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Rubik's Cube/The Cube

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Ray Calvin Baker 19:31, 18 August 2011 (UTC)

| HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                               |
|                                            (Preliminary April 20, 2007 version)             |
|                                            by Mr. Ray Calvin Baker                          |
|                                            FREE Public Domain Educational Material        . |
|                                                                                             |
| Chapter Zero - - - - - - - - - - The Cube                                                   |
|                                                                                             |
| DO NOT PEEL ANY OF THE COLORED LABELS OFF YOUR CUBE! (I call this "the chemist's method,    |
| because it involves the breaking of octillions of weak molecular bonds. This is not meant   |
| to disparage chemists at all -- one of the first books on the Cube that I read was written  |
| by a chemist, and he described elegant geometrical and mathematical moves such as "Rubik's  |
| Maneuver".)                                                                                 |
|                                                                                             |
| There are other, less damaging ways to disassemble a Cube if you really want to see what's  |
| inside it (or if you want to cheat.) -- see Chapter Two, "Using Pictures, Diagrams,         |
| Notation, and Abbreviations",  for "the Physicist's method of disassembling the Cube. Of    |
| course, if you can follow directions from a mathematician, you won't need to damage or take |
| apart your Cube at all!                                                                     |
|                                                                                             |
| From the outside, your Cube looks like it is made of 27 smaller cubes. (There's a picture   |
| in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations".) Each of these     |
| smaller cubes is called a cubie. A cube is a shape having eight corners, six flat square    |
| faces, and twelve edges. A Cube (upper case) is Rubik's Cube (or a cheap imitation of it).  |
|                                                                                             |
| When it came from the factory, each face of your Cube had nine colored labels, all the same |
| color -- six faces, six different colors. It may not seem likely to you now, but your Cube  |
| (unless it's physically damaged) can be restored to this condition.                         |
|                                                                                             |
| Your Cube consists of one central core, which holds six axles. Each axle supports one face  |
| cubie, which is free to rotate unless blocked by other components of your Cube. Each face   |
| cubie has a colored label -- six faces, six different colors. Each of the twelve edge       |
| cubies is clamped between a pair of face cubies. Each edge cubie has two different colored  |
| labels. Each of eight sets of three edge cubies clamps a corner cubie to the rest of the    |
| Cube. Each corner cubie has three different colored labels. (There are diagrammatic         |
| pictures in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations".)          |
|                                                                                             |
| Even though each of the eight corner cubies and twelve edge cubies is a separate piece,     |
| they are all clamped together into a secure and stable assembly -- the Cube. And each face  |
| of the Cube can be rotated. Of course, that rearranges the colored labels.                  |
|                                                                                             |
| YOUR MISSION, should you choose to accept it, IS TO RESTORE YOUR CUBE TO ITS FACTORY        |
| CONDITION!                                                                                  |
|                                                                                             |
| How will you be able to tell where each cubie belongs when part of your Cube is scrambled?  |
| Look at the center square on each side. This piece is firmly attached to the core of the    |
| Cube, and is only free to rotate, The central square of each side of the Cube is your       |
| "fixed landmark" around which you must arrange all of the other cubies. (Once we start      |
| arranging edge cubies, we will temporarily use other landmarks, but we will always restore  |
| the positions of the center squares.)                                                       |
|                                                                                             |
| I suggest that you skim through this paper to see how the parts are organized, and to       |
| convince yourself that there really are ways to solve the Cube. Then, you can pick up a     |
| scrambled Cube to try to follow the exact steps you will need to solve it. Don't be         |
| discouraged if your first few attempts seem only to scramble the Cube more! Start again     |
| from the beginning if you have to. You will become more familiar with your Cube, and you    |
| will, with practice, become more confident that you can master each necessary operation.    |
|                                                                                             |
# Just how many ways are there to arrange the cubies of the Cube? Let's calculate! But be     #
# aware that most calculators cannot hold numbers this large! (Some calculators may use       #
# scientific notation to express large numbers, but they almost always truncate the           #
# significant figures of the numbers, which is an approximation, not an exact number.)        #
#                                                                                             #
# There are 8 factorial (usually written "8!") ways to arrange the eight corner cubies.       #
# This is 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 = 8 * 7 * 6 * 5 * 4 * 3 * #
# 2 * 1 =  40,320. (I have used lots of calculators and computer languages which use "*" as   #
# the sign for "multiply" or "times".)                                                        #
#                            .                                                                #
# Then, There are ( (three to the eighth power) divided by three ) ways to orient the eight   #
# corner cubies. This is often written (on a computer) as "(3 ^ 8) / 3", which is             #
# (3 * 3 * 3 * 3 * 3 * 3 * 3 * 3) / 3 =  6,561 / 3  = 2,187. (We need to divide by three      #
# here, because it is not possible to rotate just one corner cubie. Whenever one corner cubie #
# is rotated clockwise, another has to rotate counterclockwise, and vice versa -- unless, of  #
# course, you physically disassemble your Cube.)                                              #
#                                                                                             #
# Next, there are 12! / 2 ways to arrange the edge cubies. This is (12 * 11 * 10 * 9 * 8 * 7  #
# * 6 * 5 * 4 * 3 * 2 * 1) / 2 =  479,001,600 / 2 = 239,500,800. ( We need to divide by two   #
# here, because it is not possible to interchange exactly two edge cubies, unless you         #
# physically disassemble the Cube.)                                                           #
#                                                                                             #
# Lastly, there are ( ( 2 ^ 12 ) / 2 ) = ( 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 )    #
# / 2 = 4,096 / 2 = 2,048 ways to orient the twelve edge cubies. (It is not possible to       #
# "flip" just one edge cubie, unless you disassemble... )                                     #
#                                                                                             #
# Multiply these four numbers together, and you get 40,320 * 2,187 *  239,500,800 * 2,048     #
# = 43,252,003,274,489,856,000, which is exactly the number of ways the cubies of Rubik's     #
# Cube can be arranged. This is a twenty digit number -- forty-three quintillion, two        #
# hundred fifty-two quadrillion, three trillion, two hundred seventy-four billion, four       #
# hundred eighty-nine million, eight hundred fifty-six thousand. (OK, one of these ways is    #
# actually the unscrambled, pristine Cube, so there are really only                           #
# 43,252,003,274,489,855,999 different scrambled Cubes, plus exactly ONE unscrambled Cube.)   #
#                                                                                             #
# Each of those four factors of 43,252,003,274,489.056,000 corresponds to one of our major    #
# goals, which are discussed in the following chapters.                                       #
#                                                                                             #
# By the way, if you do dissassemble your Cube, there are twelve distinctively different ways #
# to put it back together. Twelve = 3 * 2 * 2, and these factors correspond to those numbers  #
# by which we divided in the previous calculations. Of these twelve ways, only one will allow #
# you to solve the Cube. There will always (until you disassemble again) be something wrong   #
# with the other eleven ways of re-assembly -- an edge cubie will be flipped (disoriented),   #
# or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
# you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #
|                                                                                             |
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Ray Calvin Baker 15:01, 29 October 2011 (UTC)