The Reading Room

@bobble Seen it, haven't voted or answered yet. My TL;DR is that it's potentially a useful question with useful answers for Rubik's cube people, and especially the top answer shows how this real problem is solved in practice.

@Namaskaram Thanks :-)

There's a whole bunch now scoring 4, so we might have another tie situation at the end of next month.

I'll try to help out next month, too :P But, how are ties usually resolved?

A: Rule for selecting a topic challenge among those with the same number of votes

2

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...

Ahh, okay

@Randal'Thor I think it's an interesting question about how one can rigorously define a "random" configuration of an n^k Rubik's cube. I recently came across the probabilistic method, and was taking a look at the Erdős–Rényi model for generating random graphs... Fascinating stuff! I don't know if there's any literature on the Rubik's cube from this angle, though.

7:10 AM

I think there is some published mathematical literature on the Rubik's cube.

Like proving the maximum number of twists needed from any position to solve it (apparently around 20 for a standard 3x3x3 cube).

Oh definitely. But specifically about how to randomize, I'm not sure.

I think that the minimum number was arrived at through brute force computation, though. I remember being a tad disappointed, which is why the factoid is stuck in my head :(

There's a good deal of literature (probably mostly combinatorial) on recreational maths like games and puzzles.

I keep meaning to self-answer this question based on some searching in academic papers.

7:36 AM

@Randal'Thor Upvoted and bookmarked :) I read Gareth's answer, but the rest will still take me time. The OEIS sequence is terribly short. :( A closed-form solution may be too much to expect, but one may try to get some asymptotic results perhaps...

I'm only just starting to dive into this sort of stuff. Entering combinatorics was completely unplanned when I joined the PhD program :)

Combinatorics was about the only pure-maths course I didn't take as an undergrad. I know it just to the level of some olympiad stuff, but no further.

That's still a lot more than what I knew when I went to talk to my professor about doing a PhD under him :P

I never looked at Olympiad material seriously when I was an undergrad. At that time I had the impression that there was not much depth in the "bag of tricks" that one needs to know to solve those kinds of problems.

8:50 AM

@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.

Also ARGH!

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.

Q: Visualising a scene described in beginning of “The adventure of the three students”

9:37 AM

Btw finding an optimal solution for a random config of the 3 cube is easier in the average case than the worst case, because a typical random config can be solved in 17 or 18 face turns, while the worst is 20 turns but those configs are quite rare. (See cube20.org for the specific numbers.)

Bookworm

10:09 AM

0

“The sitting-room of our client opened by a long, low, latticed window on to the ancient lichen-tinted court of the old college. A Gothic arched door led to a worn stone staircase” This is the description of the scene as Sherlock’s team enters St. Luke’s. How do you visualise this? I can’t seem t...

arthur-conan-doyle sherlock-holmes

11:20 AM

@Bookworm I wonder what we can do with this one. There's supposedly a short silent film adaptation, but it's from 1923 and probably impossible to find. There don't seem to be modern film adapatations. There are some book illustrations, and commons.wikimedia.org/wiki/… probably depicts that sitting room, but doesn't give a really good view of it, it's the characters who are in focus.

The best solution might be to find photos of a similar room in an old college building, not necessarily the (possibly fictional) building in the story.

11:37 AM

@Spagirl I also don't understand what a "lichen-tinted court" would be; nor can I easily imagine a collage building with a great indoor hall that's four storeys tall as that description seems to imply.

So I think it's a fair question.

Admittedly commons.wikimedia.org/wiki/File:George-peabody-library.jpg is 6 storeys tall.

But how would you see the staircase for those four storeys?

Also what college building would have the rooms of three students above the room of the professor, with a sitting room on the bottom level?

It doesn't seem like a usual arrangement.

12:22 PM

Wow, that was a fast two upvotes.

@b_jonas Yes, exactly. Oxbridge colleges have a fairly distinctive architecture, and Conan Doyle was writing based on what he'd seen, even if the story was set in a fictional college.

Wait, so it's the open top courtyard from where you can see the four stories, not the indoors sitting room? That makes somewhat more sense.

Well ok, but would it really have the rooms of the three students, one per storey, above the professor's room?

I know that part of the description isn't quoted in the question, but it belongs there.

2:05 PM

@b_jonas I didn't say it wasn't a fair question, I asked for more detail as to what the difficulty the OP was experiencing was.

Where are you getting the four storey indoor hall from? I have the text on hand, but a search doesn't bring up the word 'hall' so I'm not sure which bit you mean.

Or what the difficulty is of being able to see a staircase from each floor that it serves.

@Spagirl It's not an indoor hall, I read that part wrong. They are not in the sitting room, they are in an outdoor court and looking at an indoor sitting room through the window.

Sorry, I just noticed that you had commented further

It's the same paragraph as the quote in the question comes from. It continues: “On the ground floor was the tutor's room. Above were three students, one on each story.”

en.wikisource.org/wiki/The_Return_of_Sherlock_Holmes/Chapter_9 has the full text

As I said, I have the text on hand... I just couldn't understand where you had conjured the indoor hall from, but you have explained that it was a misreading,

That's also why "lichen-tinted court" confused me. Outdoors that makes sense, indoors it would be odd for an established active university.

2:20 PM

But just to clarify > so it's the open top courtyard from where you can see the four stories: the text doesn't say you can see them from anywhere in particular, it just says that they are there. The passage is an explanation of the arrangement of the building rather than just a description of a particular view.

@Spagirl They can see the professor's room at least. All of the rooms must have a window either to one of these internal courts or to the outside, and they went to this court because that's where the professor's room opens. The students' room could perhaps face to somewhere else.

However,

later the text says
> Three yellow squares of light shone above us in the gathering gloom.
> “Your three birds are all in their nests,” said Holmes, looking up.
Which I read as meaning that the rooms or suites of all three students had a window opening to the same court.

@b_jonas There are a couple of reasons for this arrangement, having a ground floor room means that a Tutor has to spend less time and expend less effort getting to and from his rooms than if they were on a higher storey. Leave the running up and down flights to the younger, subordinate legs.

The other reason is that the presence of the Tutor's rooms next to the entrance to the stair would be expected to moderate the behaviour of the students coming and going, discouraging staying out past curfews or general rowdiness.

That part makes sense, but why are the students on three different floors, instead of on one floor next to each other?

Maybe other students are sharing floors with them.

Or there's only one room on each floor off that particular staircase.

2:41 PM

I have to admit that university buildings sometimes have very strange arrangements.

3:26 PM

@b_jonas The colleges in this type of university town were designed around the tutorial system where students were expected to most of their study in their rooms or the library and have regular tutorials in small groups in the Tutors rooms.

As a consequence the Courts (Cambs) and Quads(Oxon) are mainly constituted of accommodation for students and tutors. There are Common Rooms and Dining Rooms etc, but the situation being described is a building that is almost certainly, at least the bits accessed from that stairway, all accomodation.

I've only visited student rooms in such a college once, at Oriel, but it was exactly like that, several floors of student accomodation off corridors accessed from a stone staircase that led up from the quad.

So ACD is just clarifying that the floors above the Tutor's rooms are not library or common rooms, but rooms belonging to individual students.