Elegant Method Discussion

This is a conversation about my Elegant Rubik's Cube Solution Method, as well as its descriptions. Please send contributions to PeterATdancingDOTorg.
Please see Peter Renzland's Elegant Rubik's Cube Solution Method first -- this page is mereley commentary on the Method, not a description of the Method.

Please see Overview of Four Rubik's Cube Solution Methods (Which also includes the Minimalist Method)

[Google does not index the Standard Description (PDF). Will have to make HTML instead!]

Why does an elegant solution method have 3 different descriptions?

The standard (PDF) description provides both understanding and instruction, with considerable redundancy.
The Verbose Description avoids formal notation, and focusses on detailed instruction rather than conceptual understanding.
The Brief Description describes the solution method to thinkers in 50 words.
There is also the introductory page that lists aspects of elegance, and I may make a short video or two).

I found that it was impossible to have just one description. A concise description is elegant, but it is inaccessible to people who are unfamiliar with compact notation. A description in plain, clear English, has its own elegance. Different things can be left unsaid, depending on the reader. For some readers it is enough to say "The solution method uses 3 transforms: Corner-Swap, Corner-Twist, and Edge-Swap, which are: ...". Some readers can do one face intuitively, but others need to be shown how. Some people don't know that parentheses are used for grouping, in maths, rather than for (optional) commentary. Some people hate reading unnecessary verbosity, other people want everything stated, repeated, and explained in many different ways. Some people won't read at all. Some just read a few words here and there, imagining the rest.

How long, how many turns does this solution method take?

Lucas Etter did 42 turns in under 5 seconds. That's 8 turns per second.

This is a leisurely method, for elegance, not speed.
If we look at each of the 20 moveable cubies for 10 seconds, then move it home in 10 turns, one turn per second,
it will take 6 to 7 minutes to solve the cube at this leisurely pace.
(The total time being about 1 turn every 2 seconds, for about 170 turns.)
Step 1:   10 turns <1 minute
Step 2:   15 turns <1 minute (need to apply at most once)
Step 3:   24 turns <1 minute (need to apply 1, 2, or 3 times)
Step 4:  120 turns <4 minutes
Total:   170 turns <6 minutes.

Why are Up and Right turns the easiest?

Being right-handed, and holding the cube with the left hand, with the left thumb on top of the left corner, near my navel, Right and Back turns are physically easiest.

If I were to raise the cube to be near my nose, raising my left arm and twisting my left wrist toward me, the Back face would become the Up face, making Up and Right turns easy.

I just made a special Plumb Rule Star Wars Edition

Which colour-orientations do I like?

I like to put Red on the Right and Blue on the Back. This is obviously natural to me. On a standard cube, that places White Up, which is also natural, since White has the logo. Standard cubes have Red, Blue, White opposite Orange, Green, Yellow, respectively. (Older cubes differ.)

The general principle is always: Place the cubies to be operated on into the places where the Transforms need them to be, apply the transforms, invert (undo in reverse order) the placing moves. When remembering the placing moves, I say the Face-Colours, not the Face-Positions, because the Face-Positions do not remain fixed. Thus, I may say: Red, Big-Blue Red, Un-Green Big-Blue Red, as I move: -R 2B G. Then, after applying the transform, I apply the remembered inverse: -G 2B R. This priciple is called "conjugate".

How did I create this solution method?

In 1981 I got a Rubik's Cube and began to learn how to solve it myself. I soon discovered that most people simply went through the motions of other people's solution methods. I carefully avoided reading other people's methods, but instead set out to find the most elegant solution on my own. So, I first defined what I meant by elegant, and what a solution method required. Then I pursued such a solution method.

Being right-handed, I decided that I was interested in transforms that only involved R and B turns. I found a sufficient set of such transforms, namely edge-swap, corner-swap, and corner-twist. The process of finding this solution required me to ask others to restore the cube, or to restore the cube myself by taking it apart and re-assembling it, many times over several months.

My solution method first places the corners, using corner-swap, then orients the corners, using corner-twist, then places all the edges properly, using edge-swap.

Can we make it more elegant (or more minimalist)?

While intuition is quite sufficient -- and efficient -- for the initial steps, to rely on intuition for getting started with the Corners, seems ad-hoc and inelegant. However, while it is possible to use Corner-Swap and Corner-Twist to do the Up-Corners, this takes more understanding, more thought, more turns, and more time than the intuitive approach takes. It is easier to develop the intuition to do (the corners of) one face, than to do it non-intuitively.

