The Rubik's Cube Perfect Scramble (2024)(solutionslookingforproblems.com) |
The Rubik's Cube Perfect Scramble (2024)(solutionslookingforproblems.com) |
> ...
> 2) I can place the first 11 edge pieces onto the cube any way I want. The orientation of the last edge piece is determined by the orientation of the first 11.
> 3) I need to track how many swaps I create by placing those pieces. An even number of swaps is solvable, and odd number is not.
Would it be equivalent to say, after placing the first 10 edge pieces, the position of the 11th is mandated, and then the orientation of the 12th is too? Or if (3) is broken might it be harder to fix than swapping the 11th and 12th?
But it introduces an artificial difference between edges and corners. You'd get the same ability for corners if you did them after the edges.
The slightly counterintuitive “magic” is that you can trade a corner swap for an edge swap: for permutation parity, corners and edges are interchangeable.
For people who like powers of 2, that's "only" 2^65.2
That's within the realm of computability in practical timespans, if you can make the code fast and have $$$$$ to spend on compute. (modern CPU cores can do billions of operations per second, and that's not even considering GPUs)
The approach presented in the article is obviously far more efficient, but I wonder if anyone's done a "full search" of all possible cube positions before. I don't think there's any reason to do that, but that hasn't stopped people before (see: pi calculation records).
I wonder how many positions qualify (perhaps only 1 discounting symmetry as there is 1 fully solved cube, again discounting symmetry). But that’s a rabbit hole that I am not going down.
But I can appreciate the choice made in the article.
Reminds me a bit of the "randomart" seeing the positions and colors of the cube splayed out like that.
If you mean to use it exclusively as a real-world key transmission like with Cryptonomicon's Solitaire decks, the problem becomes finding the shortest path or whatever the protocol determines is the normalized form.
Not to rain on your parade, it's a fun approach to think about, like maybe if the properties of a specific face determine which rotations to perform next, and which face to look at next, in addition to being the nonce for decoding the next letter. But even something like that would be too complicated to expect a person to remember all of while not being complicated enough to fluster a computational approach. The nice thing about Solitaire is that it's reasonable to perform the algorithm in your head.
Also a solution looking for a problem, maybe.
I'll keep my Rubik's Cube in this position.
I don’t see how ”there is 1 fully solved cube” would even hint at “perhaps only 1”
Also, there isn’t only 1. https://www.cube20.org/: “Distance-20 positions are both rare and plentiful; they are rarer than one in a billion positions, yet there are probably more than one hundred million such positions. We do not yet know exactly how many there are”
That is, count a 180° move as two 90° moves. Which it is, though we usually don't think of it that way.
Assuming a random move length distribution, that would only leave a (2/3)^20 fraction of them, which is about 147000 positions.