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Is it possible to have an algorithm that can rotate top layer corners on the 3x3. Similar to what R' D' R D does except it is incapable of doing it when more than one of the bottom layer corners is not oriented correctly. Is there an algorithm out there that will work for this?

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    $\begingroup$ This is just OLL/PLL. Any good cubing site should have it, look up OLL and PLL online. $\endgroup$ – boboquack Mar 8 '17 at 5:04
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The easiest method to remember is this:

  1. Turn the U layer until there is a twisted corner at the UFR location.
  2. Repeat RF'R'F until the corner at URF has become untwisted. You will need to do the sequence twice to twist URF clockwise, four times for counter-clockwise.
  3. Repeat steps 1 & 2 until there are no more twisted corners in the U layer.

Even though the moves in step 2 mix up the bottom two layers, by the time all the U corners have been untwisted, the bottom two layers will be back to normal again.

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My method was developed not borrowed. As a result I orient the cube with the first layer on top and it stays there throughout without changing cube orientation. Once I finish the top 2 layers, I check the last layer corner placements. There were 3 possible corner location configurations. All in place, two in place on the same face or two in place diagonal to one another. Case 1 proceeds to the next step, while cases 2&3 need proper relocation.

In the case of 2 being properly placed in the same face, the unfinished face of the cube is faced down with the two incorrectly placed corners in the lower front corners. Turn the right face counter clockwise the rotate the bottom slice to the left. Return the right face clockwise. Rotate the front face clockwise, now torn the bottom slice to the right. Rotate the front face counterclockwise. Rotate the right face counterclockwise, the bottom slice to the right and right face clockwise. The two corners are now swapped.

The process for swapping diagonal corners is nearly identical. Tipping the cube so you can see the bottom orient a properly placed corner in the upper right corner. When the unfinished layer is now faced downward that corner is the lower right corner. Rotate the right face counter-clockwise, rotate the bottom slice left, turn the right face clockwise. Now rotate the front face clockwise, and more the bottom slice 180 degrees to the right. Turn the front face counterclockwise, the right face counterclockwise, the bottom face to the right and the right face clockwise. All corners will now be placed.

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