This document discusses permutation puzzles and their connection to group theory. It specifically examines three puzzles: the Rubik's Cube, Pyraminx, and Megaminx. For the Rubik's Cube, it provides the history, establishes notation for the sides, cubies (pieces), and basic moves, and discusses how the cube's moves form a non-abelian group with specific structure and properties. The Pyraminx and Megaminx are similarly introduced, with notation and an overview of how their moves relate to group theory.
3. 1 Introduction to Permutation Puzzles
Permutation puzzles (often referred to as twisty puzzles), have been popu-
larized around the world as toys, stress relievers, and mental challenges. A
permutation puzzle is a one player game with a finite number of possible
moves, and where a finite number of moves is always necessary to achieve a
solution. Each puzzle is made up of movable pieces connected to a mecha-
nism that allows their possible movements. In mathematics, a permutation
is defined as a bijective function which maps a set X to itself. In the case of
permutation puzzles, the moves of the puzzles are permutations on the col-
lection of puzzle pieces, rearranging those pieces in some predictable fashion.
Permutation puzzles have five important properties:
1. For some n >1 depending only on the puzzle’s construction, each move
of the puzzle corresponds to a unique permutation of the numbers in
T = {1, 2, ..., n}
2. If the permutation of T in (1) corresponds to more than one puzzle
move then the two positions reached by those two respective moves
must be indistinguishable.
3. Each move, say M, must be ”invertible” in the sense that there must
exist another move, say M-1
, which restores the puzzle to the position
it was at before M was performed.
4. If M1 is a move corresponding to a permutation f1 of T and if M2 is a
move corresponding to a permutation f2 of T then M1*M2 (the move
M1 followed by the move M2) is either not a legal move or corresponds
to the permutation f1*f2.
5. Each puzzle must be given an attainable solved position.
We will be discussing three popular permutation puzzles and the ways in
which they may each be expressed as groups. These puzzles are the Rubik’s
Cube, the Pyraminx, and the Megaminx. However, in our efforts to focus
on the puzzles’ connections to Abstract Algebra, we will not prove that the
above principles hold for each puzzle. Instead, they are given here as a
definition [8].
3
4. 2 The Rubik’s Cube
”It was wonderful, to see how, after only a few turns, the colors became mixed,
apparently in random fashion. It was tremendously satisfying to watch this
color parade. Like after a nice walk when you have seen many lovely sights
you decide to go home, after a while I decided it was time to go home, let us
put the cubes back in order. And it was at that moment that I came face to
face with the Big Challenge: What is the way home?” - Erno Rubik
2.1 History
The first puzzle that we will address is the most iconic of all: the Rubiks
Cube. The standard Rubiks Cube is a 3x3x3 cube with six faces and 26
miniature cubes, or cubies, total. There are six center cubies, twelve edge
cubies, and 8 corner cubies. The Cube, and the other permuation puzzles we
will address, is solved when all of its faces have cubies of the same color.
Figure 1: The Rubik’s Cube
Erno Rubik invented the Rubiks Cube in 1974, and he then later patented
the Cube in 1977. After its release, the Rubiks Cube quickly became popular,
with many left wondering how it could possibly be solved. Approximately
4
5. 350 million Cubes have been sold, with 1/8 of the world population having
laid their hands on a Rubiks Cube at some point. Years after its release, the
Rubiks Cube has continued to fascinate puzzle solvers [1][12].
2.2 Notation
Before venturing into the mathematics behind the Rubik’s Cube, it is essen-
tial that we first have some way to describe both its many positions, and the
permutations which take one to another. In the following subsections, we
will develop a standard notation for the sides, the cubies, and the moves of
a Rubik’s Cube. Thereafter, we will refer to this notation as we develop the
group theory behind the Cube.
2.2.1 Sides
A Rubik’s Cube has six sides, or faces. In the standard notation, the position
of the Cube as a whole is fixed with one side facing us. The sides are then
labeled Front, Back, Left, Right, Up, and Down, as shown in Figure 2. In
an abbreviated form, the set of sides can be written as {F,B,L,R,U,D}.
Figure 2: Sides of the Rubik’s Cube
The choice of which face will become the front face is completely arbitrary,
but once it is chosen, it cannot be modified. In other words, we will never
5
6. rotate the cube as a whole. The reasoning behind this decision will be further
discussed in the section on moves [4].
2.2.2 Cubies
As we inspect the standard 3x3x3 Cube, we can see that it is the conglomerate
of 26 smaller cubes. Each of these smaller cubes is commonly referred to as
a cubie. There may actually appear to be 27 cubies, but there is no solid
cubie within the center of the Cube. Instead, this is the location of the
mechanism by which the Cubes moves are made possible. Figure 3 illustrates
the positions of four individual cubies. The two on the leftmost Rubik’s Cube
are referred to as edge cubies, while the two on the rightmost Rubik’s Cube
are called corner cubies. More generally, the edge cubies are those with
visible faces on two different sides of the Cube, while the corner cubies have
visible faces on three different sides. Although none are highlighted in Figure
3, there are also six center cubies, each of which has only one visible face,
occupying the centers of each of the Cube’s six sides [2].
While we are on the topic of cubies, it may be important to note that many
notations include the concept of a cubicle. A cubicle is not a physical object.
Instead, it is the space occupied by a cubie. Cubicles are typically used
to describe how a specific cubie can move from one position to another.
However, this adds an unnecessary level of complication to our description.
Instead, when we describe the changing positions of cubies, we will simply
say that some move or series of moves transforms one cubie into another
cubie.
