# 3 color 3x3x3 algorithm thread

#### JustinTimeCuber

##### Member
how about a 2-color cube? Three red sides, three blue sides. Red opposite blue for all of them. How many PLLs would there be?

#### cuBerBruce

##### Member
if you include U, U', U2 as a setup %75 of the time, you can always reduce it to only 2 rotations of the algorithm that need to be memorized. I think that might be similar to what you were saying with the probability being reversed.
Yes, I guess you can think of dealing with AUF case and angle case at the same time rather than independently.

Taking "F2 U' L2 U F2 U" as our standard algorithm, we have the two versions:

"a" = F2 U' L2 U F2 U
"b" = R2 U' F2 U R2 U (rotated alg)

We can then solve any combination of the 4 angle cases and 2 pre-AUf cases according to the following table.

Code:
               ("Good" AUF case)      ("Bad" AUF case)
Solved       Edges mis-aligned      Edges aligned
corners on:        with F2L              with F2L

F              "b"                    U "a"
B              U2 "b"                 U' "a"
L              "a"                    U' "b"
R              U2 "a"                 U "b"
A bit more complicated than simply treating the angle case and AUF case independently, if you ask me. (And the cases using a U2 are not FTM-optimal, but cube rotations are totally avoided using this scheme.)

in that other thread, is HTM another way to say FTM?
Yes.

Mathematicians in the early days generally assumed <U,D,F,B,L,R> (face turns only) solving, so they simply referred to the metrics involving face turns as QTM (quarter-turn metric) and HTM (half-turn metric). But in the speedsolving community, we don't generally assume solving via face turns only, so we typically say "FTM" (face turn metric) to indicate anything that is a face turn counts as a single move, rather than saying HTM which doesn't describe "what type" of half-turns are allowed.

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#### supercube

##### Member
I think it is very exciting to know that there are easy and advanced methods for the 3 color cube that can both be used in speedsolving. avoiding cube rotations and U moves is definitely something to work for long term without getting bored.

#### supercube

##### Member
Code:
X INDICATES NOT SOLVED

B
0X0
L 00X R
000
F

R2 U F2 R2 F2 U

B
0X0
L X0X R
0X0
F

L2 R2 D L2 R2 U

B
XXX
L X0X R
0X0
F

R2 D L2 D' R2 U

B
X0X
L 00X R
0X0
F

L2 U F2 R2 U R2 U

B
XXX
L X00 R
000
F

R2 U' R2 U R2 U

#### supercube

##### Member
I started using this for the first time today. I should have checked it out a long time ago. the methods contained here are fantastic. instead of layer by layer OLL + PLL solving you can solve using petrus or something else. the methods used in this link are 3 move or 5 move solutions to finish the 3 color cube. I don't think I will switch to a %100 corners first solve style but I think if I can influence corner rotation intuitively during a LBL or petrus solve then I have a good chance at getting a case with finishing move length 3. also interesting is that corners first solvers can use roux for edge orientation and permutation with great success due to the limited edge identity (increased lucky edge permutation). I regret not using a speed cube but the diansheng is working reasonably well.

#### supercube

##### Member
I am now %100 convinced that the corners first method is better for fewest moves, less algorithms to memorize, shorter algorithms, easier recognition, and most likely faster times.

after all, think about this, there are only two types of unique corner pieces. like binary, each corner is correct or it is not correct. the maximum number of moves to permute (but not care to orient) one corner piece from a scrambled cube is 1 move in QTM (with one exception). there is however a very un-lucky rare case of having all 8 corners in a specific configuration that makes this statement untrue slightly. in %1.42857142857 of scrambles (4/8*3/7*2/6*1/5) you will have for example corner type A in LFU RFU RFB RFD also with corner type B in LBU LFD LBD RBD. given that in this example that RFU should contain type B as the correct corner piece, for this special case you can not move the correct corner piece into the RFU position with 1 move in QTM because LFU RFB RFD all contain type A pieces. a type B corner piece must be moved from one of the LBU LFD LBD RBD positions to the RFU position using 2 moves in QTM. also 3 moves QTM if the corner piece is taken from LBD to be placed in RFU.

