# Thread: Possible orders of Rubik's Cube positions

1. ## Possible orders of Rubik's Cube positions

A year or two ago, I set out to calculate with pencil and paper the set of all possible orders* of Rubik's Cube positions, combined with the number of positions with each order. The calculation isn't horribly difficult from a theoretical standpoint (SPOILER: details listed below), but my patience didn't last, and I put the question aside.

Today, I tried a similar calculation with a more modest goal: finding only the possible orders. I believe I have succeeded in this. I started by restricting my view to only edges and only corners respectively. The possible cycle structures of edges (or corners) correspond exactly to the partitions of 12 (or 8). I listed all of these partitions in "even permutation" and "odd permutation" columns. Then I found the order of each permutation, noting that for all permutations except a single 12-cycle of edges (or 8-cycle of corners) it is possible for some cycles to be off by a flipped edge (or twisted corner), and therefore that the orders of those permutations can be multiplied by 2 for edges (or 3 for corners). I then tabulated all possible orders for four categories:

ee (edges, even permutation): {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 35, 40, 42, 60}
eo (edges, odd permutation): {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 56, 60, 84, 120}
ce (corners, even permutation): {1, 2, 3, 4, 5, 6, 7, 9, 12, 15, 18, 21, 45}
co (corners, odd permutation): {2, 4, 6, 8, 10, 12, 18, 30, 36}

Since ee only coexists with ce and eo with co, I had Mathematica list all LCMs of an ee order and a ce, as well as LCMs of an eo with a co. Combining the two lists, I obtained the following:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 90, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 165, 168, 180, 198, 210, 231, 240, 252, 280, 315, 330, 336, 360, 420, 462, 495, 504, 630, 720, 840, 990, 1260.

These are all possible orders of Rubik's Cube positions. Their frequencies could be found by attaching to each ee/eo/ce/co its frequency (it's not too hard to find the probability of having a "bad" cycle) and carrying these over to the LCMs at the end--this last stage is where I gave up last time.

Feel free to verify my work, or to construct the full table of frequencies. I'd be glad to check your work if you can get some results.

EDIT: This calculation may well have been done before, but I haven't seen it. I googled and oeis-ed the numbers without any results. I once did, however, see a web page listing some short algs for 3x3, 4x4, and 5x5 with given orders--I can't find that page now, and I don't remember who made it, but I remember being so disappointed at the absence of order 11 that I decided to look for the shortest possible pure 11-cycle. (The best I got was L B' F2 L2 U R U L R B2 D R2 B D'.)

*"order" = number of times one must repeat an algorithm before it produces the identity. Here I am abusing notation a little by equating an algorithm with the position it generates when applied to a solved cube.

2. I wonder what an alg would be for the for the 1260?

3. Yes, this has been done before. A long time ago.

http://www.jaapsch.net/puzzles/cubic3.htm#p34

You are probably thinking of qqwref's page for the orders of 3x3x3 through 3x3x5 cubes. On that page, inner slice, double-layer turns, and cube rotations were also used, so he could have had an order 2520 alg. I see it's still exists: http://www.mzrg.com/rubik/orders.shtml

4. TheCubeMaster5000: 1260 arises only as LCM(28,45). It comes from a 5-cycle and a 3-cycle of corners, both of which must be "bad," and a 7-cycle and two swaps of edges, where at least one of the edge swaps is "bad." Playing around briefly with Cube Explorer yields the 16-move U' B2 D' F2 D L D' F2 L' F2 D F2 L' B2 D R2, which is presumably far from the shortest 1260.

EDIT: The alg above doesn't work. I must have typoed, and I can't recover it. Here's a 17-mover: F L2 F2 B D2 F2 U2 R2 U2 L' U' R L B2 D' B' D2. (even further from the shortest)

5. That's amazing. I would probably make a chart if I had the time...

6. Here's a ramdom 11-cycles I found U B U D' L' F' R' U2 D2 R (10f).

7. Originally Posted by Ravi
TheCubeMaster5000: 1260 arises only as LCM(28,45). It comes from a 5-cycle and a 3-cycle of corners, both of which must be "bad," and a 7-cycle and two swaps of edges, where at least one of the edge swaps is "bad." Playing around briefly with Cube Explorer yields the 16-move U' B2 D' F2 D L D' F2 L' F2 D F2 L' B2 D R2, which is presumably far from the shortest 1260.
The following one has been known since at least 1981: R F2 B' U B'. Probably the shortest one.

8. Originally Posted by TMOY
Originally Posted by Ravi
TheCubeMaster5000: 1260 arises only as LCM(28,45). It comes from a 5-cycle and a 3-cycle of corners, both of which must be "bad," and a 7-cycle and two swaps of edges, where at least one of the edge swaps is "bad." Playing around briefly with Cube Explorer yields the 16-move U' B2 D' F2 D L D' F2 L' F2 D F2 L' B2 D R2, which is presumably far from the shortest 1260.
The following one has been known since at least 1981: R F2 B' U B'. Probably the shortest one.
Yes, it's a minimal order-1260 sequence for either FTM or QTM.

Spoiler:
My program counted 288 5f* positions and 480 6q* positions of order 1260. So it would seem there are probably 3 distinct 5f* positions mod M+inv, and 5 distinct 6q* positions mod M+inv.

9. Ravi i sent you a message concerning this thread, hope you can reply as soon as possible. This stuff is real awesome, keep at it.

10. R y.

EDIT: Order 1260.

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