzzle is closely related to the Leech lattice, a set of “points,” or ordered lists of numbers, in a 24-dimensional space. It is known that among all sphere packings in 24-dimensional space constructed by centering 24-dimensional “spheres” on the points of a lattice, a sphere packing based on the Leech lattice is the tightest.

Of Babies and Monsters
Only four sporadic simple groups surpass Co1 in size: the Janko group J4, the Fischer group Fi24, the Baby Monster B and the Monster M. True to its name, the Monster is the largest of them all, with some 8 X 1053 elements. It was constructed in 1980 by Robert L. Griess, Jr., of the University of Michigan at Ann Arbor as the group of transformations of a certain complicated mathematical structure in 196,884-dimensional space.

We have not tried constructing puzzles on the basis of any other sporadic simple groups—although some are surely possible. But designing any workable puzzle based on the Monster would be a serious mathematical undertaking. The reason is that it is not known whether the Monster is the permutation group of any object small enough to be visualized, although, according to one conjecture, it is the permutation group of a certain 24-dimensional curved space. A successful design of a “Monster puzzle” might bring mathematicians closer to proving this tantalizing conjecture.

Note: This story was originally printed with the title, "Simple Groups at Play."