on average there will be 4 corners not permuted. I do not know if the distribution is flat or gaussian. I do know that if you plan your moves carefully that you can position 1, 2, 3, or 4 corners pieces simultaneously using 1 move in QTM. the lower bound for move length in QTM to place but not orient all 8 corners from any given scramble would likely be based on the lower bound to generate the worst case scenario given in the example above from a solved cube. I can manually try the upper bound by hand. we know it is less than or equal to the 14 move superflip provided in this thread. I do not have proof but I think given the 1 move QTM solutions to place any given corner it would certainly be around 8 plus or minus a few.

in conclusion I think that a corners first solver can orient and permute all the edges after permuting and orienting all the corners with a combination of slice moves and 180 face turns (L2 R2 U2 D2 F2 B2) that will preserve the appearance of corner sticker colors. there are indeed even more algorithms that do not conform to this convention but do preserve the appearance of corner sticker colors. it is likely that none of these algorithms are greater than 7 moves unless we have a system of solving all 12 edges at once. a 32 move speed solve is reasonably easy to achieve using this method. that number will be reduced by combining edge orientation and edge permutation into a one look. we can also use edge influence when solving corners intuitively since most corners first solves of a 3 color cube have many equally efficient alternate mirrors. that would bring a speed solve down to 24 moves.

#### supercube

##### Member
the most important alg you will need for solving with permuting the corners as step 1. I use this to swap two bad corners. if you have 4 bad corners on the same face you can always turn that face a quarter turn in either direction. having only 2 bad corners looking at the entire cube is 5 moves HTM. 1 move QTM vs 5 move HTM. the difference is huge.
Code:
   B
XXX
L X00 R
000
F

R2 U' R2 U R2 U
8 bad corners can be solved by

Code:
U D
U D'
U' D
U' D'
L R
L R'
L' R
L' R'
B F
B F'
B' F
B' F'
the optimal solution for 6 bad corners is unknown to me at this time but you can use D if 4 of the bad corners are in the D layer. then you will have 2 bad corners in the U layer. see above.

also very interesting is that all of these moves preserve corner orientation so you could in theory orient corners as step 1 before permuting them. or you can permute them as step 1. there is absolutely no difference here. right now I am permuting corners first and using sune after cube rotations with color neutral LL. I do this because I am very good at corner permutation recognition but not very good at corner orientation when picking up a scrambled cube. they all look the same almost.

#### supercube

##### Member
switched back to layer by layer but almost never start with a cross or cross hunting exactly. I have found that color neutral solving is of great importance. I also found that one can not be strictly petrus or strictly beginner/CFOP/fridrich. the 3 color cube has very little in the way of %100 unique pieces (cubies?). it does have many alternate pieces with the same identity or interchangeable identity. the flexibility of the solver in combination with the flexibility of the cube itself will exponentially yield more efficient solves. I use edge influence with the corner slot pair. I do OLL for the corners only. then I use the modified PLL in this thread. I keep my eyes on it while executing the OLL slow so I can execute the PLL in a fluid motion. memorizing all reflections, rotations and inverse of the modified PLL has proved to be very valuable with very little difficulty. ladies and gentlemen this is in my opinion the fastest way to solve this puzzle.

#### supercube

##### Member
I have a new question. Starting from a solved 3 color cube, what are the possible 2 move, 3 move, 4 move scrambles (HTM) with the 48 redundant symmetries removed.

for example, in 2 moves, there is only

Code:
U R
U R2
U D
Notice there is a lot less than 18*15 unique 2 move HTM scrambles before we remove redundant symmetries on a 6 color cube. A 3 color cube will not ever start a scramble with a 180 face turn. Nor will the second term be a 180 face turn on an opposite face from the first term. Therefor, before symetries are removed, a 3 color cube can have only 12*14 scrambles 2 moves long. And only 3 possible states with unique properties. In 1 move HTM there is only one scrambled state of a 90 face turn.

So far we have Fibonacci 1,1,3 starting from solved to scrambled. Any good programmers want to cover some new territory here? If God's number is 14 then maybe it would be similar to the 15th term of the Fibonacci sequence? Is the rate of growth perfectly logarithmic?