Figure 3: Edge Cubies and Corner Cubies
In order to assign unique identifiers to each of the 26 cubies, we will refer to
the cubies by the faces that they touch. Each edge cubie touches two faces,
6
7. corner cubies touch three faces, and center cubies touch only one face. As an
example, the cube on the left in Figure 3 has two cubies highlighted. These
are denoted as UF (Up, Front) and UB (Up, Back). The two cubies high-
lighted on the rightmost cube are denoted FLU (Up, Front, Left) and FRU
(Up, Right, Right). Notice that the three sides contacted by these corner cu-
bies are listed in clockwise order, starting with the highest (vertically). If two
of the sides have the same vertical height, as in the cubie FDL, the frontmost
side is listed first. This is a common convention for permutation puzzles, but
it is completely unnecessary for our purposes [2]. The justification typically
given for such an ordering is that, when discussing every possible position
of the Cube, we must take into account the fact that different orientations
of the same cubie lead to distinct positions. For instance, the faces of FLU
may be colored red, blue, and yellow, or they may be colored blue, yellow,
and red. This is a valid point, but because we are not attempting to solve
the Cube, it turns out that an abstract notation for cubies, without taking
orientation into account, will be sufficient. Therefore, the faces touched by
cubies will be listed in any arbitrary order throughout this paper.
Along those same lines, it is easy to see that any one corner cubie has three
unique orientations, while any edge cubie has two unique orientations. This
”oriented” idea of a cubie is necessary in many situations, including the
actual solving of the Cube [2]. After this explanation, the reason for which
we fix the position of the overall cube becomes obvious. If we defined any
move which could change the positions of the faces, such as a rotation of the
entire cube around its center, we would no longer be able to identify each
cubie accurately.
2.2.3 Moves
The moves of the Rubik’s Cube are tied directly to its six sides. That is to
say that there are exactly six basic moves, and that each move is a clockwise
rotation of one of the six sides of the Cube by 90°. Figure 4 illustrates these
six moves. For another perspective, refer to Figure 2 and imagine rotating
each of the highlighted sides. Like the cubies, these six basic moves will be
denoted by the sides of the Cube that they rotate. The set of basic moves of
the Rubik’s Cube is then {F,B,U,D,L,R}. To differentiate between the six
moves and the six center cubies, we will explicitly identify abbreviations as
moves or cubies when necessary. Furthermore, most of the possible moves of
7
8. the Rubik’s Cube are not one of the six basic moves, but instead combinations
of those moves. These compound moves will be denoted by listing the basic
moves from which they are constructed, in the order in which they must be
applied. For instance, the move consisting of a 180°clockwise rotation of the
top side, followed by a 90°clockwise rotation of the front side is denoted TTF
[2].
Figure 4: Moves of the Rubik’s Cube
There are a few interesting things to notice about the basic moves of the
Rubik’s Cube. The first is that each of the basic moves has an inverse.
These inverses follow trivially from reversing the move, returning all cubies
to their starting positions with a counterclockwise rotation by 90°. We will
denote the set of inverses of the basic moves by {F’,B’,U’,D’,L’,R’}. The
second is that none of them move the center cubies. Because all possible
positions of the Rubik’s Cube are reachable only through combinations of
the six basic moves, this means that the center cubies never move at all.
In fact, this meshes flawlessly with our decision to fix the orientation of the
Cube’s sides. To understand this, it is first important to understand that the
rotation of the Cube’s central ”side” or ”column” (Figure 5) is analogous to
a rotation of the entire Cube about its center, followed by equal and opposite
rotations of the two sides which are parallel to the central ”side.” The key
8
9. concept here is the fact that we have indirectly performed a rotation of the
Cube as a whole. This is exactly what we committed ourselves to avoiding
when we fixed the front side of the Cube. Another way to explain it is that
the only thing uniquely identifying a specific side of the Cube is the color of
its center cubie. When that center cubie moves, the side moves with it.
Figure 5: A Rotation of the Cube’s Center
The overarching reason for avoiding rotations of the Cube as a whole is
simplicity. We all agree that a solved Rubik’s Cube remains solved regardless
of its orientation. The only requirement is that each side is comprised of
faces with only one uniform color. More generally, all possible positions of
the Cube are identical to infinitely many other positions, up to rotation of
the entire Cube around its spatial center. To put this into mathematical
terms, let S be the set of all possible distinct positions of the cubies and let
R be the group of all distinct three-dimensional rotations of the entire Cube
about its center. Furthermore, let us restrict R to be the group generated by
90°rotations about the x, y, and z axes. This way, R has order 6, as opposed to
having infinite order. Now let A be the subset of R consisting of all positions
achievable by fixing the sides of the cube (not allowing rotations in R to be
applied). Then, we can say that S is partitioned into o(A) equivalence classes,
each with 6 elements, which happen to correspond to the six different sides
9
10. of the cube which could be chosen as the front side. By fixing one side as the
front side, we effectively chose one element from every equivalence class of
S, using it to represent the other elements within that class. The results are
identical if we do not restrict the elements of R as well, but each equivalence
class holds infinitely many elements.
3 The Rubik’s Cube and Group Theory
It should not be surprising that the Rubik’s Cube, by its very nature, is a
deeply mathematical construct. Specifically, the Cube can be used as a tool
to demonstrate the occurrences of many of the key concepts of group theory
in the real world. True to its strong ties to abstract algebra, the Cube cannot
be viewed as a group physically. Instead, the permutations (moves) of the
Cube form the group which we will study. However, since this fact may not
be obvious, we will first give the formal definition of a group, followed by a
proof that the permutations of the Rubik’s Cube form one.
