/
The Mathematics of  R ubik’s Cubes The Mathematics of  R ubik’s Cubes

The Mathematics of R ubik’s Cubes - PowerPoint Presentation

celsa-spraggs
celsa-spraggs . @celsa-spraggs
Follow
362 views
Uploaded On 2018-03-08

The Mathematics of R ubik’s Cubes - PPT Presentation

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

elements set conjugacy blocks set elements blocks conjugacy cycles element inverse cube cycle group closed equivalence identity cubes state operation move parity

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "The Mathematics of R ubik’s Cubes" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

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