It's possible to use 2 transforms, or even 1, instead of 3. But that would make the Method much more difficult and inelegant. It's possible to use shorter transform, but they would be more complex. It's possible to use more powerful transforms (with multiple simultaneous effects, but that would be less elegant and easy than the simple effects of the 3 transforms.

The Corner Swap Transform can be applied twice to twist 2 corners. Therefore, the 2 transforms -- Corner Swap and Edge Pair Swap -- would be sufficient.
However, the Corner Swap Transform is difficult to apply for twists, which the solution method less elegant.

My original hope was to construct a solution method from 2 transforms: A 4-turn Corner Pair Swap (which can also Twist 2 corner pairs), and a 6-turn Edge Pair Swap (which can also Flip an edge pair). R U R- U- and R2D2 * 3. Unfortunately the Corner Pair Swap Transform is not suffient.

What other elegant solutions are there?

If you know of any, please let me know. I am not aware of any.

General Thoughts on Comparing Solution Methods

We can apply 4 metrics to solution methods:
1. Number of transforms
2. Complexity of transforms
3. Complexity of description
4. Required knowledge (memorization)
5. Required judgement (intelligence)
Thus, the extreme method "Just use Q-turns" is trivially simple on 1-4, but impossibly complex on 5. The extreme method "Just look up the 20-turn solution sequence in the list of 43 quintillion states" is trivially simple on 3 and 5, very complex on 2, and impossibly complex on 1 and 4. AFAIK, my method scores lowest of published methods on 1-4, and low (but not lowest) on 5. Possible challenges when doing the edges: 1. Keeping Centres aligned; 2. picking a suitable helper-pair; 3. remembering the inverse of the line-up sequence.

Video Outline

Video 1. Rubik's Cube Concepts and Terminology. (2 min)

6 Faces, 8 Corners, 12 Edges; Face Positions / Colours. 20 Cubies. Cubicle. Facelet.
Q-turn. Half-turn. Un-turn. Cube-turn. Edge-flip, Corner-twist.

Video 2. Four-step Solution Method Overview. (2 min)

Do White Corners, Place Yellow Corners, Orient Yellow Corners, Do Edges.
If we know how to swap 2 corners, we can put any corner where it needs to be.
If we know how to twist corners, we can orient all the corners.
If we can swap 2 edges, we can send all the edges home.

Video 3. The 3 Transforms: Corner-Swap, Corner-Twist, Edge-Swap. (2 min)

Corner-Swap: Swaps the Left-Down corners. Hold these two corners with thumb and index fingers, and do: URUR URUR URUR URU.
Corner-Twist: Twists the 3 Back corners, other than Up-Right-Back, right: ULDR ULDR ULDR.
Edge-Swap: Swaps the Edge-Pair Up-Front and Up-Back. It also swaps Right-Front and Right-Back: UURR UURR UURR.

Video 4. Solving the 8 Corners. (Steps 1, 2, 3) (2 min)

Step 1: Do the White Corners intuitively.
Step 2: Place the Yellow Corners (without orienting them), using the Corner-Swap Transform.
Step 3: Orient the Yellow Corners, using the Corner-Twist Transform. Now all the corners are done.

Video 5. Solving the 12 Edges. (Step 4) (3 min)

Step 4: Do the Edges, without disturbing the corners, using the Edge-Swap Transform.

(I'll probably just make 2 videos: 1-4, and 5.)

[Please read the PDF web page instead of the following text, which is here only because Google does not search/index PDF web pages.]

Overview of Four Rubik's Cube Solution Methods

A. The Simple Minded Method. Uses 4 Transforms.

Place Corners; Twist Corners; Place Edges; Flip Edges.

1. Two Corner Swap: UR *7.5 {disturbs edges, centres}
2. Two Corner Twist: F R' F 'U' R' U L U' R U F R F 'L'
3. Two Edge Swap: U2R2 *3 {also swaps a helper edge-pair}
4. Two Edge Flip: U F' U F R B L U' F U' F' L' B' R'

B. The Elegant Method. Uses 3 Transforms.

Do White corners intuitively; place Yellow corners; twist Yellow corners; do Edges.

1. Two Corner Swap: UR *7.5
2. Three Corner Twist: U[r] *12
3. Two Edge Swap (& Flip): U2R2 *3

C. The Minimalist Method. Uses 2 Transforms.

Do Corners, do Edges.

1. Two Corner Swap (& Twist): F2U2[r] UR *7.5 B2R2
2. Two Edge Swap (& Flip): U2R2 *3

D. The Superhuman Method. Uses 1 Transform.

Apply the correct sequence of 20 Face-Turns. Utterly Impractical! See also Peter Renzland's Elegant Rubik's Cube Solution Method.

[ peter [at] dancing [dot] org ]