3.1 Proof that the Cube’s Moves Are a Group
Definition of a Group: A set G, equipped with a binary operation , called
the product or group operation is called a group provided that
1. a (b c) = (a b) c, for all a, b, c ∈ G. (associativity)
2. There is an element e ∈ G such that a e = e a = a for all a ∈ G.
(Existence of identity)
3. Given a ∈ G there is some b ∈ G such that a b = b a = e. (Existence
of inverse)
Before the proof, a single point is in order. Within this section, we will refer
to the Rubik’s Cube Group as (G, ). The first step in our proof should be
to verify that a binary operation exists, and that (G, ) is closed under it.
This is simple, as we already know that the elements of (G, ) are permuta-
tions (moves). Therefore, the binary operation is the composition of moves.
Furthermore, it is trivial to see that the composition of two moves is another
move (in fact, we have already defined it this way in Section 2.2.3). Let’s
put this into more formal terms. Suppose we have a move M1 which takes a
10
11. cubie C to M1(C). Now suppose that we have another move M2 that takes
the cubie M1(C) to M2(M1(C)). Then
(M1 M2)(C) = M2(M1(C)) (1)
Since (G, ) is closed under the composition of its permutations, we can begin
to prove the requirements.
The associativity of composition of moves is easy for us to understand be-
cause, in the physical world, moves in a sequence can only be performed
from left to right. We can use the move sequence LRT as a simple example.
Whether we write this sequence as L(RT) or (LR)T, the moves L, R, and
T must be performed in that order to return the desired result. Of course,
we will need to transform this intuition into a formal statement. We want
to show that ((M1 M2) M3)(C) = (M1 (M2 M3))(C). Starting with
the left hand side, and utilizing our formal conclusion from the previous
paragraph, we can see that
((M1 M2) M3)(C) = M3((M1 M2)(C)) = M3(M2(M1(C))) (2)
Moving on to the right hand side, we can see that
(M1 (M2 M3))(C) = (M2 M3)(M1(C)) = M3(M2(M1(C))) (3)
It follows that associativity holds.
The existance of the identity does not require a complicated proof. In this
case, the identity move is simply no move at all. We will denote this move,
which takes all cubies to themselves, as I. If M1 is a permutation of the
Rubik’s Cube and C is a cubie, it is easy to see that
(M1 I)(C) = I(M1(C)) = M1(C) = M1(I(C)) = (I M1)(C) (4)
Therefore, an identity exists in (G, ).
Finally, the existance of an inverse for every move is a fact that we have
already covered in Section 2.2.3. Although we only defined the inverses of
the six basic permutations of the Cube, it should be clear that the inverse of
any move consisting of a sequence of basic moves is a sequence consisting of
the inverses of those basic move, applied in reverse order. For instance, the
11
12. inverse of the permutation LTLB is B’L’T’L’. Formally, suppose we have the
move M1 M2, where M1 and M2 are basic moves. Furthermore, suppose
we have the move M2 M1 . Notice that, for any cubie C,
((M1 M2) (M2 M1 ))(C) = (M2 M1 )((M1 M2)(C)) =
(M2 M1 )(M2(M1(C))) = M1 (M2 (M2(M1(C)))) = M1 (M1(C)) = C
(5)
From the other direction, we can see that
((M2 M1 ) (M1 M2))(C) = (M1 M2)((M2 M1 )(C)) =
(M1 M2)(M1 (M2 (C))) = M2(M1(M1 (M2 (C)))) = M2(M2 (C)) = C
(6)
A simple induction argument proves that this holds for sequences of basic
moves of any length. Since any permutation of the Rubik’s Cube may be
expressed as a sequence of basic moves, it follows that every element of (G, )
has an inverse [2].
3.2 Basic Group Structure
Now that we have expressed the Rubik’s Cube as a group, we can begin
to develop the structure of this group. To begin, recall that the elements
of (G, ) are the valid permutations of the cubies on the Cube. The word
”valid” is important here, because not all permutations on the set of cubies
are physically possible. The Rubik’s Cube is made of 48 oriented cubies.
There are eight corner cubies and 12 edge cubies on the Cube. Additionally,
each corner cubie has three possible orientations, while each edge cubie has
two, giving us our sum of 48. At this point, we may be tempted to say
that the Cube group is S48. However, this is where our mention of the valid
permutations becomes relevant. No permutations on the Rubik’s Cube can
transform an edge cubie into a corner cubie or visa versa, so all that we can
say is that (G, ) is a subgroup of S48 [13].
3.3 Group Generators
Because we know that all permutations in (G, ) are formed from the com-
position of six basic permutations, we can say that (G, ) has generators F,
B, U, D, R, and L. We can then write (G, ) as F,B,U,D,R,L .
12
13. 3.4 Non-Abelianess
Next, we can determine whether or not (G, ) is an abelian group, by checking
the commutativity of the group operation. Here, we can look at two moves
in particular: D and R. We will observe their transformation on the cubie
FDL. We can see that
(D R)(FDL) = R(D(FDL)) = R(FRD) = URF (7)
On the other hand,
(R D)(FDL) = D(R(FDL)) = D(FDL) = FRD (8)
Clearly, the two moves D and R do not commute, and therefore (G, ) is not
abelian. It also directly follows from this that (G, ) is not a cyclic group.
