## Tuesday, July 12, 2016

### [pcuviwfu] 1x1 Rubik's cube puzzles

First, we enumerate ways of describing moves of a standard color scheme 1x1x1 Rubik's cube (an object typically presented as a joke).  Rotating any face of the cube of course rotates the entire cube.

Standard move notation: Rotate the {Up, Down, Left, Right, Front, Back} face by a {quarter, half, three-quarter} turn clockwise.

Rotate the {White, Yellow, Blue, Green, Red, Orange} face by a {quarter, half, three-quarter} turn clockwise.

Rotate the cube by a quarter turn so that the {Up, Down, Left, Right, Front, Back, White, Yellow, Blue, Green, Red, Orange} face takes the place of the {Up, Down, Left, Right, Front, Back, White, Yellow, Blue, Green, Red, Orange} face.

Next, a series of puzzle templates of increasing difficulty:

The cube has {U,D,L,R,F,B} face of color {W,Y,B,G,R,O} and {U,D,L,R,F,B} face of color {W,Y,B,G,R,O}.  What is the color of the {U,D,L,R,F,B} face?  Or, where is the {W,Y,B,G,R,O} face?

The cube has {U,D,L,R,F,B} face of color {W,Y,B,G,R,O} and {U,D,L,R,F,B} face of color {W,Y,B,G,R,O}.  We do the following sequence of moves {move possibilities listed above}.  After this sequence, what is the color of the {U,D,L,R,F,B} face?  Or, where is the {W,Y,B,G,R,O} face?

The cube has {U,D,L,R,F,B} face of color {W,Y,B,G,R,O}. We do the following sequence of moves {move possibilities listed above}.  After this sequence, the {U,D,L,R,F,B} face has color {W,Y,B,G,R,O}.  We then do another sequence of moves {move possibilities listed above}.  After this sequence, what is the color of the {U,D,L,R,F,B} face?  Or, where is the {W,Y,B,G,R,O} face?

The constraints could be harder: The {U,D,L,R,F,B} face at time point T1 in this sequence of moves ... has {the same, a different} color as the {U,D,L,R,F,B} face at time point T2.  Or dually, the {W,Y,B,G,R,O} face at time point T1 is in {the same, a different} location as the {W,Y,B,G,R,O} face at time point T2.

All of these are easily solvable by considering all 24 possible orientations of the initial cube.  Doing 3-dimensional rotations in one's head is the challenge.