Catch 22 ? Catch-24. You Can’t See Me.

[swipymam]

1. It's not about the found sequel to Joseph Heller's absurdist novel Catch-22, which ranks #11 on the Top 200 Novels in the United Kingdom list.
2. It's not about the music album «You Can't See Me» by wrestler John Cena, the author of the eponymous gesture.
3. And it's not about the "You Can't See Me" gesture made to his teammate, 22-year-old basketball star Catlin Clark, with the number 22.
4. It's not about the mental health series on Apple TV+ hosted by Oprah Winfrey and Prince Harry.

It's a saga about some of the symmetries of a cube.

You've probably come across the number of possible scrambles of the Rubik's cube:
43,252,003,274,489,856,000 = 12! * 2^12 * 8! * 3^8 * 1/12
Here:
Edge positions = 12! - the number of possible placements of the 12 edge elements (hereinafter referred to as edges).
Edge flips = 2^12 - the number of orientations of all edges of the cube.
Corner positions = 8! - the number of possible placements of the 8 corner elements (hereinafter referred to as corners).
Corner rotations = 3^8 - the number of orientations of all corners of the cube.

As for the last factor in this formula 1/12, many of us probably don't have enough knowledge to justify the reduction by 12 mathematically, let's just say that dividing by 12 eliminates the number of impossible positions (edge flips, corner flips, parity) for a correctly solved Rubik's Cube.

It is generally accepted that the number 43,252,003,274,489,856,000 is the total number of possible permutations that can result from any number of moves (rotations of the cube's faces).
However, will all these positions be different? - Only formally.

As is known, there are only 13 axes of symmetry of a cube:

By rotating the entire cube along one or a combination of two axes of symmetry, 24 different orientations of the cube in space are achieved, the same check that is used at the beginning of solving the cube and does not count as a move. This can be called the "Catch-24" because no matter how we rotate the cube, all the solutions, up to the last one, will remain the same, only the solution symbols will change. Such cubes are called equivalent. Therefore, to get the actual number of different positions, we can reduce our number by 24:
43,252,003,274,489,856,000 / 24 = 1,802,166,803,103,744,000
"You Can't See Me" - this new number tells us!
You can surprise your math teacher now, by writing on the board in class: 3x3x3 = 1,802,166,803,103,744,000

The minimum number of nonequivalent positions was calculated by D. Hoey (the true number of cube provisions =
901.083 404 981 813 616), which reduced “our number” by almost half (1.99999999992498), but unlike the Catch-24 we discussed, it is not so trivial.

For example, for God's number (26q*), found in 2014 the researchers write: "We divided the positions into 2,217,093,120 sets of 19,508,428,800 positions each. We reduced the number of sets we needed to solve to 55,882,296 by using symmetry and set covering." In this way, they reduced the number 43,252,003,274,489,856,000 not by 24, but by almost 40 times!

Incidentally, the number 55,882,296 has 22 divisors — numbers that divide the desired number evenly, not including 1 and the number itself. The circle is closed, gentlemen!

P.S. While I was writing this post, I found solvers for the superflip composed with four spot (26q*), which was mentioned by some reputable researchers long ago.
Oddly enough, there were 22 equivalent solutions (and this is also a “Catch-22” ).
Interestingly, the solutions to this position are also mirror-symmetrical.
Feel the difference in the effectiveness of these solutions:

Feel the difference in the efficiency of each pair of these solutions:
1)——>
F B U' R L' U' D R' F2 L' R' U D B' R2 U D' F2 L' U2 D2 (21f, 26q*, 18t)
D2 U2 L F2 D U' R2 B D' U' R L F2 R D' U L R' U B' F' (21f, 26q*, 18t)
<——1a)
F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' U2 D2 (21f, 26q*, 18t)
U R L' D F U R' F B D R' F' R' D U2 L R F2 U2 D2 L R (22f, 26q*, 19t)
D R L' U B D R' F B U R' B' R' U D2 L R B2 D2 U2 L R (22f, 26q*, 19t)
U2 F U2 R’ L F2 U F’ B’ R L U2 R U D’ R L’ D R’ L’ D2 (21f, 26q*, 20t)
B R L' F U B R' U D F R' U' R' F D2 U2 F2 L R U2 L R (22f, 26q*, 20t)
F R L' F U B R' U D F R' U' R' B L2 R2 B2 D U R2 D U (22f, 26q*, 20t)
R’ F U F L' U' D' F R' U' L' F' B L' D' U' B2 D' U' L2 B2 F2 (22f, 26q*, 20t)
L’ F U F L' U' D' F R' U' L' F' B R' B' F' U2 B' F' R2 U2 D2 (22f, 26q*, 20t)
R D' R F' L' U R B' U B' D B' U' F R' F2 L2 U2 L2 U' D2 (21f, 26q*, 25t)
R U' R B' L' D R F' D F' U F' D' B R' B2 L2 D2 L2 U2 D' (21f, 26q*, 25t)
Time Robot Metric - TRM (t), defined as follows: quarter turn of any number of faces located on one axis is counted as 1 turn.

 

[bcube]

Therefore, to get the actual number of different positions, we can reduce our number by 24:
43,252,003,274,489,856,000 / 24 = 1,802,166,803,103,744,000

By no means I am expert on cube symmetry (link 2, link 3) but you should by dividing by a factor a little bit less than 24 here.