3.5 Group Cardinality
One standard notation for permutations is cyclic notation. To refresh our
minds, the definition of a cycle is the following:
Let a1, ..., ak be a sequence of distinct elements of X. Then the cycle σ
associated to the sequence is the map σ : X → X given by
σ(x) =
ai+1 if x = ai and 1 ≤ i < k
a1 if x = ak
x if x /∈ {a1, ..., ak}
We say that the cycle σ has length k, and we denote it by σ = (a1, ..., ak).
Now that we are up to speed, we can express each of the generators of (G, )
as the product of two cycles of length four. These cycles act on the the corner
cubies and the edge cubies connected to whichever side the move represents.
For instance, U = (URF,UFL,UBL,URB)(UF,UL,UB,UR). The cyclic nature
of this move is illustrated in Figure 6.
13
14. Figure 6: A Rotation of the Cube’s Top Side
Because every generator of (G, ) can be written as (C1, C2, C3, C4)(C5, C6, C7, C8)
where Ci are the relevant edge and corner cubies, they can be rewritten as the
product of transpositions (C1, C2)(C2, C3)(C3, C4)(C5, C6)(C6, C7)(C7, C8). Since
there are obviously an even number of transpositions here, every generator
of (G, ) is an even permutation. It follows that every permutation in (G, )
is even, because the product of even permutations is always even. This is
an important fact, because Hannah Provenza from the University of Chicago
gives a nice proof showing that exactly half of the permutations in the group
of all permutations of all cubies are even [10]. Note that this group is not
(G, ), since (G, ) contains only those permutations that are physically pos-
sible. Since there are 12 edge cubies with two orientations each and eight
corner cubies with three orientations each, the group of all permutations of
all cubies contains 12! × 212
× 8! × 38
. By dividing this number by two, we
are one step closer to discovering the true number of elements in (G, ).
The final step in calculating the cardinality of (G, ) is the realization that
the orientation of each cubie has a hand in determining the orientation of
other cubies. The reason is that, at any one time, a cubies is fixed either
in a column or a row of the cube, so that by applying a permutation to it,
we are simultaneously applying that permutation to every other cubie within
that same column or row. Our current calculation automatically takes into
account the fact that, when the positions of all but one edge or corner cubie
have been chosen, the position of the final edge or corner cubie is induced (no
choice is left for its position). However, this combinatorial fact is also true
14
15. for the orientations of the edge and corner cubies. When the orientations
of every edge or corner cubie have been chosen, the orientation of the last
one is also induced. Therefore, we must subtract one from the exponents of
the two and the three in our calculation, leaving us with 12!×211×8!×37
2
as the
cardinality of (G, ) [4].
3.6 Subgroups
Due to the unfathomable size of (G, ), the only viable way to gain an under-
standing of the group is to understand its subgroups. Unfortunately, there
is an astronomically large list of possible subgroups to check, and it cannot
be easily shortened. For instance, we cannot narrow the list by looking for
subsets of (G, ) with a certain order. Lagrange’s Theorem tells us that the
order of a subgroup of a finite group must divide the order of that group. By
factoring our equation for the order of (G, ) into its prime factors, we get the
following expression: 227
×314
×53
×72
×11. Basic counting techniques then
lead us to the conclusion that the number of possible orders for subgroups
in (G, ) is
(27 + 1) × (14 + 1) × (3 + 1) × (2 + 1) × (1 + 1) = 10, 080 (9)
As a result, we cannot begin to classify the subgroups of (G, ) within this
paper, but we will touch on two of the more important ones.
The first of the two subgroups of (G, ) that we will discuss is Co, the group
of permutations which leave every block fixed, but might change their orien-
tations [13]. To prove that this is a subgroup, we must notice a few important
facts. First, the move I is a member of Co, because it leaves every block fixed
(an element of Co is not required to change the orientations of any cubies).
This means that Co is non-empty. Furthermore, if two permutations do not
change the positions of any cubies, the composition of the two (equivalent to
performing both moves, one after another) has the same effect. Therefore,
Co is closed under the binary operation of (G, ). Finally, we know that, if
a permutation M is an element of Co, that permutation’s inverse must not
change the positions of any cubies, because M did not do so. If M−1
were
to change the positions of some cubies, then the move M M−1
would obvi-
ously change the positions of some cubies. Then M M−1
= I, and we have
reached a contradiction.
15
16. As it turns out, Co is not just any subgroup of (G, ); it is a normal subgroup.
Suppose that C ∈ Co and that G ∈ (G, ). We will also say that, for two
cubies C1 and C2, G(C1) = C2 and G−1
(C2) = C1. Then
(G C G−1
)(C1) = G−1
((G C)(C1)) = G−1
(C(G(C1))) = G−1
(C(C2)) =
G−1
(C2) = C1
(10)
Obviously, G C G−1 does not change the positions of any cubies, and so
G C G−1
∈ Co.
The second important subgroup of (G, ) on which we will enumerate is Cp,
the group of permutations in (G, ) which do not alter the orientations of
any cubie, but might transform their positions [13]. The proof that Cp is a
subgroup of (G, ) is nearly identical to the same proof for Cp above. The
identity is clearly in Cp, the composition of two moves which do not alter
the orientations of any cubies cannot possible alter the orientations of any
cubies, and if the inverse of a move in Cp were to modify the orientations of
even one cubie, a contradiction would result. However, Cp is not a normal
subgroup. Suppose we have the moves G ∈ (G, ) and C ∈ Cp. We can
actually choose G to be the move that transforms the positions of cubies,
such that (G C)(C1) = C1 for some cubie C1, but that the orientation of
C1 has been changed. When we are evaluating the move G C G−1
, after
applying both G and C to the Cube, there is no guarantee that G−1
will
undo the change in orientation for C1. Therefore, G C G−1
∈ Cp, and Cp
is not a normal subgroup of (G, ).
