Calculating the Number of K4 "PLLs" on the nxnxn Cube

[Christopher Mowla]

Hey guys, what's up?

This is a follow-up thread to Calculating the Number of K4 "OLLs" on an nxnxn Cube.

TLDR of that thread:

1. @cuBerBruce ultimately was the one to find a method to find the number of OLLs there are on the nxnx in the 14th post.
2. I wrote this paper to help to visualize his method.
3. I wrote this paper to create a closed formula. (Some inputs and output Wolfram|Alpha links can be found in this post. And, not that it matters here, I eventually made an alternate closed formula which doesn't contain step functions.)

Well, I want to find out the same thing for K4 PLLs!

I have been wanting to do this for years, but what triggered this was, recently this topic on rCubers was started. (OP asked how many K4 PLLs are there on the 4x4x4 - 7x7x7.)

As you can see, one of the commenters stated:

4x4: i think probably something like more than 4!*4!*4!/4/4=864
5x5: more than 4!*4!*4!*4!/2/4/4=10368
6x6: more than 4!*4!*4!*4!*4!/4/4=497664
7x7: more than 4!*4!*4!*4!*4!*4!/2/4/4=5971968

Taking the 4x4x4 case for example,

• 1st 4! represents the number of oriented wings on "one half" of the set of wing edges in a face.
• 2nd 4! represents the "..." in the other half.
• 3rd 4! represents number of corner permutations.
• 1st 4 means to assume that 1 corner is solved. (4! / 4 = only consider 6 corner permutations.)
• 2nd 4 means we divide by 4 to approximate the symmetries. (So the actual number will be > 864.)

I posted this PDF in that thread, which very much closely resembles the first one I made to illustrate cuBerBruce's OLL method, and I arrived at 1,639 unsolved 4x4x4 PLL cases versus the 911 unsolved PLL cases for the 4x4x4 that another commeter claimed he generated in a spreadsheet in Microsoft Excel.

Interestingly, I calculated 10,831 unsolved PLL cases for the 5x5x5 (which is very close to the approximated 10,368 in the quote above). And my wing edge case symmetry counts agreed with the other poster who came up with the 911 for the 4x4x4. (He didn't carry out the same computation for the 5x5x5 - 7x7x7.)

I was hoping to get a second opinion on this result for 4x4x4 PLLs (and maybe for the nxnxn). Preferably if someone can look at my PLL PDF and see "where I went wrong"? (Most ... if not all... of the work should already be done for you.... I just want to get this issue resolved. I was wrong with calculating the number of OLLs, and I could be wrong again for the PLLs. That doesn't matter. I just want to get this resolved.)

The only portion of my PLLs PDF that I (initially) was unsure of was the corner permutation table (for even cubes) on page 1. I state my reasons/justification for it below.

In 4! = 24 permutations of 4 objects, there are:

• 8 3-cycles
• 6 4-cycles
• 3 2 2-cycles
• 6 2-cycles
• 1 solved state

We can adjust the upper face (AUF) to solve 2 of the 6 4-cycles, 6 of the 3-cycles (just a 3-cycle and its inverse will remain), 3 of the adjacent 2-cycles, and one of the 2 diagonal 2-cycles.

We can then cleverly use the 3-cycle + AUF to take care of the remaining 4-cycles, as well as the adjacent 2-cycle.

So we are now left with 1 diagonal 2-cycle, a 3-cycle and its inverse, and the solved state.

(I used an E-perm instead of a diagonal corner swap, but you can convert one into another by AUF.)

That is, I assumed that the 4x4x4 wing edge PLLs cannot be reduced by AUF. That it's just the corners which we need to try to reduce (as I attempted to in the explanation above... and of course on page 1 of my PLLs PDF) by AUF.

Additional Notes:

• There are 167 unsolved 4x4x4 wing edge PLLs...ignoring corners completely. (I literally show an image of each case in my PDF.)
  • cuBerBruce helped me to arrive at that number in the past. (There is no need for anyone to verify that aspect of it.)
  • The other respondent in the reddit thread confirmed my distribution on their symmetries to what I show in the PLLs PDF as well. (So yet another reason why there is no need to verify the wing edge cases aspect.)

I want to get my PDF correct (if it isn't) so that I can attempt to make a closed formula for the number of "PLLs" of the nxnxn cube for the community. (Make a "Part 2" paper deriving a closed formula.)

Any help (or confirmation that I'm correct) would be greatly appreciated!

 

[Deleted member 36445]

k4 is nice

 

[abunickabhi]

Interesting study Mowla, thanks a lot for sharing your calculations.