Rubic Cube Basics and Groups.pdf

FUN WITH RUBIK’S CUBE
Adopted from Tom Davis 1
(1) Do NOT jumble your physical cube, I DO NOT KNOW HOW TO SOLVE IT!
(2) Download the Rubik computer program at http://www.geometer.org/rubik/
(3) Basics of Rubik’s cube
• The small cubes which make the Rubik cube are called cubise. How many
visible and invisible cubies are in Rubik’s cube?
• How many cubies show only one face, they are called face cubies or center
cubies)?
• How many cubies show two faces, they are called called edge cubies (edgies)?
• How many cubies some show three faces, they are called the corner cubies
(cornies)?
• We may call facelets the little faces on the cubies. How many visible facelets
are in Rubik’s cube?
(4) Move Descriptions
• The letters: U, L, F, R, B and D correspond to quarter-turn clockwise twists
about the up, left, front, right, back and down faces, respectively. Clockwise
refers to the direction to turn the face if you are looking directly at the face.
If you hold the cube looking at the front face, the move B appears to turn the
back face counterclockwise. The lower-case letters, u, l, f, r, b and d, refer to
quarter-turn counterclockwise moves about the respective faces.
• Inverse Operations.
1 http://www.geometer.org/rubik/group.pdf
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– Apply an F move, you can undo that by turning the same face a quarterturn
counter-clockwise by doing an f move.
– Apply move F followed by R. How can you undo this move?
– In mathematics, an operation that undoes a particular operation is called
the inverse. This is usually denoted by writing a −1 exponent. We can
write F −1 in place of f , or f = F −1 . In algebra we write 1 2 = 2−1 .
– What is (FR) −1 =? Explain the double reversal idea.
– Find an inverse of the sequence f f RuDlU, and check if it works using
the Rubik computer program (be VERY careful if you use your physical
cube).
• Powers.
– We will write FFF = F 3 , the 3 in the exponent indicates that the operation
is repeated three times.
– Write FRFRFRFRFR as a power.
– What is F 9 =?
– We call the operation of doing doing nothing the identity operation and
will label it here as 1 (like when you multiply a number by 1, you do
nothing.
– What is F 0 =?.
• Commutativity and Non-Commutativity.
– The order in which you apply moves makes a difference. Take your
physical cube and apply an FR to it and have your friend apply RF to
his/her cube. FR ≠ RF.
– In ordinary arithmetic 7 × 9 = 9 × 7. When the order does not matter,
as in multiplication of numbers, we call the operation commutative.
If the order does matter, e.g. for division of numbers (give an example)
then we say that the operation is non-commutative.
– Find in your cube moves that are commutative. Find number divisions
that are commutative.
• Order.
– Perform any move you want 4 times. The resulting cube returns to a
solved state. We can write F 4 = 1.
– How many times do you need to apply FR to return to the solved cube?

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– The smallest number of repetitions for a move, or for a sequence of
moves, to go from a solved cube to a solved cube, is called the order of
this move, or of a sequence of moves.
– Find the order of FFRR using a physical cube (dont make a mistake).
– Check your answer with Rubik. Here is how: Reset the cube to solved
and type FFRR into the window labeled CurrentMacro. Then press the
Macro Order button just above the window in which you just typed, and
Rubik will pop up an information window showing you the order that
it calculated.
– Apply the FFRR operation repeatedly until the cube is solved to check
the answer. Here is how: With the cube in Rubik solved and FFRR
in the CurrentMacro window, click the Apply Macro button. Click the
same Apply Macro button again and again until the cube returns to
solved.
– Reset the cube again, use the same FFRR is in the Current Macro window,
click in that Current Macro window with the mouse just before the
first F and press the return key repeatedly on your keyboard. Rubik
then twists the cube faces as you watch.
– Explore Rubik.
– Start with a solved cube. Try a different sequence repeatedly using
Rubik, will you get back to the solved cube?
(5) Important Property of Rubik’s Cube: Any cube operation has an order. This means
that if you begin with a solved cube and repeatedly apply the same move, or a
sequence of moves, you will return to a solved cube after a finite (could be 4,
1000, or more) number of steps.
• Solve this problem: The number of possible rearrangements of facelets is
smaller than what number?
• Recall the Pigeonhole Principle. If more than K pigeons are to be placed in
K pigeonholes then there must be at least two pigeons in one pigeonhole.
• What number of repetitions of any operation has to get you through two identical
states of Rubik’s cube? What are the pigeon and pigeonholes here?
• If you need K repetitions to go between two identical states, how many repetitions
do you need to go from a solved cube to a solved cube?
(6) Useful Macros
• As you know FFRR has order 6. The number 6 is divisible by small numbers
2 and 3. What does (FFRR) 2 do to Rubik’s cube? How about (FFRR) 3 ?

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• Try a different Macro. Find its order. Find small prime divisors of the order,
like 2, 3 or 5. If k is such a divisor, see if your Macro repeated k times is
useful.

What is a Group?
A group G consists of a set of objects and a binary operation ∗ on those objects satisfying
the following four conditions:
1. The operation ∗ is closed. In other words, if g and h are any two elements of the
group G then the object g ∗ h is also in G.
2. The operation ∗ is associative. In other words, if f, g and h are any three elements
of G, then (f ∗ g) ∗ h = f ∗ (g ∗ h)
3. There is an identity element e in G. In other words, there exists an e ∈ G such
that for every element g ∈ G, e ∗ g = g ∗ e = g.
4. Every element in G has an inverse relative to the operation ∗. In other words, for
every g ∈ G, there exists an element g −1 ∈ G such that g ∗ g −1 = g −1 ∗ g = e.
Examples of Groups
The integers under addition,
The rational numbers under addition,
The rational numbers except for 0 under multiplication,
The real numbers under addition,
The real numbers except for 0 under multiplication.
Rubik cube moves. Why?
All of these groups are infinite and commutative.
(Commutative means that a∗b = b∗a for every a and b in the group.)
The natural numbers (= {0, 1, 2, 3, . . .}) under addition do not form a group. Why?
We can’t include zero in the rational or real numbers under multiplication. Why?
In the modular arithmetic, if the operation is addition modulo n, the n elements
0, 1, . . . , n − 1 form a group under that operation.
For any particular geometric object, the symmetry operations form a group. A symmetry
operation is a movement after which the object looks the same. For example, there are 4
symmetry operations on an ellipse whose width and height are different:
1: Leave it unchanged
a: Rotate it 180◦ about its center
b: Reflect it across its short axis
c: Reflect it across its long axis

The group operation consists of making the first movement followed by making the
second movement. Clearly 1 is the identity, and each of the operations is its own inverse.
We can write down the group operation ∗ on any pair of elements in the following table:
∗ 1 a b c
1 1 a b c
a a 1 c b
b b c 1 a
c c b a 1
The group of symmetries of an equilateral triangle consists of six elements. You can
leave it unchanged, rotate it by 120 ◦ or 240 ◦ , and you can reflect it across any of the
lines through the center and a vertex.
In the same way, the group of symmetries of a square consists of eight elements: the four
rotations (including a rotation of 0 ◦ which is the identity) and four reflections through
lines passing through the center and either perpendicular to the edges or the diagonals.