Now that we are familiar with the two subgroups Co and Cp, we can show
why they are considered to be interesting. First, notice that the only permu-
tation of the Rubik’s Cube which transforms neither the cubies’ positions,
nor their orientations is the identity (or the empty move). In more formal
language, Co ∩ Cp = I. Furthermore, it is easy to see that, by combining
every permutation in Co with every permutation in Cp, we get every permu-
tation in (G, ). This is obvious because both subgroups cover each others’
defeciencies. As an example, although Co contains no permutations which
transform the positions of the cubies, Cp contains every permutation which
does so. The reverse condition is also true. Therefore, CoCp = (G, ). Since
we already know that Co ¡ (G, ), we can conclude that
(G, ) = Co Cp (11)
16
17. Of course, epressing (G, ) as a semidirect product of two subgroups that we
do not fully understand is not particularily important on its own. What is
important, however, is that we also know how to break Co and Cp down into
direct and semidirect products of simple groups. For instance, there are eight
corners and 12 edges on the Cube, having three possible orientations and
two possible orientations respectively. It is known that the rotation groups
on these edges and corners (the groups of permutations which change their
orientations but not their positions) are abelian, and this fact is quite easy to
visualize. Therefore, we can say that these rotation groups are isomorpic to
Z2 (the edges) and Z3 (the corners) [13]. The intersection of these rotation
groups is clearly the identity, our previous proof may be reapplied to show
that they are all normal subgroups, and their product is clearly equal to Co,
since only the edge and corner cubies can undergo orientational changes. All
of this shows us that
Co = Z7
3 × Z11
2 (12)
The reason that Z3 occurs only seven times instead of eight is the same reason
that we used when calculating the number of elements in (G, ). After all but
one corner of the Cube have been given orientations, the last orientations is
induced. This also applies to the exponent on Z2. Although the explanation
is too complicated to include here, it is also well known that
Cp = (A8 × A12) Z2 (13)
In conclusion, we have broken the massive group (G, ) down to its simple
components in the following way:
(G, ) = (Z7
3 × Z11
2 ) ((A8 × A12) Z2) (14)
This makes the group much easier to study and to understand [13].
17
18. 4 The Pyraminx
4.1 History
Figure 7: The Pyraminx
Although the Rubik’s Cube is by far the most popular of the permutation
puzzles, it is by no means the only one. The second permutation puzzle
that we will address is the Pyraminx. The Pyraminx was invented by Uwe
Meffert in 1971, before the invention of the Rubiks Cube. Thinking that no
one would be interested, Meffert was reluctant to try to market the Pyraminx.
However, after seeing the success of the Cube, Meffert decided to pitch the
idea to Tomy Toys, the third largest toy company in the world. Tomy (the
owner) loved the Pyraminx and decided to try to sell it. As it turns out, the
puzzle was wildly successful. By the following year, Tomy Toys had sold ten
million units [3].
18
19. 4.2 Notation
4.2.1 Sides and Corners
The Pyraminx is a permutation puzzle in the shape of a tetrahedron. This
means that it has four sides and four corners. Like the Rubik’s Cube, the
Pyraminx is best visualized with one side fixed as the front side; rotation of
the Pyraminx as a whole is ill-advised. However, we will see that this front
side is not as easily identifiable as it was on the Cube.
Now we can see that the Pyraminx has four main sides: front (F), down
(D), left (L) and right (R). Additionally, the four corners of the Pyraminx
can then be labeled as up (U), back (B), right (R), and left (L). Although
these notations contains significant overlap, it is sometimes important to
comply with the standard (other times, it is not). Therefore, we will explicitly
differentiate between corners and sides in context. The corners are illustrated
in Figure 8, assuming that the green side is fixed as the front side. You may
have noticed that we had to develop a notation for both corners and sides
here, while an understanding of the Cube required only the sides. This is
because we will be using the sides to denote only some of the cubies, while the
corners will be used to denote others, as well as the moves of the Pyraminx.
Although it may seem like a strange convention, this is how the Pyraminx
is generally understood, and a more detailed justification shall be given in
Sections 4.2.2 and 4.2.3 [11].
Figure 8: The Corners of the Pyraminx
19
20. 4.2.2 Cubies
Although each piece in the Pyraminx puzzle is either an octahedron or a
tetrahedron, it is quite convenient to continue to call them cubies. In fact,
due to the popularity of the Rubik’s Cube, the word ”cubie” is commonly
used to describe the individual pieces of almost every mechanical permutation
puzzle. Once again, we can start by using the sides that cubies touch to
denote them. For example, the top corner cubie could be denoted as FLR,
because its faces lay on the front, left, and right sides of the Pyraminx. This
system works identically in the case of the edge cubies, each of which touches
two faces. However, there exist three other cubie faces on each side as well.
These faces belong to the center cubies, which pose a threat to our current
understanding of these puzzles.
