I’m not sure if this was done before, but I thought I’d give it a try.

The first step in doing 4BLD is to choose an orientation for the cube.

Usually this step is done trying to maximize the number of solved centers to reduce both memorization and execution.

However, there are 24 different possible orientations for the cube and it’s not always trivial to see which solution is best, especially if there is no bar on any face, or if there are multiple ones.

So, in order to do this automatically, for a given center scramble, I made a program to check for each of the 24 orientations, which was the best, and how many centers were solved for every orientation.

But this lead me to think about the following problem:

What is the most common number of centers solved for any given scramble. Since there are no restrictions for centers, the number of possible combinations is the colossal number of 24!/4!^6 = 3246670537110000 (since centers of the same color are equivalent) which is a very big number to check all cases. I didn’t want to go deeper in weeding out isomorphic scrambles. I just thought I’d test with random (or pseudo-random) scrambles with the only restriction is being different from one another.

The sample I used had 100 million (100.000.000) different scrambles, which is ridiculously small compared to the total number of possible scrambles (namely 0.00000308% of the whole thing).

For each scramble, it was tested what was the highest number of solved centers for all possible orientations, and in the end scrambles with the same ammount of maximum centers solved were summed to see what was their frequency.

Here are the results I achieved:

Spoiler: Frequency Chart (Line-based)
Since the values are of very different order of size, I created a logaritmic chart:

Spoiler: Logaritmic Chart
Spoiler: Distribution Chart (Pie-based)

From the results obtained, it's easy to see that the most common optimal 4BLD solved centers is 8, very close to 7.

Another thing you can see is you always have more than 4 centers solved for any given orientation, the minimum being 5.

And having 18 centers solved (6 unsolved) is so rare for a random (or pseudo-random) case that only happened 4 times in 100 million.

**Introduction:**The first step in doing 4BLD is to choose an orientation for the cube.

Usually this step is done trying to maximize the number of solved centers to reduce both memorization and execution.

However, there are 24 different possible orientations for the cube and it’s not always trivial to see which solution is best, especially if there is no bar on any face, or if there are multiple ones.

So, in order to do this automatically, for a given center scramble, I made a program to check for each of the 24 orientations, which was the best, and how many centers were solved for every orientation.

But this lead me to think about the following problem:

**Problem:**What is the most common number of centers solved for any given scramble. Since there are no restrictions for centers, the number of possible combinations is the colossal number of 24!/4!^6 = 3246670537110000 (since centers of the same color are equivalent) which is a very big number to check all cases. I didn’t want to go deeper in weeding out isomorphic scrambles. I just thought I’d test with random (or pseudo-random) scrambles with the only restriction is being different from one another.

**Results:**The sample I used had 100 million (100.000.000) different scrambles, which is ridiculously small compared to the total number of possible scrambles (namely 0.00000308% of the whole thing).

For each scramble, it was tested what was the highest number of solved centers for all possible orientations, and in the end scrambles with the same ammount of maximum centers solved were summed to see what was their frequency.

Here are the results I achieved:

Number of Solved Centers | Number of Scrambles |

5 | 165268 |

6 | 7698835 |

7 | 31876637 |

8 | 33138395 |

9 | 17870290 |

10 | 6680225 |

11 | 1965825 |

12 | 482093 |

13 | 100858 |

14 | 18330 |

15 | 2833 |

16 | 359 |

17 | 48 |

18 | 4 |

**Discussion:**From the results obtained, it's easy to see that the most common optimal 4BLD solved centers is 8, very close to 7.

Another thing you can see is you always have more than 4 centers solved for any given orientation, the minimum being 5.

And having 18 centers solved (6 unsolved) is so rare for a random (or pseudo-random) case that only happened 4 times in 100 million.

Last edited: Sep 22, 2013