"How many turns should I make to completely shuffle the surface of the Rubik's cube?" Is a very difficult problem in mathematics
Popular around the worldRubik's CubeIs a three-dimensional puzzle that is so difficult for beginners to have one plane, but the fastest record in the world that has all six planesAbout 4 secondsThe research has been done by skilled players every day. But,Monash UniversityAccording to Associate Professor Tim Garoni of the Department of Mathematics, it is more difficult to completely shuffle the surface of a Rubik's cube than to prepare a Rubik's cube in the world of mathematics.
How hard is it to scramble Rubik ’s Cube?
Many mathematicians have studied "shuffle to create randomness". Stanford University mathematician Professor Parsi Diaconis and Columbia University mathematician Professor Dave Bayer published a research paper on “ Rifle Shuffle '' in 1990.Announcementdoing. Rifle shuffle is a way to shuffle cards like the following movie, Professor Diaconis and Professor Bayer said, “ It is necessary to repeat riffle shuffle seven times to make the order of cards completely random '' Prove that.
(embed) https://www.youtube.com/watch?v=f6ZD1lDbW3M (/ embed)
Also, you can find out what happens if you stir the cards on your desk by reading the following article.
There are as many as 4325 Kyo 2003 trillion 247,489,866,000 patterns of Rubik's cube, but in 2010, it can be said that “ Rubik's cube can always solve any pattern within just 20 moves ''Proofit was done. The number "20 hands" is called "God's number" in the Rubik's cube world. On the other hand, it is a very difficult problem to say, "How much work is required to randomly move a Rubik's cube to be completely shuffled," says Garoni.
The Rubik's Cube can rotate one of the three rows in the X, Y, or Z axis by 90, 180, or 270 degrees. In order to shuffle the surface of the Rubik's Cube, it is necessary to repeat the act of randomly performing one of 15 operations (the first time, 18 operations). According to Associate Professor Garoni, “ to completely shuffle the Rubik's cube '' is mathematically “ the probability of occurrence of all patterns approaches the limit as close as 1 in 4325 Kyoto 2003 trillion 274,489,885,6000 It means that this state is called "probability distribution is uniform".
According to Garoni, the operation of the Rubik's cube is "Markov chain"It can be explained with a probability statistical model called. And "How much manipulation is required to make the probability distribution infinitely uniform in the Rubik's cube"Is a sampling of the probability distribution by the algorithm"Markov chain Monte Carlo method". However, performing the Markov chain Monte Carlo method with a general Rubik's cube of 3 blocks × 3 blocks × 3 blocks requires too much calculation, so Galoni Associates that it is an unsolved problem at the time of article creation Professor says.
Therefore, Associate Professor Galoni uses a "Pocket Cube", a 2-block x 2-block x 2-block Rubik's cube. The total number of pocket cube patterns is 3,674,160, and the number of Gods is 11.
The following graph shows the probability distribution of a pocket cube based on a simulation. T on the horizontal axis is the number of steps, and d (t) on the vertical axis means “ how far away from the uniform probability distribution '', and the smaller d (t), the more shuffled Indicates that According to Garoni, the line that is generally "well shuffled" is when d (t) is below 0.25, and for pocket cubes when t is 19. In other words, if you do not operate at least 19 times, it can not be mathematically said that "the pocket cube was shuffled firmly".
Also, as the number of moves increases, the probability distribution becomes more uniform. It seems that d (t) is 0.092 when operated 25 times, 0.0012 when operated 50 times, and 0.00000017 when operated 100 times.
The Markov chain Monte Carlo algorithm for the pocket cube isGitHubPublished in.