Figure 9: A Face of one of the Pyraminx’s Center Cubies
In Figure 9, the cubie face marked with a ”Z” belongs to one of the center
cubies. Obviously, each center cubie must lie directly underneath a corner
cubie, ”underneath” being a very relative term here. Unlike the rest of the
cubies in the puzzle, the center cubies are octahedrons [14]. This means that
they have eight faces. As it turns out, three of these faces are exposed, with
one appearing on each of the same three sides also touched by the corner
cubie above. Another three of the eight faces are connected to tetrahedron-
shaped edge cubies. The final two sides of the center cubies are hidden within
20
21. the Pyraminx. One is connected to a corner piece, while the other is at the
very center of the Pyraminx itself. What we are getting at here is that
there are exactly four center cubies in the Pyraminx: one for each corner.
Therefore, each will be denoted by the corner that it touches (i.e. B or L).
These and other cubie notations can be observed in Figure 10. It is now that
we begin to see that the corners of the Pyraminx are largely more important
for notational purposes than are the sides [11].
Figure 10: The Cubies of the Pyraminx
4.2.3 Moves
As we have stated, the basic moves of the Pyraminx are denoted using its cor-
ners. Specifically, each move twists either one or two layers of the Pyraminx
about an axis which runs through one of the four corners. The letter used to
denote that corner is then used again to denote the move. The Pyraminx is
constructed from three layers of cubies, emphasized in Figure 9. Using one of
the corners as a rotational axis and fixing the layer farthest from that corner
(which happens to be the side opposite the corner), we can rotate either the
corner itself or the corner and central layer together. The rotation of the
corner piece will be denoted with a lower case letter, while the rotation of
both the corner and central layer will be denoted with an upper case letter.
21
22. Table 1 enumerates these moves.
U rotates the upper corner and corresponding middle layer 60 degrees clockwise
B rotates the back corner and corresponding middle layer 60 degrees clockwise
R rotates the right corner and corresponding middle layer 60 degrees clockwise
L rotates the left corner and corresponding middle layer 60 degrees clockwise
u rotates the upper corner cubie 60 degrees clockwise
b rotates the back corner cubie 60 degrees clockwise
r rotates the right corner cubie 60 degrees clockwise
l rotates the left corner cubie 60 degrees clockwise
Table 1: Moves Notated with Respect to the Corners
As it was with the Rubik’s Cube, each of these moves has its own inverse.
The set of inverses to the basic moves of the Pyraminx is {U,B,R,L,u,b,r,l}.
Furthermore, we can once again define I to be the empty move, or the move
in which no change in the permutations of the cube takes place. I can be
considered as not making any move on the Pyraminx at all, or as a set of
compositions of moves from which no change in the Pyraminx occurs. Some
examples of this would be UUU, RRR, LLL, BBB, uuu, rrr, lll, bbb, UU,
RR, LL, BB,and URRU [11].
It is true that we could duplicate the two-layer rotation by rotating the side
opposite the corner. However, the movement of the center cubies that such
a move initiates is quite difficult to understand. Therefore, although the
rotation of two layers simultaneously may not appear to be a very ”basic”
move, we will definine it as such anyway.
Now that we have seen how the moves of the Pyraminx transform its cubies,
we can see the real reason that the Pyraminx’s center cubies differentiate it
from the Rubik’s Cube. There is no cubie in the entire structure which cannot
be transformed by a basic move, including the center cubies. Although their
positions never change, their orientation does. So, although we used the
center cubies to determine the faces of the Rubik’s Cube, we cannot repeat
this here. Then how can we say that we are fixing any particular face as
the front face of the Pyraminx? Note that we must be able to say this with
certainty. Were this not to be the case, we would be including rotations of
the Pyraminx as a whole in our set of moves. The answer to this question
is the corners. Each corner has three sides, and thus has three colors on it.
22
23. Since there are four colors on the Pyraminx, we know that the side opposite
a given corner must be covered with the one color not represented on that
corner in its solved state. In other words, each side is defined by its opposite
corner. This is another reason to rotate two layers at once as a basic moves.
The alternative rotation of a side actually moves the Pyraminx’s corners
to different positions, inadvertently changing which side is which, and is
therefore equivalent to a rotation of the Pyraminx as a whole. Our current
basic moves change only the orientations of the corner cubies, never their
locations.
4.3 The Pyraminx and Group Theory
Because we have already covered the group structure embodied by one per-
mutation puzzle (the Rubik’s Cube) extensively, this section will be more
of a summary. The fact is that all permutation puzzles have moves which
populate groups with near-identical properties. Therefore, let us begin with
a condensed version of the proof that (G, ), the set of permutations of the
Pyraminx, is a group.
Just as it was for the set of permutations of the Rubik’s Cube, composition
of functions is the obvious choice here for the binary operation of (G, ). Be-
cause the Pyraminx is a closed system, no combination of moves can create
something which is not a move, so (G, ) is closed under its binary operation.
Furthermore, associativity holds in (G, ) for the same reason associativity
held for the Rubik’s Cube group. Moves composed together must always be
physically performed from left to right, regardless of any parenthesis posi-
tioning. Finally, we have already defined both the identity (I) and the set of
inverses to the basic moves of the Pyraminx. Just as it did for the Rubik’s
Cube group, the existence of an inverse for any basic move of the Pyraminx
proves the existence of an inverse for any move at all. As we can see, (G, )
is most certainly a group.
Of course, the simple knowledge that (G, ) is a group is not insightful, nor is
it particularily helpful. We would like to know what kind of group (G, ) is.
The first step is to look at any large groups which may contain it. Like the
Cube, the Pyraminx is a permutation puzzle, and so it is easy to determine
which symmetric group it is a subgroup of. The Cube was a subgroup of S48.
