# Simplest Rubik's cube “algorithm” with cycle length 3

A regular Rubik's cube has a staggering number of possible states. However, that number is most decidedly finite; after all, a quintillion is just as far from infinity as seven is. From this finiteness it immediately follows, that any sequence of rotations, twists and turns, however complex, will have a cycle.

What I mean by that is that if you perform the sequence again on the resulting cube, and just keep on repeating, the cube will at some point return to the original position. (The cube will eventually run out of states that weren't already visited.)

These sequences are usually called algorithms. For the purpose of this puzzle, I'll redefine "algorithm" in terms of real world appearance of the cube:

An algorithm is any sequence of physical manipulation of the cube that results in the cube being in the same shape as before, in the same place as before, and at least one of the sticker positions now showing a different colour. $^*$

You are to keep this definition as a given, even if it might contradict with any other definitions, including the one you think is the correct one.

Now then, all algorithms can be categorised by the length of their cycle.

For example, the algorithm typically denoted with U (a quarter turn of the topmost layer) will return the cube to the original position after exactly 4 repetitions. Therefore, U has a cycle length of 4.

Some examples of algorithms with cycle length 2 would be

• 180° turn on the left face (L2)
• swapping any two pieces while preserving orientation
• turning the cube upside down
• flipping any given pair of edges
• ..and so on and so forth.

It's pretty easy to come up with algorithms with a cycle length of any N, as long as N is a power of two. Finding cycles of other lengths is a bit more challenging. Especially the ones that have a cycle length of 3 are not so common, and the first ones that spring to mind are usually quite complicated. So here, finally, we get to the puzzle:

Find the simplest algorithm (as defined above) that returns the original colours to all sticker positions when performed exactly 3 times.

The intended solution is simple both in terms of the physical movements involved, and speed, as in "time required to perform on a cube initially lying on a timer pad".

$^*$: You probably already noticed why it's important to disallow "null algorithms" and twists that are not a multiple of 90°. Also, you probably agree that even though the definition basically allows disassembling and reassembling the entire cube, it doesn't really matter here.

If you allow a solution that uses only whole cube rotations, the shortest is

a 120 degree rotation around a corner, which can be expressed as the two quarter turns of the whole cube $R_c\ U_c$.

Using only face turns, the shortest I know of is:

$R2\ U2\ R2\ U2$, which is 4 moves, though it is 8 quarter turns.

• The inteded solution is, indeed, a single 120 degree turn around a main diagonal, which can be achieved by the simple motion of picking up the cube in a certain way and setting it down on another side. I think this is what @micsthepick also meant, but I wasn't able to read his answer as anything else than a two-phase move. I of course upvoted his answer, so maybe if a couple of you guys did the same, the damage from the possible misunderstanding would be undone. – Bass Dec 18 '17 at 17:07
• Since it seems to be a conditional in every answer, I hope it was obvious that such a move was allowed. The third example in the 2-cycle list should have made it clear beyond any doubt. – Bass Dec 18 '17 at 17:10

Thinking of simple Rubik’s cube algorithms, there is one with 7 turns, or 10 quarter twists of outer faces, the algorithm for rotating 3 edges. If rotating the cube counts, then you can rotate around a corner, which would count as 6 slice moves. Rotate the whole cube around any two faces one quarter turn each in any way, and that can be repeated twice again to get back to the original state.

Here is an algorithm with only 6 moves.

U R L U R' L'

And here is a link to a visualization: alg.cubing.net

Not sure if this counts: 3 moves

R u' z'

And here is a link to a visualization: alg.cubing.net

This one was listed on Michael Gottlieb's website: Rubik's Cube Orders Table