The title is a little general, but it should be, as I have a lot to share about forming several types of 2-cycle inner-layer odd permutation algorithms.

To begin, why not start with the "Pure Edge Flip" (my favorite, obviously).

I would first like to show you how to derive common OLL parity algorithms by hand. To do this, I have taken the time to make tutorial videos for three popular algorithms. Each of the three algorithm's derivations are two videos long.

Note: I recommend you to watch the derivation on the Standard Algorithm first before you watch the other videos.

The Standard Algorithm

r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2

Derivation

Part I

Part II

Lucasparity

r U2 r U2 r' U2 r U2 l' U2 l x U2 r' U2 x' r' U2 r'

Derivation

Part I

Part II

Another algorithm which consists of all U2 face turns

r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 l r2 U2 r'

Derivation

Part I

Part II

Please watch the videos before reading this spoiler:Spoiler:

In addition, I have chosen to release...

God's Algorithm?

For my main method for pure edge flip algorithms, I have made a video on the best/briefest algorithm I have ever found in block quarter turns (BQTM) that works for all cube sizes (Very few low move count algorithms work for all cube sizes).

The Holy Grail

From now on, I will abbreviate BQTM with just q, and BHTM with h.

(Also note that BHTM is commonly called btm. So h = BHTM = btm).

On the 4X4X4: 19q/18h

z d' m D l' u' 2R' u l u' l'2 b' 2R' b r' R' 2U y' m' u x2 z'

On the 5X5X5: 19q/18h

z d' 3m D 3l' u' 2R' u 3l u' 3l'2 b' 2R' b r' R' 2-3u y' 3m' u x2 z'

On the inner-orbit of the 6X6X6: 20q/19h

z 3d' 4m D 3l' 3u' 3R' 3u 3l 3u' 3l'2 2R' 3b' 3R' 3b 3r' R' 2-3u y' 4m' 3u x2 z'

On the outer-orbit of the 6X6X6: 19q/19h

z 3d' 4m D 3l' 3u' 2R' 3u 3l 3u' 3l'2 3R' 3b' 2R' 3b 3r' R' 2-3u y' 4m' 3u x2 z'

=

z 3d' 4m D 3l' 3u' 2R' 3u 3l 3u' 4l 3l 3b' 2R' 3b 3r' R' 2-3u y' 4m' 3u x2 z'

On the inner-orbit of the 7X7X7: 20q/19h

3d' 5m D 4l' 3u' 3R' 3u 4l 3u' 4l'2 2R' 3b' 3R' 3b 3r' R' 2-4u y' 5m' 3u x2 z'

On the outer-orbit of the 7X7X7: 19q/19h

z 3d' 5m D 4l' 3u' 2R' 3u 4l 3u' 4l'2 3R' 3b' 2R' 3b 3r' R' 2-4u y' 5m' 3u x2 z'

=

z 3d' 5m D 4l' 3u' 2R' 3u 4l 3u' 5l 4l 3b' 2R' 3b 3r' R' 2-4u y' 5m' 3u x2 z'

I point out in the latter portion in the video that its average between quarter and half turns is also less than all other algorithms which currently exist.

Here is a link to the following formula in Wolfram|Alpha (the simplified formula shown near the end of the video)

Just substitute an integer greater than or equal to 4 for n to obtain the average for a cube of sizen.

Formula Derivation

Spoiler:As you can see, there are many more cases for an average of 19.5 than for 18.5 and 19.0.

But, the overall average still is less than 19.5.

Based on the fact that even and odd cubes have the same number of orbits (e.g. both the 6X6X6 and 7X7X7 have two inner-layer orbits), the floor function can be omitted and we can take the limit as the cube size gets large.

This means, no matter how largengets, the average will come arbitrarily close to, but it never reaches 19.50. (This is apparent already without calculus, but I thought a "second opinion" would further verify this statement.)

My 23q/16h "cmowlaparity" obviously has an average of 19.5, but the Holy Grail beats it slightly.

x' r2 U2 l' U2 r U2 l x U2 x' U r U' F2 U r' U r2 x

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