However, we know that the number of possible states of the Pyraminx is not
23
24. 48. The Pyraminx contains four center cubies with three orientations each,
six edge cubies with two orientations each, and four corner cubies with three
orientations each. Since 4 × 3 + 6 × 2 + 4 × 3 = 36, this leads us to conclude
that (G, ) is a subgroup of the symmetric group S36.
We don’t need to stop after finding a group which contains (G, ). Instead,
we should now focus on the internal structure of (G, ). This starts with
knowledge about the generators of (G, ). We already know that the basic
moves which we defined earlier generate every other permutation of (G, ),
so it is clear that (G, ) = U, B, R, L, u, b, r, l . Next, we will want to know
whether or not (G, ) is abelian. As with the Cube group, we only need one
counterexample to prove that it is not. In this case, we will use the moves U
and L. With an actual Pyraminx handy, we would be able to see that
(U L)(FR) = L(U(FR)) = L(FL) = FD (15)
and that
(L U)(FR) = U(L(FR)) = U(FR) = FL (16)
It is therefore obvious that (G, ) is not abelian.
By simply looking at the Pyraminx, we can see that the cardinality of (G, )
must be smaller than that of the Cube. Of course, we can verify this by
applying the same type of calculation. First, let us consider a single move
M1 of the Pyraminx. Suppose that M1 is a rotation of one of the corners.
In that case, the permutation M1 may be written as a single cycle of length
three. For instance, by denoting the three possible orientations for the upper
corner cubie FRL as O1, O2, and O3, the move u may be written as the
cycle (O1,O2,O3). Furthermore, this cycle may be written as a product of
transpositions in the following way: (O1,O2)(O2,O3). Clearly, this move is an
even permutation. One the other hand, M1 might be one of the moves which
rotates two layers at once. In this case, the cyclic notation for M1 is more
complicated. The corner cubie after which the move is named is permuted in
the same way that it would be were this a move which rotated only that cor-
ner cubie. However, the cubies in the second layer of the Pyraminx are also
transformed by the permutation, as so must be represented in the product of
cycles. Interestingly, the center cubie is permuted in exactly the same fashion
as the corner cubie, so the cycle that represents the permutation’s effect on
the center cubie can be written as (C1,C2,C3), where the Ci’s are the three
24
25. possible orientations of that center cubie. Finally, the edge cubies are per-
muted in a straightforward fashion. After these observations, we can see that
the move U may be written as the cycle (FR,FL,RL)(O1,O2,O3)(C1,C2,C3).
Each of the three cycles in this product may be written as a product of two
transpositions, so this is also an even permutation.
Because the proof that exactly half of the permutations of the Rubik’s Cube
are even also applies to the Pyraminx, it may seem as though we can use
the same calculation to determine the cardinality of (G, ). However, the
corner and center cubies the Pyraminx do not change positions. Only their
orientations are transformed by moves. Therefore, the factorial terms dis-
appear. Furthermore, each move affects only the corner and center cubies
located on its axis of rotation, so the orientations of three of those cubies
do not determine the orientation of the last. Therefore, the number of fea-
sible orientations of the four corner cubies and the four center cubies of the
Pyraminx is 24
× 24
= 28
. It is now clear that the number of elements in the
Pyraminx group is 6!×25×38
2
.
5 The Megaminx
5.1 History
The final permutation puzzle that we will discuss is called the Megaminx.
The Megaminx was first invented and manufactured in the 1980s by several
independent groups. Later, Uwe Meffert bought the rights to some of the
patents and began to sell the Megaminx in his shop. The Megaminx is a
symmetric twisty puzzle, with mechanics very similar to those of the Rubik’s
Cube [5].
25
26. Figure 11: The Megaminx
5.2 Notation
5.2.1 Sides
The Megaminx is a dodecahedron. It therefore has 12 sides, each containing
a center piece, 5 corner pieces and 5 edge pieces. Fixing the Megaminx so
that it does not rotate as a whole is straightforward. We will choose one side
to be the front side fixing its position permanently. Unlike the Pyraminx,
each side has an immovable center cubie, which can be used to identify that
side. We can then observe that the Megaminx has twelve sides: Front (F),
Back(B), Up (U), Down(D), Left (L), Right (R), Down-Right (DR), Down-
Left (DL), Up-Right (UR), Up-Left (UL), Back-Right (BR), and Back-Left
(BL). As they were with the Rubik’s Cube, the sides will be used to denote
both the cubies of the Megaminx and its moves. These sides are shown in
Figure 12 [6].
26
27. Figure 12: The Sides of the Megaminx
5.2.2 Cubies
Now that we have the sides, we are able to label the cubies. The procedure
is exactly identicle to that of the Rubik’s Cube. Cubies will be denoted with
respect to the sides that they touch. For example, we have already mentioned
the twelve center cubies which do not touch any other cubies. These cubies
will be named after the side on which they are fixed. There are also 30 edge
cubies, whose faces lie on exactly two sides. These cubies will be labled as
usual. For instance, the cubie that touches the front and lower-right sides
will be denoted FDR, or any other permutation of the three letters. Finally,
there are 20 corner cubies, each of which touches three sides. All of these
notations can be observed in Figure 13. It is important to note that the
cubies of the Megaminx are often denoted with numbers. This is useful in
many cases, but we believe that this notation emphasizes the fact that the
faces of the Megaminx are fixed in a certain orientation. Simply numbering
the cubies does not aid the reader in any sort of understanding [6].
