Finding Probability of Solving a Rubik Cube with 3 Pieces

[Feynmanfan]

Dear friends,

I need some help with this problem.
"What is the probability of building a solvable rubik cube only by ordering randomly all pieces but three"(let us suposse that these 3 pieces are vertices)
That is you first get a random cube without three vertices and then you try to build a solvable cube by choosing a correct position for these three vertices.

What I got is that only 1/12 of all posible random cubes are solvable. I think that all posible random configurations are 8!*3^8*12!*2^8. But the thing is that in this problem 3 vertices are correctly places. I don't know what to do.

Thanks for your help

 

[Feynmanfan]

just a hint, please

 

[wais]

.



[Name]

Thank you for reaching out for help with this problem. The probability of building a solvable Rubik's cube with only three pieces in a random order is a bit tricky to calculate, but I will do my best to explain the process.

First, let's define some terms for clarity. A "solvable" Rubik's cube is one that can be solved by following the standard algorithms and moves. A "random cube" refers to a cube that has been scrambled or mixed up in a completely random way.

To calculate the probability, we need to consider the total number of possible random cubes and the number of those that are solvable. As you mentioned, the total number of possible random configurations is 8!*3^8*12!*2^8. This is because there are 8 corner pieces that can be arranged in 8! ways, 12 edge pieces that can be arranged in 12! ways, and each of these pieces can be rotated in 3 ways. Additionally, the center pieces can be flipped in 2 ways.

Now, let's focus on the number of solvable cubes. When we have a cube with three correctly placed vertices, we essentially have a starting point from which we can solve the rest of the cube. This means that the remaining pieces (9 corners and 9 edges) can be arranged in any order and still result in a solvable cube. Therefore, the number of solvable cubes would be 9!*9! = 362,880.

To find the probability, we divide the number of solvable cubes by the total number of possible random cubes. This gives us a probability of 362,880 / (8!*3^8*12!*2^8) = 1/12. So your initial intuition was correct!

I hope this explanation helps you understand the problem better. Let me know if you have any further questions or if I can assist you in any other way.