# Move Sequence Repetition Problem



## nickvu2 (Jan 17, 2010)

I'm trying to figure out mathematically how many moves it would take repeating the sequence (R F L B) to get from a solved state back to a solved state. I don't even know where to start. Anyone up for the challenge?

Interesting observation: U edges never enter the B layer and B edges never enter the U layer.

Edit:
Oh dear, can't believe I wrote the wrong alg. You'd think I could keep track of 4 moves. RFLB, not RULB.


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## Inf3rn0 (Jan 17, 2010)

I remember a while ago stumbling upon a website where you put in a sequence and it would tell you how many times it had to be repeated untill you reached a solved cube again. I shall try find it.


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## LewisJ (Jan 17, 2010)

Look at it in terms of cycles and flips/twists of the edges/corners. 

If you're familiar with 3OP BLD - make your edge cycles and corner cycles then check which edges are flipped and which corners are twisted and in what way. From there you can find how many repetitions it takes for each orientation and permutation group to go back to solved and by finding the LCD of all those you find the number of RULB reps it takes to get to solved.


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## Cride5 (Jan 17, 2010)

Inf3rn0 said:


> I remember a while ago stumbling upon a website where you put in a sequence and it would tell you how many times it had to be repeated untill you reached a solved cube again. I shall try find it.



http://solvethecube.110mb.com/tools.html#order


Its fairly easy to do algorithmically, but calculating it mathematically is probably a bit more of a challenge. As a quick stab in the dark you could maybe look at the number of iterations each individual cubie takes to get back to its home position (with correct orientation), then find the lowest common multiple of all the cubies' periods.

EDIT: I think that works. Taking RU' as an example. Each corner requires 9 iterations to get back to home, and each edge requires 7. LCM of 9 and 7 is 63, which is correct according to Joel's tool. Another interesting question though, which alg has the highest order? ... or maybe slightly easier: What is the highest possible order of any algorithm? UR'F has quite a high order (360)..

Edit2: An attempt to figure out the alg order upper bound.


Spoiler



I think the key to answering this question is examining the period length of cycles created by an alg.

Period of a cycle: A cycle's highest possible period is the number of positions/orientations each cubie can occupy within the cycle, so a corner 3-cycle's max period is 3*3 = 9, a 5-cycle's max period is 15 and an 8-cycle's max period would be 24. Similarly for edges a 3-cycle's max period is 3*2 = 6, and a 12-cycle has a max period of 24. 

So, the highest possible order is a blend of periods of each cycle which gives the highest LCM. However, its not really a straight forward calculation because using as many cycles as possible means the period is very short. On the other hand, using fewer longer cycles means there are less periods to combine to produce the LCM.

So I guess by figuring out all possible periods for each possible corner and edge cycle it would by possible to find the answer..


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## Inf3rn0 (Jan 17, 2010)

Cride5 said:


> Inf3rn0 said:
> 
> 
> > I remember a while ago stumbling upon a website where you put in a sequence and it would tell you how many times it had to be repeated untill you reached a solved cube again. I shall try find it.
> ...


Yup that's it. I was having some trouble finding it


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## Zane_C (Jan 17, 2010)

252, I just had a look on http://solvethecube.110mb.com/tools.html#order


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## Swordsman Kirby (Jan 17, 2010)

Cride5 said:


> what is the highest possible order of any algorithm?





Spoiler



1260

If you allow for cube rotations, the shortest algorithm that has this order is (z U). Of course, there are algorithms that don't require rotations. I believe the number is higher if you allow for center cycles.


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## Tim Major (Jan 17, 2010)

Zane_C said:


> 252, I just had a look on http://solvethecube.110mb.com/tools.html#order



Isn't it 315?

Edit: I see he edited his original post.


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## Zane_C (Jan 17, 2010)

ZB_FTW!!! said:


> Zane_C said:
> 
> 
> > 252, I just had a look on http://solvethecube.110mb.com/tools.html#order
> ...



Hmm, maby. I tried it again and it said 315, then I tried it another time and it said 252. I must be doing something wrong.
Yeah it must be 315.


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## nickvu2 (Jan 17, 2010)

Wow, thanks guys! Still learning new things about the cube even after years =)

BTW, I somehow managed to botch the alg I was using when I wrote the post; it was RFLB rather than RULB. Never the less, from LewisJ and Cride5's explanation I get the idea.

Here's what I got:
7 edges are in a 7 cycle
5 edges are in a 5 cycle
(is this just a coincidence or will they always match like this?)
*35 cycles for edges*

3 corners in a 9 cycle
1 corner in a 3 cycle
1 corner in a 3 cycle
1 corner in a 3 cycle
1 corner in a 3 cycle
1 corner in a 3 cycle
*9 cycles for corners*

*LCM of 9 and 35 is 315*

And Joel's calculator confirms! Woohoo! This is so cool =)


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## Lucas Garron (Jan 17, 2010)

nickvu2 said:


> I'm trying to figure out mathematically how many moves it would take repeating the sequence (R F L B) to get from a solved state back to a solved state. I don't even know where to start. Anyone up for the challenge?


Not that you should have known, but it's probably a better idea to ask if this has been considered before, and then for how to understand it. Chances are, if you ask about a well-known problem as if you're posing a brand-new question, people won't be as inclined to answer so nicely.

Also, your title doesn't say very much. Mind if I change it to something like "Move Sequence Repetition Problem"?
(Although the commonly used terms would be "algorithm" and "order".)



nickvu2 said:


> Interesting observation: U edges never enter the B layer and B edges never enter the U layer.


False, unless you mean "stickers," in which case that's very well known to anyone who's ever worked with edge orientation.

Anyhow, if you want some fun, try L2 F' D' F U2 F' D' B R2 D2 F D2 B' U2 D2


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## manyhobbyfreak (Jan 22, 2010)

i got a great alg for longest amount and it is : U' R small L


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## Stefan (Jan 22, 2010)

manyhobbyfreak said:


> i got a great alg for longest amount and it is : U' R *small* L


Huh?


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