Also, if you want to get a truly different configurations (450,541,810,590,509,978), you should by dividing by another factor a little bit less than 2 for inverse move sequence, also known as anti-symmetry.

 

[swipymam]

By no means I am expert on cube symmetry (link 2, link 3) but you should by dividing by a factor a little bit less than 24 here.

Also, if you want to get a truly different configurations (450,541,810,590,509,978), you should by dividing by another factor a little bit less than 2 for inverse move sequence, also known as anti-symmetry.

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 increase of this factor of 24 is not pertinent to this discussion.

 

[Filipe Teixeira]

What's your point? @swipymam

 

[swipymam]

What's your point? @swipymam

Please give more context here.

 

[bcube]

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 increase of this factor of 24 is not pertinent to this discussion.

Just to be clear on what I was saying: there are 43,252,003,274,489,856,000 different configurations and 24 whole-cube rotational symmetries. But when reduced by the whole-cube rotations, there are not exactly 43,252,003,274,489,856,000 / 24 unique configurations.

 

[swipymam]

Just to be clear on what I was saying: there are 43,252,003,274,489,856,000 different configurations and 24 whole-cube rotational symmetries. But when reduced by the whole-cube rotations, there are not exactly 43,252,003,274,489,856,000 / 24 unique configurations.

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, and this is mentioned in the post. But please don't write that the coefficient should be LESS than 24.

@bcube:
...but you should by dividing by a factor a little bit less than 24 here.

That is what's incorrect!

 

[Fred]

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, and this is mentioned in the post. But please don't write that the coefficient should be LESS than 24.

That is what's incorrect!

Please no shooting each other on the forums.

 

[Renslay]

Most of the cube states has 24 symmetries (48 if you count inverses). All of them are different from each other. If this were true for all states, the number of unique states would be indeed N / 24 (with N being the number of total cube states).
However, some cube states have symmetries that are equivalent to the original state. For example, a simple U2 turn. All of its symmetries are equivalent, including the inverses.
Hence, the number of unique states are not N / 24 unique cube states. Ergo, to get the number of unique cube states, you have to divide N with a number different than 24 (probably not even an integer).

 

[bcube]

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, and this is mentioned in the post. But please don't write that the coefficient should be LESS than 24.

Let´s recap what we know:

1. the number of symmetry-reduced configurations for the 3x3x3 is 901,083,404,981,813,616, so the ratio of cube configurations to symmetry-reduced cube configurations for 3x3x3 is 43,252,003,274,489,856,000 / 901,083,404,981,813,616 which is approximately 47.9999998172897 - but you are not interested in reflections so let´s move on
2. the number of symmetry-and-anti-symmetry-reduced configurations for the 3x3x3 is 450,541,810,590,509,978 - but you are not interested in anti-symmetry so let´s move on
3. the number of whole-cube-rotation-reduced configurations for the 3x3x3 is greater than 43,252,003,274,489,856,000 / 24

How do we know that point 3 is true? That is because some configurations have symmetry and don't have 23 other equivalent configurations. For example, R is only equivalent to L, U, D, F, and B (and itself), so only 6 configurations rather than 24.

Most of the cube states has 24 symmetries (48 if you count inverses).

You meant reflections (also called mirrors), right?

 

[swipymam]

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 = 901.083 404 981 813 616), which reduced “our number” by almost half (1.99999999992498), but unlike the Catch-24 we discussed, it is not so trivial.”

Now, to summarize the entire post: Yes, as D. Hoey indicated (with a cultural nod to Professor Preobrazhensky © M. Bulgakov), the number of equivalent positions is less than the number 1,802,166,803,103,744,000. But this post doesn't assert the contrary either.

This post only says that the universally declared figure of 43,252,003,274,489,856,000 can be easily (without throwing the baby out with the bathwater) divided by 24. And one can solve just the standard WCA orientation, without needing to rotate the whole cube another 23 times. If you have other substantiated facts, please present them to the community.

 

[qwr]

Use The Lemma that is not Burnside's

 

[SpeedCubingLord]

The only thing I understand here is tht there aren't 43 quintillion combinations. How could you lie to me jperm.

 

[SpeedCubingLord]

OHHH... so are you saying relfections of the same cube in different orientations are not counted as scambles. But isn't that why center orientation, white top green front, always stays the same,and each of 43 quintillion different States are in fact unique?

It seems this deep cube theory and im just dumb...

 

[bcube]

That is because some configurations have symmetry and don't have 23 other equivalent configurations.

Let me rephrase this a little bit: That is because while most of the configurations are completely asymmetrical and thus have 24 equivalent configurations, some configurations have symmetry and don't have 23 other equivalent configurations. The more symmetrical the configuration is, the less equivalent configurations it has.

Therefore, the number of whole-cube-rotation-reduced configurations expressed as 43,252,003,274,489,856,000 / 24 is an underestimate, not an overestimate as you are implying by:

the number of equivalent positions is less than the number 1,802,166,803,103,744,000. But this post doesn't assert the contrary either.

 

[qwr]

OHHH... so are you saying relfections of the same cube in different orientations are not counted as scambles. But isn't that why center orientation, white top green front, always stays the same,and each of 43 quintillion different States are in fact unique?

It seems this deep cube theory and im just dumb...

That's why I said...

Use The Lemma that is not Burnside's