08:50
@bobble “The ついにゆく poem from Narihira has 8 syllables rather than 7 in the last line and the first line in Komachi’s 花の色は has 6 syllables rather than the standard 5 right?” I can understand how that one annoys someone enough to post a question. I still have to get back to Rácz István's translation of the Kalevala, which has 16 syllables in almost every line, which makes the few counterexamples very annoying.
I'll have to borrow two editions of it from the library, and then find (at least some of) the counterexamples, and ask about them.
@Namaskaram I don't see why it would be difficult to rigorously define a uniform random configuration. The 3x3 cube is just a finite group with about 2**33 elements, and it's easy to describe an element with the permutations and orientations of the vertexes and edges.
So to generate a random configuration, you randomly permute the vertexes, then randomly permute the edges but assure that the two permutations have the same parity, then you randomly twist the vertexes (keeping the modulo 3 invariant), then randomly flip the edges.
The big cubes are slightly more interesting in that the observed configuration is not a group, but an equivalence class of a group divided by a subgroup that isn't normal (because you can't observe how centers with identical stickers are permuted). But that doesn't make it harder to define.
Now to actually shuffle a cube uniform randomly, you have to not only generate a random configuration, but also solve it so that you can put a physical cube into that configuration. You don't even necessarily need an optimal solution, and on big cubes you almost certainly won't have one, because it doesn't matter if you're doing a few more rotations for the shuffle than necessary.
For a 3x3 cube a current computer can compute the optimal solution, but not easily; but generating a close to optimal solution with a computer is very easy, and well studied because it's interesting on its own right.
That said, there is an interesting question hidden here: namely that if you don't have a computer or don't want to solve the cube, can you just make random face turns and how quickly it converges to the uniform distribution. The stochastic guys over in the university study that kind of thing, but I don't know much about it.
“Entering combinatorics was completely unplanned when I joined the PhD program” sounds funny by the way
I'm not active on puzzling so I don't have an opinion on the original question whether the question is on topic by the way.
I meant 2**65 elements for the 3x3 cube, not 2**33 elements obviously.
Btw, if you want to sound pretentious about it, like if you write a grant application for cube shuffling research, then call it “Haar measure for discrete topology” instead of just a uniform distribution on the finite set of configurations. We'll laugh at you but know how some grant applications work.
I propose: pick the oldest suggestion.
Using a criterion based on number of downvotes or upvotes will quite often result in a tie: if two answers are both sitting at (for example) +4 / -1, then we're right back in the same situation needing to find a tie-break criterion. The criteria based on age...
The phrase “one person’s modus ponens is another’s modus tollens” is a popular philosophical motto that reminds us that a chain of logical reasoning works in both directions: if the truth of some premises entail the truth of a conclusion, then the falsity of the conclusion entails the falsity of ...
