Rubik's Cube (HAW)

8/3/2019 Rubik's Cube (HAW)

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Advanced Artificial IntelligenceCS 702 ± Fall ( 2011 / 2012)

Rubik¶s Cube

Presented by (Presented by (HAW):HAW):

HatimHatim Al Sum Al Sum Wael IbrahimWael Ibrahim

Ahmed Faisal Ahmed Faisal

PresentedPresented to:to:

Dr Dr.. HeshamHesham ElEl--DeebDeeb



Agenda

(1) Introduction History

What is Rubik¶s Cube

How the Cube Works

Puzzle Objective

Puzzle Complexity

(2) The Algorithm Notations

Manual Solution Steps

Optimal Solution

(3) Demo Rubik¶s Cube Solver Robot ± Videos

Rubik¶s Cube Solver Simulator ± Application



Agenda

(1) Introduction(1) Introduction History


How the Cube Works

Puzzle Objective

Puzzle Complexity


Manual Solution Steps

Optimal Solution





History

Rubik's Cube?! Who is Rubik¶s and why the name?

A 3-D mechanical puzzle

invented in 1974 by Hungarian

sculptor and professor of

architecture Ern Rubik.

Originally called the "Magic Cube"



History

The puzzle was licensed by

Rubik to be sold by Ideal Toy

Corp. in 1980 and won the

German Game of the Year special

award for Best Puzzle that year.

As of January 2009, 350 million

cubes had been sold worldwide

making it the world's top-sellingpuzzle game.

It is widely considered to be the

world's best-selling toy.




In a classic Rubik's Cube,

each of the six faces is covered

by nine stickers, each of one of sixsolid colors, (traditionally white,

red, blue, orange, green, and

yellow).

A pivot mechanism enables

each face to turn independently,thus mixing up the colors.



Rubik¶s Cube Puzzle

Initial State Goal State



Initial State Goal State

In order to solve the Rubik Cube puzzle; each face must

be returned to consisting of one color.

Puzzle Objective



How the Cube Works

Taken apart, one can see that the cube consists of 6 centers, 8 corners,

and 12 edges:

The centers are fixed on a central mechanism, so they don¶t move relative

to each other

NoteNote::

TheThe color of a center color of a center determinesdetermines whatwhat color that facecolor that face will be when thewill be when the

cube is completedcube is completed



Puzzle Complexity



Puzzle Complexity



Number of Configurations

The original (3×3×3) Rubik's Cube has eight corners and twelve edges.

There are 8! (=40,320) ways to arrange the corner cubes.

Seven can be oriented independently, and the orientation of the eighth

depends on the preceding seven, giving 3^7 (=2,187) possibilities.

There are 12!/2 (=239,500,800) ways to arrange the edges, since an odd

permutation of the corners implies an odd permutation of the edges as well.

Eleven edges can be flipped independently, with the flip of the twelfth

depending on the preceding ones, giving 2^11 (=2,048) possibilities



43,252,003,274,489,856,000

If one were to make one move every second, it would take1,370,000,000,000 (1.37 trillion) years to reach every case,assuming no repeats

That amount of time spans the age of the universe ten times

If this person began to twist from the moment the dinos died out (65MYA), he/she would have reached only 4.7% of all the cases

If a cube existed for every configuration, and all were lined up nextto each other, they would circle the earth five times

That is 124,508 miles, or about half the distance from Earth to themoon



Rubik¶s Cube Importance












Agenda



How the Cube Works Puzzle Objective

Puzzle Complexity

(2) The Algorithm (Planning)(2) The Algorithm (Planning) Notations

Human Planning: Manual Solution (Steps with Example)

Computer Planning: Optimal Solution (Advanced Algorithms)


Rubik¶s Cube Solver Simulator ± Applications



Solve by Planning

In Rubik's cubists' parlance,

a memorized sequence of moves that has a desired effect on the cube

is called an algorithmalgorithm.

This terminology is derived from the mathematical use of algorithm,

meaning a list of well-defined instructions for performing a certain task from

a given (initial state), through well-defined successive states, to a (desired

end-state).

Each method of solving the Rubik's Cubeemploys its own set of algorithms, together with descriptions of what the

effect of the algorithm is, and when it can be used to bring the cube closer

to being solved.



Notation

Everyone needs to know the notation used for cubing!

Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by David

Singmaster to denote a sequence of moves, referred to as "Singmaster

notation"

For example, what do these mean:

F (R U R¶ U¶) F¶

(R2 U)(R U R¶ U¶)(R¶ U¶)(R¶ U R¶)

(M¶ U)*4 (M U)*4

These are algorithms. Each letter represents a move to be carried out in

sequence, and represents a face of the cube.

There are 6 faces, named (F)ront, (B)ack, (U)p, (B)own, (R)ight, (L)eft.

There are also 3 slices, named (M)iddle, (E)quator, (S)ide



Notation



Notation

F = front face B = back face R = right face L = left face

U = up face D = down face

In addition to a letter, each move may be accompanied by an apostrophe or the number two: A letter by itself means turn that face 90 degrees clockwise (eg. F).

A letter followed by an apostrophe means turn that face 90 degrees anti-clockwise (eg. F').

A letter followed by the number 2 means turn that face 180 degrees (direction is irrelevant),

(eg. F2).

So R U' L2 is shorthand for "turn the right face 90 degrees clockwise, then

turn the up face 90 degrees anti-clockwise, then turn the left face 180

degrees". When thinking whether to turn clockwise/anti-clockwise, imagine

that you are looking directly at the particular face you are turning.



Notation: U and D

U

U¶

U2

D

D¶

D2



Notation: F and B

F

F¶

F2

B

B¶

B2



Notation: R and L

R

R¶

R2

L

L¶

L2



Human Planning

There are many different methods for solving the

Rubik's cube.

They can be divided into two broad categories ±

1) Layer methods

2) Corners first methods

(and there are sub-categories within these broad categories).

The following method is a layer based method. This

beginner method requires memorizing only a few

algorithms, and when done efficiently can achieve solves

of 60 seconds or faster.



Beginners Plan

The method you will learn consists of 6 major steps, and each stepis broken down into smaller steps

1. Top edges

2. Three Top Layer corners

At this point, we have all but one top layer cube solved

3. Three Middle Edges

Now we (almost) have 2 layers done. Flip over the cube so the last layer (LL) is on top.

4. Solve the remaining corner and edge

Now we have 2 layers (of three) solved. 5. Orient Last Layer (OLL)

Edges first, then corners

6. Permute Last Layer (PLL)

Again, edges first, then corners



The Solution, Step by Step

1. Top Edges 4. Finish 2 Layers

2. Top Corners 5. Orient LL

3. Middle Edges 6. Permute LL



Top Edges

Also called the cross

We will use orange as our top face

This is the first step towards completing

the top layer Complete one edge at a time

Experienced cubers usually solve the cross

consistently in 6 moves, < 2 seconds

Reasonable average is 15 moves, 10 seconds with some practice



Cross - Details

Oriented vs. non-oriented edge

Your first goal is to locate an edge that makes up the cross

This means that it has orange as one of its two stickers

Remember that the center of a face determines what color it will be:

<-- Bad edge

Good edge -->

<-- Bad cross

Good cross -->



In case of a bad cross«

«do the following:

Rotate the top layer until at least two of the edges are in place

(you will always be able to get two)

Match your cube with one of the following cases Execute the sequence to place the edges that are out of place

These two edges

need to switch

places

These two edges

need to switch

places

R¶ U¶ R U R¶ L R¶ U2 L¶ R



Corners

Solving the corners completes the top

layer

This step is easier than the cross since

there are fewer possibilities



Solving a Corner

The only thing to keep in mind is you can¶t mess up the

cross

To make progress, you must temporarily disturb what you have

already solved, then restore it Hopefully, when you restore a few solved pieces, progress gets

made

To solve a corner, find one that belongs in the top layer

Move it directly beneath where it belongs

Corner belongs here

Move corner to here



Finding a Corner

Remember that the centers don¶t move

So the corner (which has three stickers) depends on the color of three

centers:

So, the corner that belongs in this spot must be orange, green, and white

This is the process used to figure out where a piece belongs

Use it to figure out where any piece belongs

This must be orange

This must be greenThis must be white



Three Possibilities

Once you have a corner in the right spot in the bottom layer, you

must determine it¶s orientation.