27
28. Figure 13: The Cubies of the Megaminx
5.2.3 Moves
Now that we have done this twice before, naming the moves of the Megaminx
should be trivial. Unlike the pyraminx or the Rubik’s Cube, the Megaminx
has no moves which can move the center cubies (center rotations for the Cube
and side rotations for the Pyraminx). There are only 12 moves possible on the
Megaminx, and each of them is the rotation of one of the 12 sides. Although
these moves leave the center cubies fixed (albeit rotating around their own
centers), the edge and corner cubies are all permuted. Table 2 explains these
12 moves in detail, while Figure 14 illustrates a rotation of the upper face of
the Megaminx.
28
29. Figure 14: A Rotation of the Megaminx’s Upper Face
F rotates the front face 108 degrees clockwise
B rotates the back face 108 degrees clockwise
U rotates the upper face 108 degrees clockwise
D rotates the lower face 108 degrees clockwise
L rotates the left face 108 degrees clockwise
R rotates the right face 108 degrees clockwise
DR rotates the lower-right face 108 degrees clockwise
DL rotates the lower-left face 108 degrees clockwise
UR rotates the upper-right face 108 degrees clockwise
UL rotates the upper-left face 108 degrees clockwise
BR rotates the back-right face 108 degrees clockwise
BL rotates the back-left face 108 degrees clockwise
Table 2: Moves of the Megaminx
Once again, it follows that each basic move has its own inverse. The set of in-
verse moves will be {F’,B’,U’,D’,L’,R’,DR’,DL’,UR’,UL’,BR’,BL’}. Finally,
we once again define I to be the empty move [6].
5.3 The Megaminx and Group Theory
As done with the Rubik’s Cube and the Pyraminx, we will now prove that
the Megaminx Group is, in fact, a group. This will be done concisely as the
proof is similar to that of the Cube. For simplicity, The Megaminx Group
will be referred to as (G, ) from here on out. To begin, the elements of (G, )
29
30. are permutations, so the binary operation will be the composition of moves.
It is obvious to see that two moves composed together will still be a move,
so (G, ) is closed under its binary operation. Next, the moves of (G, ) are
associative, as they must always be performed from left to right. Thirdly, an
identity exists because, for the empty move I and an arbitrary move M1, we
see that I M1 = M1 = M1 I. Finally, there is the existence of inverses as
every clockwise move can be composed with its matching counterclockwise
move to produce I. Therefore, we can see that (G, ) is a group closed under
composition.
Now that the Megaminx has been proven to be a group, we can begin to look
at its structure. There are thirty edge cubies with two possible orientations
each, and twenty corner cubies with three possible orientations each. From
this, we can see that the Megaminx supports 120 unique oriented cubie po-
sitions. Therefore, we know that the Megaminx must then be a subgroup of
S120.
All moves of the Megaminx can be written as a composition of our twelve
basic moves defined above, so we can define the group generators. As a result,
(G, ) is a group with generators F, L, R, U, DL, DR, B, BR, BL, UR, UL,
and D. It follows that we can write (G, ) in the following way:
F, L, R, U, DL, DR, B, BR, BL, UR, UL, D
To check commutativity of the Megaminx, we will look at two different moves
and the effects they will have on a cubie when performed in a different order.
In particular, we will compare (F B)(DLF) and (B F)(DLF). We observe
that
(F L)(DLF) = L(F(DLF) = L(LF) = LDL (17)
We can also see that
(L F)(DLF) = F(L(DLF) = F(DLF) = LF (18)
From this example, we can clearly see that these two elements of (G, ) do
not commute, and thus (G, ) is not abelian.
In order to accurately count the number of elements in (G, ), we must first
determine whether its permutations are even, odd, or a mixture of the two.
30
31. This process is easier than it was for the Pyraminx, due to the fact that every
feasible move is the rotation of a side. We can see that each permutation
may be written as the product of two cycles of length five. This is because
there are five corner cubies and five edge cubies on each side. As an example,
the move F may be written as
(FRU, FRDR, FDLDR, FLDR, FLU)(FR, FDR, FDL, FL, FU) (19)
This, in turn, may be written as the following product of transpositions:
(FRU, FRDR)(FRDR, FDLDR)(FDLDR, FLDR)(FLDR, FLU)(FR, FDR)
(FDR, FDL)(FDL, FL)(FL, FU)
(20)
Clearly, every move is an even permutation. It follows that calculations for
the cardinality of the Megaminx are nearly identical to that of the Rubik’s
Cube. However, the Megaminx instead contains thirty edge cubes with two
possible orientations, and twenty corner cubies with three possible orienta-
tions. Our calculation for cardinality of the Megaminx Group will then be
30!×229×20!×319
2
.
6 Conclusion
In conclusion, we have studied the group structures formed by the permuta-
tions of three different popular puzzles. The Rubik’s Cube, the Pyraminx,
and the Megaminx may be the most well-known permutation puzzles, but
they are centainly not the only ones. In fact, it is true that every permuta-
tion puzzle, by its very nature, has a group structure similar to those that
we have discussed here. This group structure may even be expanded trivially
to the many sizes and variations of each puzzle (with the exception of the
Pyraminx), and non-trivially to the extra-dimensional counterparts to these
puzzles. We also discovered how we can break these massive subsets of the
symmetric group down into simple groups, allowing us to understand their
structures more fully. Although this paper encompassed quite a lot, there
is still much room for improvement and expansion, especially into the topic
of general, n-dimensional permutation puzzles. We hope that our work has
provided a strong platform from which others might discuss these topics, and
learn about a fascinating application of abstract algebra.
31
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