Self Solving Algs

[fw]

the shortest one I can think of is r L'
does that count?

Sure it does.. Actually, if you talk about the "order of an algorithm", you mean the order of the permutation the algorithms corresponds to (so of course two different algorithms (different moves, etc.) which correspond to the same permutation have the same order). Your algorithm is the identity permutation, which has order 1, and so has your algorithm

 

[cuBerBruce]

I got that to be 360. Any larger ones?

There is a permutation in the cube group which has order 1260 and it is proven that there is no cyclic subgroup of the cube group which has a higher order

The maximum order of 1260 applies to the cube group with fixed centers. If you allow non-fixed centers (cube rotations, inner slice turns, double-layer turns) as mentioned at the link Lucas mentioned (http://mzrg.com/rubik/orders.shtml), I can find at least an order 2520 position (misoriented corner cycles of 3 and 5, misoriented edge cycles of 4 and 7, and a 4-cycle of centers).

Note the position "cF U" mentioned at the link has to become "L D R U" if you limit to fixed centers, and (L D R U) has order 1260/4 = 315.

The position mentioned by Joyner in "Adventures in Group theory" is R U2 D' B D' (attributed to J. Butler). That is a fixed-centers position of order 1260.

 

[fw]

The maximum order of 1260 applies to the cube group with fixed centers. If you allow non-fixed centers (...)

Yes, you are right, I didnt think about that. But why would someone want to do that? The group gets much bigger if you do it like that and you do not gain any interesting information from giving up the fixed-center restriction, do you?

 

[Derrick Eide17]

lol wow those are some cool things. WOW i never knew there were such bigs ones before. also VERY funny Johannes lol yes that alg DOES work but i have an even SHORTER one: Just dont buy a Rubik's cube

 

[cuBerBruce]

The maximum order of 1260 applies to the cube group with fixed centers. If you allow non-fixed centers (...)

Yes, you are right, I didnt think about that. But why would someone want to do that? The group gets much bigger if you do it like that and you do not gain any interesting information from giving up the fixed-center restriction, do you?

This thread was referencing algorithms that did not leave the centers in the same positions. In particular, the web page Lucas mentioned used several algorithms that don't leave the centers fixed. You can think of M as being equivalent to L' R as they produce the same position in terms of the cube group with respect to fixed centers. But if you look at M U and L' R U, you see that M U and L' R U do not produce the same cube position. So in that sense, M is not the same as L' R. Basically, when you start using cube rotations, inner layer moves, or double layer moves for algorithms for the 3x3x3 cube, you are implying the use of the larger cube group where centers are no longer considered fixed. That group is 24 times bigger than the group usually referred to as the Rubik's cube group.

One can also look at M U and L' R B which seem to create the same cube position (with respect to fixed centers). Note that the order of M U is 8, while the order L' R B is 180. So if you want to talk about the order of a position (as opposed to the order of an algorithm), you really need to be clear about whether or not you are considering the centers to possibly have moved with respect to their start positions.

Another way to look at this. If you use algorithms that use cube rotations, inner layer moves, or double layer moves; the move U doesn't necessarily move cubies around same colored center that the move U would do at the start. In the usual Rubik's cube group, however, U always moves cubies around the same center. When you execute an algorithm repeatedly (and the algorithm contains cube rotations or other moves affecting centers), the move U will not necessarily move cubies around the same center in the 2nd execution of the algorithm (for simplicity, assume the algorithm only contains U once) as it did the first time the algorithm was executed. So the usual Rubik's cube group simply isn't adequate when dealing with repeated algorithms containing the types of moves mentioned at the beginning of this paragraph.

I also want to mention that within a finite group, the order of a position (element) of that group is constant, regardless of what algorithm might be used to reach that position. But not all puzzles have positions that satisfies the defining properties of a group. For example, the positions of the 4x4x4 cube do not make up a group (unless you mean the 4x4x4 supercube). Several 4x4x4 supercube positions may correspond to the same position of the ordinary 4x4x4 because centers of the same color can be permuted without affecting the apparent position. But these 4x4x4 supercube positions may not have the same order. Thus, the same position for an ordinary 4x4x4 may have an ambiguous "order." Any specific 4x4x4 algorithm will generate a specific supercube position, so it makes sense to talk about the order of a 4x4x4 algorithm while it does not make sense to talk about the order of a (non-supercube) 4x4x4 position. The same goes for larger cubes (non-supercubes).

Finally, I would mention that even-size cubes (2x2x2, 4x4x4, etc.) do not have a fixed relationship of the (centermost) centers, so unless you restrict moves sufficiently (such as using only moves that keep a specific corner fixed), you have multiple solved positions (with respect to the layers referred to by U, R, F, etc.). You could have the U face white and the F face green, or you could have the U face blue and the F face orange, but still have a solved cube in either case. Similary, odd-size cubes where centers are allowed to be moved also have this issue. So the group can have 24 elements corresponding to a solved cube. But, by definition, a group can only have one identity element. Also, mathematically speaking, the order of a group element is the power the element must be raised to in order to reach the identity element (in puzzle terms, how many times the algorithm for the position must be applied to get back to the starting position). So you might reach one of the 23 other "solved positions" before the start position.

For example, Sam's algorithm (r L'), in terms of the 24x larger cube group, creates a solved position, but it is not the identity of that group. Technically in terms of the 24x larger cube group, r L' has order 4. In that group, Sam's algorithm produces x, not the identity, and x has order 4.

 

[Cuc]

Obviously all inversion/refletios/ rotations are also valid?
Are there any others that are 12 or less that you can think of?

