Sean Rogers Possibilities 43252003274489856000 possible states Depends on properties of each face Thats a lot Model each as a set Define R 0 as the solved state r1 r2 r4 r9 b1 b2 b3 b9 w1 ID: 642861
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Slide1
The Mathematics of Rubik’s Cubes
Sean RogersSlide2
Possibilities43,252,003,274,489,856,000 possible states
Depends on properties of each face
That’s a lot!!
Model each as a set
Define
R
_0 as the solved state
{r_1, r_2, r_4 …, r_9, b_1, b_2, b_3 … b_9, w_1 …}
So every set has 54 elements Slide3
Functions
Define f:
R
_x
R
_y
as this:
We have a special name for this: L
Similarly, we have R, U, B, D, R^2, L’, R’, etc.
These functions are
bijections
from one set to another
Obvious- one-to-one correspondence,
|
R
_
x
|=|
R_
y
|Slide4
How to get from A to B
R
_7
R
_6
R
_5
R
_4
R
_3
R
_2
R_1
R_0Slide5
AlgorithmsWe collect these
bijections
into algorithms (macros) to get from one set to another (when you know the properties of the 2 sets required)Slide6
Groups
A group G is (G, *)
G is a set of objects, * is an operator acting on them
4 axioms:
Closed (for any group elements
a
and
b
,
a*b ∈ G)Operation * is associativeFor elements a, b,
and c, (a*b)*c=a(
b*c)There exists an identity element e ∈ G s.t.
e*g=g*e=gEvery element in G has an inverse
relative to * s.t.
=e
Note that commutatively is not necessarily property
Slide7
Examples
Integers are closed under addition
Identity element is 0, inverse of integer
n
is
-n
Rational numbers are closed under multiplication (excluding 0)
Identity element is 1, inverse of x is
Slide8
To Rubik’s Cubes
Our group will be
R,
all possible permutations of the solved state (remember there are ~43 quintillion)
* will be a rotation of a face (associative so long as order is preserved)
Inverse is going the opposite directionSlide9
Cycles and Notation
Cycle- permutation of the elements of some set
X
which maps the elements of some subset
S
set to each other in a cyclic manner, while fixing all other elements (mapping them to themselves)
(1)(2 3 4)
1 stays put, 2, 3, and 4 are cycled in some manner
Ex. {1,2,3,4}
{3,4,1,2} is a cycleYou can’t just switch 2 blocks- permutations are products of 2-cycles
Ex. (1 2 3)=(1 2)(1 3)Analogue- Prime factorizationsSlide10
Importance of Cycles
Parity- amount of 2-cycles that make up a cycle
E
very permutation on the cube
has an even parity
Means you can never exchange just two blocks
We
use at least 3-cycles to reorder blocks in the wrong place
Can now quantify the behavior of different blocks on the cube
Let’s use Ψ
So
describes cycle structure of corners
for edge blocks, and so on
Slide11
Conjugacy
Conjugacy
≈ equivalence relations
Let
A
be some algorithm (macro) that performs an operation on the cube, like a cycle of 3 corner pieces.
Now for some legal cube move
M
,
is the conjugation of
M by AEx. if M
=RUR’U’, then the conjugate of M by F=FRUR’U’F’Do something, do something else, undo the first thing
Conjugacy is an equivalence relationInstead of equivalence classes, we have conjugacy classes
So if we know the conjugacy class of a few blocks, and how they move (Ψ), we have a way of getting from point A to point B (or set A to set B, if you prefer)
Slide12
The CubeSeveral methods to solve
They make even bigger, harder cubes
You don’t need this math though- its just a rigorous way of defining a puzzle
Invented in 1974 by
Ernő
Rubik