F D F¶ R¶ D¶ R R¶ D R F D2 F¶



The Other Corners

Follow this step for the other three corners

When 3 corners are done, it should look

like this:

It¶s important to have

the ³T´ on each face



Middle Layer

Flip over the cube so the top layer becomes the bottomlayer

Find an edge that belongs in the middle layer

Hold cube so

the edge willend up here,

in FR

Make sure theempty corner is

below the edge

you want to

solve



Inserting an Edge

There are two flips of an edge

Thus there are two cases for inserting an edge

U F¶ U¶ F R U¶ R¶

Follow these steps until three of the middle edges are

solved



The Other 2 Edges

Always hold the cube so the edge that you wish to solvewill end up in FR

This means you will have to change the orientation of thecube in your hands

You will also need to rotate the bottom layer so theempty corner is just below where the edge is to beplaced



Top Layer

At this point, 2/3 of the cube is done

Last Layer Strategy:

1. Orient Edges

2. Orient Corners

3. Permute Edges

4. Permute Corners

After orientation After permutation



Orient Edges

Three cases

[A] = F U R U¶ R¶ F¶ [B] = F R U R¶ U¶ F¶ [A] U2 [B]



Orient Corners

Seven cases

R B L¶ B¶ R¶ B L B¶ R2 D R¶ U2 R¶ D R U2 R L¶ B¶ R¶ B L B¶ R B

L¶ U R U¶ L U R¶ R U¶ L¶ U R¶ U¶ L



Orient Corners

R¶ U2 R2 U R2 U R2 U2 R¶ R¶ U¶ R U¶ R¶ U R¶ U R¶ U2 R¶



Permute Edges

Rotate the top layer until you have one of the following cases

for edge permutation:

These two cases show that only one edge is permuted.

It is possible that you won¶t encounter either of these cases inany particular solve. There is one other case:



Permutation

You must learn at least one of the first two cases:

Breaking down each sequence into groups of fewer moves makes it

easier to learn long sequences

These two sequences are broken into groups (indicated by

parentheses) that make learning them easier.

(R2 U) (F B¶) R2 (F¶ B) (U R2) (R2 U¶) (F B¶) R2 (F¶ B) (U¶ R2)



The Other Case

In case you want to learn an extra

sequence, here is the solution

(R B¶ R¶ B)(F R¶)(B¶ F)(R¶ B R)(F2 U)



Permute Corners

There are four possibilities for corner permutation

Do not rotate the top layer like you did for edges (that would mess

up the edges!)

Here are the possibilities:

Case 1 Case 2 Case 3 Case 4



The Final Step«

You only need to learn one of these cases

(either c1 or c2)

They have similar solutions

R¶ F R¶ D2 R F¶ R¶ D2 R2 R B¶ R F2 R¶ B R F2 R2



You Are Done!

You now know the way to avoid nearly all of the 43

quintillion combos of a cube

Now you need to learn how to become faster by learning

the following:

Finger tricks/triggers

Lubing your cube

Solve from bottom -> top

Learn all 57 orientations!

Learn all 21 permutations!



Agenda




Puzzle Complexity

(2) The Algorithm (Planning)(2) The Algorithm (Planning) Notations

Human Planning: Manual Solution (Steps with Example) Computer Algorithms: Optimal Solution (God¶s Algorithms)


Rubik¶s Cube Solver Simulator ± Applications



Computer Algorithm

There are many algorithms to solve scrambled Rubik's Cubes.

The maximum number of face turns needed to solve any instance of

the Rubik's cube is 20.

This number is also known as the diameter of the Cayley graph of

the Rubik's cube group.

An algorithm that solves a cube in the minimum number of moves isknown as God's algorithm.

http://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik's_Cube



A History of God's Number

http://www.cube20.org/



Search Technique

God¶s Algorithm Is based on IDA* which is a variant of

the A* search algorithm which uses iterative deepening

to keep the memory usage lower than in A*.

IDA* is a depth-first search that looks for increasingly

longer solutions in a series of iterations, using a lower-

bound heuristic to prune branches once a lower bound

on their length exceeds the current iterations bound.

It is an informed search based on the idea of the

uninformed iterative deepening depth-first search.

Iterative deepening search



Iterative deepening search

algorithm



How It Works



Why God¶s Algorithm

Complete?

Yes since it always find a solution

Time Complexity?

O(n/m), m is the amount of memory

Space Complexity?

O(n)

Optimal?

Yes, (n^2 / log(n)) moves is enough (20 moves for 3x3 cube)



Agenda




Puzzle Complexity


Manual Solution Steps Optimal Solution

(3) Demo(3) Demo Rubik¶s Cube Solver Robot ± Videos




Rubik¶s Cube Robot



Rubik¶s Cube Simulator



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