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Cubic Insanity


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Cubic Insanity

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Cubic Insanity

  1. 1. Cubic Insanity © 2011 Tofique Fatehi Page 1 CUBIC INSANITY PSYCHO-ANALYSIS OF THE RUBIK'S CUBE May 21, 2011 Tofique Fatehi CUBIC INSANITY A solved Rubik's Cube, in a "done" state may be deemed to be in a state of sanity. Make a move - turn any face through a right angle (90°) and we get a cube in the first stage of insanity. A move consists of only turning a face by 90° - either clockwise or anticlockwise. Turning a central layer by 90° cannot be deemed to be a move, but rather, two moves - a move each for each of the two sandwiching faces. Similarly, turning a face by 180° is to be considered as two moves of 90° each. Since there are six faces and each face may be moved either clockwise or anticlockwise, there can be twelve ways in which the cube can be in the first stage of insanity, from a state of sanity. Make another move, and we reach the second stage of insanity. Well - not quite - because the second move may nullify the first move and get us back to sanity. Now, for each of the twelve first moves, there are twelve second moves, of which one is a nullifying move. So, there are, in all 132 (12 x 11) moves with a potential to move to the second stage of insanity. Again - not quite - since there are some second stage insanities which can be arrived at in two ways. This happens if the two moves are on two opposite faces - in which case, the first and second moves can be interchanged with the same result. (Moving face A and then face B, is the same as moving face B first, and then face A, if the two faces are opposite to each other). This can happen in twelve ways. So from the potential 132 moves, twelve are eliminated as duplicates, leaving us with 120 (132 - 12) possible second stages of insanity. "There are two common ways to measure the length of a solution. The first is to count the number of quarter turns. The second is to count the number of face turns. A move like F2 (a half turn of the front face) would be counted as 2 moves in the quarter turn metric and as only 1 turn in the face metric." "Optimal Solutions for Rubik's Cube" Wikipedia, the Free Encyclopedia. < _solutions_for_Rubik%27s_Cube>
  2. 2. Cubic Insanity © 2011 Tofique Fatehi Page 2 Proceeding further, every subsequent move either increases the stage of insanity by one or decreases it by one. The latter happening if the new move nullifies a previously made move, or a series of moves. This can also happen if the same face is moved by three quarter turns (270°), which would be equivalent to one quarter turn in the counter-direction. Hence the third quarter turn actually reduces the insanity by one. THE CRUX If a cube is in the nth stage of insanity, then it can be brought back to sanity (or solved) in at least n moves. This would be the most efficient way of solving the cube, where each and every move always reduces the insanity of the cube. THE CORE What is the highest stage of insanity - from which any move made always reduces the insanity? This would mean that this is the maximum number of moves strictly required to get back to sanity. Most "standard" algorithms used to solve the cube use a much greater number of moves. Sheer inefficiency, I would say. After years of research, it has been found that the highest degree of insanity is 24. So, a cube can always be solved in at the most 24 moves. Or, if the face-turning convention is used, in twenty moves, since four of the twenty moves are double-moves of 180°. This has led to the assertion that "God's number for the Rubik's cube is twenty". ENIGMA There are more than one stages of sanity. Believe me there are. See for yourself. With a soft, erasable pencil, draw a line on the white face of a "done" (sane) cube - parallel to any one side of the white face, in the middle layer, but slightly off- centre. Jumble up the cube and solve it again. Is the straight line still a straight line? If yes, jumble it up, again and again, and with luck, you will find the line broken. And yet, the cube has attained sanity, but of a different kind. A question arises as to how many such enigmatic stages of sanity are there? COUNTING We saw that there are twelve ways in which a cube can be in the first stage of sanity, 120 ways for the second stage, making a total of 132 ways for the first two stages of insanity. If we continue counting - right until the highest 24th stage of insanity, we ought to get a total figure which is equal to all the possible permutations and combinations of the cube - and if we do not get that, then what?
  3. 3. Cubic Insanity © 2011 Tofique Fatehi Page 3 And remember, the enigma of the sane cube is also inherent in each and every one of the total permutations and combinations. For a further discussion, please read “Sanity Restored”. < sanity-restored> Except as otherwise expressly permitted under copyright law, no copying, redistribution, retransmission, publication or commercial exploitation of this material will be permitted without the express permission of the copyright owner. In the event of any permitted copying, redistribution or publication of copyright material, no changes in or deletion of author attribution, trademark legend or copyright notice shall be made. You acknowledge that you do not acquire any ownership rights by downloading this copyrighted material.