12 Matching Annotations
  1. Apr 2023
    1. To Solve the Rubik’s Cube, You Have to Understand the Amazing Math Inside<br /> by Dave Linkletter

      suggested by Matt Maldre's annotations

    2. Only small tidbits of math remain unresolved for Rubik’s Cube. While God’s number is 20, it’s unknown exactly how many of the 43,252,003,274,489,856,000 combinations require a whole 20 moves to be solved.

      We've got solutions for the number of configurations there are to solve a Rubic's cube with from 1 move up to 15, but we don't know how many cube configurations there are that can be solved with 16-20 moves.

      • Example: the number of positions that require exactly one move solve them is 18, which is counted by multiplying the six faces and each of the three ways they can be twisted.
    3. The first major insight on God’s Number was by Dr. Morwen Thistlethwaite in 1981, who proved it was at most 52. That means he proved every scrambled cube can be solved in 52 moves or less.Progress continued through the 1990s and 2000s. Finally, in June 2010, a team of four scientists proved that God’s number is 20.

      With respect to the Rubic's Cube, God's number is defined as the minimum number of turns to solve a randomly scrambled cube. Morwen Thistlethwaite proved that it was at most 52 moves in 1981. By June 2010 a group of four had a proof that God's number is 20.

    4. Instrumental to this era of speed cubing was Dr. Jessica Fridrich, who in 1997 developed a method for solving the cube faster than ever. Most of the fastest cube solvers nowadays use some version of the Fridrich method.
    5. Dr. David Singmaster, who wrote the famous guide “Notes on Rubik’s ‘Magic Cube’” and developed a writing method for describing turns of the cube’s faces. That notation has become the standard, and is now known as Singmaster notation.
    6. the total number of ways you can scramble a Rubik’s cube: 43,252,003,274,489,856,000. Written in a more mathematical way, that number is (388!)(21212!)/12.

      The total number of ways one can scramble a cube is

      (3^8 * 8!)(2^12 * 12!)/12

      the first term is for how the 8 corner pieces can be rotated by three before returning to their original orientations and the eight! places they can be placed<br /> the second term is for the two ways that 12 edges pieces can be inserted into their 12 locations The bottom 12 is for the 322 ways that algorithms can't solve for a single rotation of a corner piece, or the other algorithms can't swap a pair of corners or a pair of edges.

    7. no algorithm can flip a single edge in place
    8. There aren’t any algorithms that can swap only a pair of corners, nor only a pair of edges.
    9. swapping a pair of edges and a pair of corners cancel each other out, since there’s an algorithm to undo that.
    10. If you break it apart and reassemble the cubies randomly, there’s actually only a 1 in 12 chance that it’s solvable.

      A Rubic's cube taken apart and put together randomly only has a 1 in 12 chance of being solvable.

  2. Jul 2021