Rubik's Cube/The Cube

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

  Ray Calvin Baker 15:01, 29 October 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.   |
 * 1) Just how many ways are there to arrange the cubies of the Cube? Let's calculate! But be     #
 * 2) aware that most calculators cannot hold numbers this large! (Some calculators may use       #
 * 3) scientific notation to express large numbers, but they almost always truncate the           #
 * 4) significant figures of the numbers, which is an approximation, not an exact number.)        #
 * 5) There are 8 factorial (usually written "8!") ways to arrange the eight corner cubies.       #
 * 6) This is 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 = 8 * 7 * 6 * 5 * 4 * 3 * #
 * 7) 2 * 1 =  40,320. (I have used lots of calculators and computer languages which use "*" as   #
 * 8) the sign for "multiply" or "times".)                                                        #
 * 9) Then, There are ( (three to the eighth power) divided by three ) ways to orient the eight   #
 * 10) corner cubies. This is often written (on a computer) as "(3 ^ 8) / 3", which is             #
 * 11) (3 * 3 * 3 * 3 * 3 * 3 * 3 * 3) / 3 =  6,561 / 3  = 2,187. (We need to divide by three      #
 * 12) here, because it is not possible to rotate just one corner cubie. Whenever one corner cubie #
 * 13) is rotated clockwise, another has to rotate counterclockwise, and vice versa -- unless, of  #
 * 14) course, you physically disassemble your Cube.)                                              #
 * 15) Next, there are 12! / 2 ways to arrange the edge cubies. This is (12 * 11 * 10 * 9 * 8 * 7  #
 * 16) * 6 * 5 * 4 * 3 * 2 * 1) / 2 =  479,001,600 / 2 = 239,500,800. ( We need to divide by two   #
 * 17) here, because it is not possible to interchange exactly two edge cubies, unless you         #
 * 18) physically disassemble the Cube.)                                                           #
 * 19) Lastly, there are ( ( 2 ^ 12 ) / 2 ) = ( 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 )    #
 * 20) / 2 = 4,096 / 2 = 2,048 ways to orient the twelve edge cubies. (It is not possible to       #
 * 21) "flip" just one edge cubie, unless you disassemble... )                                     #
 * 22) Multiply these four numbers together, and you get 40,320 * 2,187 *  239,500,800 * 2,048     #
 * 23) = 43,252,003,274,489,856,000, which is exactly the number of ways the cubies of Rubik's     #
 * 24) Cube can be arranged. This is a twenty digit number -- forty-three quintillion, two        #
 * 25) hundred fifty-two quadrillion, three trillion, two hundred seventy-four billion, four       #
 * 26) hundred eighty-nine million, eight hundred fifty-six thousand. (OK, one of these ways is    #
 * 27) 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.)  #
 * 1) Each of those four factors of 43,252,003,274,489.056,000 corresponds to one of our major    #
 * 2) goals, which are discussed in the following chapters.                                       #
 * 3) By the way, if you do dissassemble your Cube, there are twelve distinctively different ways #
 * 4) to put it back together. Twelve = 3 * 2 * 2, and these factors correspond to those numbers  #
 * 5) by which we divided in the previous calculations. Of these twelve ways, only one will allow #
 * 6) you to solve the Cube. There will always (until you disassemble again) be something wrong   #
 * 7) with the other eleven ways of re-assembly -- an edge cubie will be flipped (disoriented),   #
 * 8) or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
 * 9) you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #
 * 1) Multiply these four numbers together, and you get 40,320 * 2,187 *  239,500,800 * 2,048     #
 * 2) = 43,252,003,274,489,856,000, which is exactly the number of ways the cubies of Rubik's     #
 * 3) Cube can be arranged. This is a twenty digit number -- forty-three quintillion, two        #
 * 4) hundred fifty-two quadrillion, three trillion, two hundred seventy-four billion, four       #
 * 5) hundred eighty-nine million, eight hundred fifty-six thousand. (OK, one of these ways is    #
 * 6) 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.)  #
 * 1) Each of those four factors of 43,252,003,274,489.056,000 corresponds to one of our major    #
 * 2) goals, which are discussed in the following chapters.                                       #
 * 3) By the way, if you do dissassemble your Cube, there are twelve distinctively different ways #
 * 4) to put it back together. Twelve = 3 * 2 * 2, and these factors correspond to those numbers  #
 * 5) by which we divided in the previous calculations. Of these twelve ways, only one will allow #
 * 6) you to solve the Cube. There will always (until you disassemble again) be something wrong   #
 * 7) with the other eleven ways of re-assembly -- an edge cubie will be flipped (disoriented),   #
 * 8) or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
 * 9) you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #
 * 1) or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
 * 2) you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #