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R L F B R L F' B' R' L' F' B'

Re L U B’ U’ R L’ B R’ F B’ D R D’ F’ One of my favorite patterns. I call it Ouroboros--the snake swallowing its tail. Randelshofer calls it Anaconda.

Remarkable. Does that include the time needed to calculate the depth 15 optimal database? How many CPU years did that take?

Perhaps I misunderstood, but it was my impression that not all cosets were actually solved. Say one has a coset S which has been determined to have a maximum depth of 19. One can define a daughter coset by multiplying all members of the coset by a face turn: t • <S> = S2 <S> • t = S3 These...

One can map the states of the cube to the integers 1 to 43,252,003,274,489,856,000. An integer in that range thus may be deemed a representation of a cube state. If one sets up a computer to count from 1 to 43,252,003,274,489,856,000, on completion the program would have represented all states...

I did a search of the posts on the old Domain of the Cube Forum and found a report of a god's algorithm calculation out to depth 15f. I don't know if anyone has done the calculation out to a greater depth. So as for "seeing" as a computer representation all cube states have been "seen" out to 15f.

12! • 2^11 = 980,995,276,800

Still playing around with this, I partitioned the 1728 patterns into 78 equivalence classes by conjugation with the 24 rotation symmetries. I then picked an element from each class having the two solved blocks on the Front and Right faces to represent the class. This element actually defines a...

I sat down and wrote some code to count the number of ways to have two solved 2x2x1 blocks on adjacent faces. On a face there are four 2x2x1 blocks which may be turned four ways to give 16 patterns. For two adjacent faces the blocks may be combined in 144 ways which are disjoint. That is the...

For a set of two corners and four edges the number of permutations is given by: (24 * 21) * (24 * 22 * 20 * 18) = 95,800,320 So for the example you give the odds are 1 in 95,800,320. With rotation and mirror symmetry there are 48 equivalent patterns of this type. However there are a number...

The number of arrangements of a particular set of four edges: 24 * 22 * 20 * 18 = 190,080 For each face, four of these meet your criteria and there are six faces. So: 4 * 6 / 190,080 or 1 in 7,920 The odds would be slightly less than this since cubes with more than one solved cross are...

Best to stick with the righteous BOY cube. Wouldn't want to mess with a sinister YOB cube. 😀

What is the usual way colors are applied to the 3x3 cube. Usually one has Yellow opposite White, Blue opposite Green and Red opposite Orange. In the Wiki database here this is done one way, but on W. Randelshofer's pretty pattern site the colors are applied in the mirror image of what is used...

Ok, so you need to convert a generator turn sequence into a string representation suitable for input into Kociemba's 2 phase solver. 1. Represent the face turns as S54 permutations of the facelet positions. Kociemba gives the order he lists the facelets in the enums.py file. 2. Convert the...

I'm not sure what you mean by Singmaster notation. Kociemba's Cube Explorer accepts input in the form of a permutation of the identity string; "UF UR UB UL DF DR DB DL FR FL BR BL UFR URB UBL ULF DRF DFL DLB DBR" This representation is composed of twenty groupings which name the twenty...

I would argue that when a face (layer) is solved that defines the color scheme of the cube. For a second layer to be solved it must conform to that color scheme. If two parallel faces are solved wrt two different color schemes then they can't both be solved. One is solved and the other is one...

The nine cases with two opposing solved layers not color aligned with one another are counted as 1 solved layer in my count. If the layers aren't color aligned I deem one of the layers unsolved. If they are aligned then one has the identity cube. I was aware of those cases (I mentioned them...

I did some testing since my last post and found there are indeed a lot of cases I was missing. The problem was with states with solved faces not color aligned with the UFR reference cubie. I thought that by looking at all the rotation conjugates I would catch these cases but that is not true...

Well I'm fairly confident in my counts. Generating all members of the group is straightforward using radix encoding to compress the position permutation and the orientations to a number between 0 and 3,674,160. And testing for a solved face is trivial. I can not see how I could have under...

You're right. The sets do overlap. The identity cube would be in all six sets for example. I don't know if the probability would be reduced significantly by this. This is a fairly small group. One could generate all the states and count how many have at least one solved face. I'll see what...