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# 4-axis Rubik's Cubes

## Question

Four modified 1x1x1 Rubik's cubes are sticking with each other by magnetics (as shown) forming the 2x2x1 puzzle block.
Starting with all the same colors on the same face to ensure solvability, shuffle by turning any two side by side
cubes on the axis that connects their centers. Few times will do.
To solve or play just apply the same rule of turn to rearrange for same colored sides. Maybe a different configuration.
Note that the 2 cubes cannot rotate on edges or face axis..just on the axis where both centers are.(90 or 180 deg)

If by then only one of the cube is disoriented, how many turns more shall it take to complete the puzzle?

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I see, and the best solution is

Spoiler

Five moves for the one position that cannot be solved in four.

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How to make 2 cubes stick together on 24 positions with aligned edges?

Spoiler

Using 1x1x1 and the regular 3x3x3 Rubik's cube.

Edited by TimeSpaceLightForce
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Spoiler

Four.*

Starting with a solved state, label the four cubes as such: (top-down view)
AB
CD

1 - Rotate AC
2 - Rotate CD
3 - Rotate AC (reverse of move 1)
4 - Rotate CD (reverse of move 2)

Cube B is not moved. Cubes A and D are rotated twice, with the second rotation returning the cube to its initial orientation. Cube C is rotated four times, and is the only cube disoriented in the final position. This new position, an example of the position in question, can be returned to the solved state in four moves. QED.

* This does not prove that all such positions can be solved in four moves, though I suspect any that cannot be solved in four moves may be unsolvable. (Open question.)

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Spoiler

Let an arrow pointing from front  face to bottom edge be the reference Position 1.Then designate all  other positions ..2-8 (low) ,9-16 (mid) ,17-24 (high).

Any cube (A , B ,C or D) can be rotated to all 24 positions via 2-axes. But not all 24 position is possible for the disoriented cube if it played  from a solved state.

@Logophobic
Four moves cannot solve all possible positions of the disoriented cube. How many one cube disorientation  can be solved with four ?

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Spoiler

Of the 24 orientations of the 4th cube relative to the other three:
- One is the solved position
- Ten can be solved in 4 moves
- One can be solved in 8 moves
- Twelve cannot be solved

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All true..yet you maybe missing something like the the odd no. of moves?

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What are you getting at? If what I've posted is 'all true', then there is no other solution.

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Spoiler

The possible position of the disoriented cube that can not be done with 4 moves is one.
Now we search for the least no. of moves to solve that (bad position)..the best answer so far is 8.

For the closing question.. would it be possible from your model:
1 - Rotate AC
2 - Rotate CD
3 - Rotate AC
4 - Rotate CD
5 - Rotate AC
6 - Rotate CD (D back in correct position as B)
7 - Rotate AC (A back in correct position as B)?

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Yes,that's the least..and it helps to know. But i can only guess, the solution for all possible shuffled position is 8 maybe  9.

Nice job !

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