Yes, any rotations (where you put the first turn to the end or vice versa) and inversions are valid as well.
Proof:
If for any algs X and Y, X Y = I, then
- Y X = Y X I = Y X (Y Y') = Y (X Y) Y' = (Y I) Y' = Y Y' = I, and
- (X Y)' = I' = I.
EOP.

Some identity (self-solving) algs were mentioned elsewhere, but here are some more.
I write [X, Y] for the commutator X Y X' Y' and {X, Y} for the conjugation X Y X'.

Consider close calls:
- [U, R] F' U L [F, U] L' U' F (14), duh!
- [U, R] (F' D F) [R, U] (F' D' F) (14), proof that [U, R] and (F' D F) commute! Visually evident.

4-gen <B,L> excluded
D2 (F2 U D F2) U2 (R2 D U R2) (10f)
U F R2 D R D' F' R2 U' R' (10f)
U F2 U2 R2 (U D) R2 D2 F2 D (10f)
(U D) R2 D2 F2 (U D) F2 U2 R2 (10f)
(U D) F2 U2 R2 (U D) R2 D2 F2 (10f)

[U', R2] D' [U, F2] D (10f)
U F2 U2 F2 U' D R2 U2 R2 D' (10f)
U F2 D2 F2 U' D R2 D2 R2 D' (10f)

3-gen <B, F, D> excluded:
U R2 U2 L U2 R2 U' R2 U2 L' U2 R2 (12f)
U' R2 U2 L U2 R2 U R2 U2 L' U2 R2 (12f)
(R2 U2)2 L (U2 R2)2 U2 L' U2 (12f)

other
(L U2 L) (R' F2 R') (L' F2 L') (R U2 R) (12f)
[U, {R2 U2, L}] = U R2 U2 L U2 R2 U' R2 U2 L' U2 R2 (12f)
[{U2 R2, U2}, L'] = U2 R2 U2 R2 U2 L' U2 R2 U2 R2 U2 L (12f)

[M', S'] [E', M] [S, E] (12s)
[M, S2] [E, M2] [S, E2] (12s)

Curiosity:
2-gen
U2 R U R U' R U' R' U' R2 U R U R' U R' U' R' (20q*) ~ {{U, R'}, {R', U2}} {{R, U'}, {U', R2}}
U2 R' U' R' U R' U R U R2 U' R' U' R U' R U R (20q*) ~ {{U', R}, {R, U2}} {{R', U}, {U, R2}}
(~ means: modulo rotations)

 

[FastCubeMaster]

Yes, any rotations (where you put the first turn to the end or vice versa) and inversions are valid as well.
Proof:
If for any algs X and Y, X Y = I, then
- Y X = Y X I = Y X (Y Y') = Y (X Y) Y' = (Y I) Y' = Y Y' = I, and
- (X Y)' = I' = I.
EOP.

Some identity (self-solving) algs were mentioned elsewhere, but here are some more.
I write [X, Y] for the commutator X Y X' Y' and {X, Y} for the conjugation X Y X'.

Consider close calls:
- [U, R] F' U L [F, U] L' U' F (14), duh!
- [U, R] (F' D F) [R, U] (F' D' F) (14), proof that [U, R] and (F' D F) commute! Visually evident.

4-gen <B,L> excluded
D2 (F2 U D F2) U2 (R2 D U R2) (10f)
U F R2 D R D' F' R2 U' R' (10f)
U F2 U2 R2 (U D) R2 D2 F2 D (10f)
(U D) R2 D2 F2 (U D) F2 U2 R2 (10f)
(U D) F2 U2 R2 (U D) R2 D2 F2 (10f)

[U', R2] D' [U, F2] D (10f)
U F2 U2 F2 U' D R2 U2 R2 D' (10f)
U F2 D2 F2 U' D R2 D2 R2 D' (10f)

3-gen <B, F, D> excluded:
U R2 U2 L U2 R2 U' R2 U2 L' U2 R2 (12f)
U' R2 U2 L U2 R2 U R2 U2 L' U2 R2 (12f)
(R2 U2)2 L (U2 R2)2 U2 L' U2 (12f)

other
(L U2 L) (R' F2 R') (L' F2 L') (R U2 R) (12f)
[U, {R2 U2, L}] = U R2 U2 L U2 R2 U' R2 U2 L' U2 R2 (12f)
[{U2 R2, U2}, L'] = U2 R2 U2 R2 U2 L' U2 R2 U2 R2 U2 L (12f)

[M', S'] [E', M] [S, E] (12s)
[M, S2] [E, M2] [S, E2] (12s)

Curiosity:
2-gen
U2 R U R U' R U' R' U' R2 U R U R' U R' U' R' (20q*) ~ {{U, R'}, {R', U2}} {{R, U'}, {U', R2}}
U2 R' U' R' U R' U R U R2 U' R' U' R U' R U R (20q*) ~ {{U', R}, {R, U2}} {{R', U}, {U, R2}}
(~ means: modulo rotations)

That's really cool and all, but.. chances are that @philkt731 won't be seeing that reply, as his message was 10 years ago. Could spark a new conversation though so it won't always be such a bad thing

 

[Cuc]

That's really cool and all, but.. chances are that @philkt731 won't be seeing that reply, as his message was 10 years ago. Could spark a new conversation though so it won't always be such a bad thing

Indeed! But you read it, and I read the OP. It intrigued me and I spent weeks figuring the algs out. And so, where else to put it? Anyway, for those who find this, it'll be some kind of treasure. I don't mind

Although I like identity algs in their own right, they seem to have an application in speeding up the search for optimal solutions [the identity algs were called "shortcuts", if I remember correctly]. But I've lost the thread to that. Feedback on the uses of these algs is appreciated.