# rubik's cube inside a rubik's cube

Once I completed solving Rubik's cube, a different pattern came into my mind. It was-

A B B
A B B
A C C


Here, same letters represent same colors. So is it possible to get this pattern in a Rubik's cube on all of it's faces?

• ...without dismantling it, correct? – Kingrames Aug 26 '15 at 15:51
• @Raystafarian I think it's perfect here. Rubik's cube is a puzzle. And if you have a answer, post it, – Rohcana Aug 27 '15 at 1:49
• @Kingrames, you can dismantle it. I've tried alot to get this pattern even by dismantling it but I couldn't. – Ahmad Aug 27 '15 at 5:50
• My intuition says it's impossible but I think it is very hard to prove – Ivo Beckers Aug 27 '15 at 10:07

This will be our model cube (just so we're all on the same page of the face (center block) colors) -

I've been trying to work on this, but I don't know enough math to formulate a rule. But, given the pattern -

A B B
A B B
A C C


We can note that each block of 4 that match in the upper right, it has 3 matches to its left (depending on which way you hold the cube). Let's make our blocks of 4 and indicate which matches are to the left -

Note that the colors I picked are arbitrary, it will work out the same way if you choose different corners.

As you can see, all of the triples to the left match with another triple to the left, thereby requiring two corner pieces that match 2 colors and an edge of those matching colors.

Pairs

yu
vx
wz


We also know what color pairs those corners can't have so they don't match their own face -

zw - yellow, red
vx - green, white
uy - orange, blue


The corners we have left:

BOY
GRY
RBY
BOW
GRW
GOW


How they can match to our pairs -

zw - BOY, BOW, GRW, GOW
vx - BOY, GRY, RBY, BOW
uy - GRY, RBY, GRW, GOW


So the pairs that are possible are

zw - BOx, GWx, OWx
vx - BOx, BYx, RYx
uy - RYx, GWx, GRx


With the blocks that remain, every color is present on exactly 3 corners and 2 edges.

Let's give a variable name to the other matches that there needs to be -

I don't see any combination of the cubes we have left that will meet the required matching criteria. But I can't formulate a proof of this. Maybe someone else can?

I see that there is a similar situation listed here:

http://ruwix.com/the-rubiks-cube/rubiks-cube-patterns-algorithms/ (Twisted cube in the big cube)

However it's not a match, you have two faces with swapped colors.

Hopefully this link will assist in your efforts. If nothing else, maybe someone can add some more relevant information. I confess I am no expert on Rubik's cubes (I know how to do the checkerboard thing though!)

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