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A beautiful conclusion to my research, which I believe was not redundant and helps to better understand the (categories) classes of symmetry. Can one more question about non-spatial symmetries be clarified? It says: "Only 79 780 positions of distance-24 in the quarter-turn metric are known. Of...

Reducing 12 actual positions, each one move away from the initial state, to just two is quite a puzzle, but I suppose I can agree with you, if we call the 'two distinct positions' two distinct categories of positions that are solved using mirrored or inverse algorithms.

For the "superflip composed with four spots" position alone, 92 optimal solutions can be written, and that's not including interchanges of move pairs on opposite faces. It's no wonder it's confusing. This is yet another reason why a standard is needed for encoding a position by more than just a...

Yes, the cubes (ccccccchhhhhhhhttt) & (bdbdbdbtetetetehhh) are equivalent. The imperfect solver was overlooking an additional R<->L swap, which is why some of the solutions converged while others did not. Thank you very much for your time and attention. Let's summarize the search for 25q...

Following your advice, I have visualized all of these positions, without a doubt that they have long been known to you. Could you please explain something? I see that the top two positions are one move (D, D') away from the single 26q* position located in the center. But for the bottom two...

Perhaps. I was checking against your findings (https://www.cube20.org/distance20s/): “There are essentially only two known distance-25 positions, both immediate neighbors of the above distance-25 positions; scrambles for them are: U1U1F1U1U1R3L1F1F1U1F3B3R1L1U1U1R1U1D3R1L3D1R3L3D3 Scramble...

Using the "Rubik's Cube Signature" (R'CS) program, 2 more positions at a distance of 25q have been discovered:

A team of engineers from Sony Computer Science Laboratories Inc. and the NeuroPiano Institute in Kyoto has developed a robotic exoskeleton that helps pianists improve their playing technique. The exoskeleton, worn on the hand, guides the musician's finger movements during training, which helps...

Sarcasm is not math's best friend.

I'm not the Head of this department. If you deem it necessary, proceed with the transfer.

A code using Base64 would provide the shortest representation at 11 characters. Why not consider it? If you read the postscript, you would see that the goal of the Catch-24 code is about more than simply minimizing the character count.

browser-based UNLICENSE program https://vibcatch24.github.io/R-CS/

The longest personal name belonged to Hubert Blaine Wolfeschlegelsteinhausenbergerdorff Sr. His full name, registered in the Guinness Book of World Records, consists of 26 given names (one for each letter of the English alphabet) and an exceedingly long and meaningful surname. In total, the name...

Both matrices and quaternions serve to represent elements of the rotation group SO(3) (the group of rotations in three-dimensional space). Quaternions (more precisely, the group of unit quaternions SU(2)) provide a "double cover" of SO(3) - here you are absolutely correct. Matrices offer a...

A unit quaternion is a quaternion of norm one. Do you read Russian? https://ru.wikipedia.org/wiki/Кватернион

Yes, the parity of the sum is not the parity of the terms.

I was quite surprised that, half a year after publication, readers of the forum were drawn to this post (though not particularly attentive ones). I quote myself from that post: “The minimum number of nonequivalent positions was calculated by D. Hoey (the true number of cube provisions = 901.083...

This reminds me of a line from the musical 'Chicago' – 'He didn't quite hear me. So, I took the shotgun from the wall and fired two warning shots—right into his head.' Are you trying to say that there are fewer than 1,802,166,803,103,744,000 equivalent positions? Well, I don't object to that...

Please give more context here.

On the factor of 24: for those not deeply familiar, try to visualize (assuming your imagination is up to the task) recoloring any given position in line with the 24 possible orientations of the cube; you will find this yields an identical set of solutions. To my great regret, any further...