Let G be the group of moves on Rubik's group with operation of concatenation of moves. G is generated by the subgroup {F,L,R,D,U,B} where those are clockwise twist of the front, left, right, down, up and back faces of the cube, respectively.

I want to know how to figure out what is the order of the move DR (DR is the move where you first do D,then R). (Figure out without manually twisting the cube and counting). (The order of D alone and R alone is 4. (If you do D 4 times you get back to the starting point.))

migrated from mathoverflow.net May 25 '14 at 19:32

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  • 4
    Method 1: Write $D$ and $R$ as permutations of the little faces on each "cubelet" and then decompose $DR$ as a product of disjoint transpositions to compute its order. You're not going to get the answer intuitively without making some computation. Method 2: Actually carry out the combined move $DR$ repeatedly on a solved cube until you get back to a solved cube. But be patient: the order is over 100. – KCd May 25 '14 at 15:13
  • 1
    I've voted to migrate to mse. I didn't downvote. – Benjamin Steinberg May 25 '14 at 16:07
  • Thanks KConard. Can you explain method1 in more details? – George May 25 '14 at 17:25
  • You can gain some information by manually twisting the cube without having to repeat hundreds of moves. After 7 applications of DR the edges will be fixed. After 8 more (15 total) the corners will be fixed. So Order(DR)=LCM(7,15)=105. – Josh B. May 26 '14 at 17:26

I solved this after reading the chapter on the symmetric group at the notes at Group Theory and the Rubik's Cube

Here is the solution:

Let's name each cubie (the 26 small cubes in the Rubik's cube) by its visible faces in clockwise order (E.g., looking at the front face, the left corner cubie in the top row is flu).

Now, let's write D, R and DR moves as disjoint cycle decompositions of the permutation of the cubies when apply the move.

D = (dlf dfr drb dbl)(df dr db dl)

R = (rfu rub rbd rdf)(ru rb rd rf)

Now, computing DR.

D(dlf)=dfr

R(dfr)=R(rdf)=rfu

=> DR(dlf)=rfu

D(rfu)=rfu (because D won't affect the rfu cubie)

R(rfu)=rub

=> DR(rfu)=rub

D(rub)=rub

R(rub)=rbd

=> DR(rub)=rbd

...

=>DR = (dlf rfu rub rbd dbl)(drf)(df rf ru rb rd db dl)

Got that DR can be decomposed to 3 disjoint cycles.

The first cycle has 5 supports, but this cycle need to be applied 3 times because each cycle (5 times performing DR) will twist the colors. So, need to perform a multiple of 5*3=15 DR moves for returning the cubies in the first cycle to the starting point.

The second cycle has 1 support and need to be applied a multiple of 3 times for colors to be oriented as in the starting position.

The third cycle has 7 supports and can be applied 1 time because it don't changing the orientation of the colors.

Thus, the total order of DR is the least common multiplier of 15,3, and 7. This equals 105.

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