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A flexi puzzle is a fun little toy, consisting of wood blocks and a string that runs through it.

Flexi puzzle

The pattern of colors is repeated twice; if a color is found at position $N$, it is found a second time at position $N+6$. For example, in the picture above, the color red occurs at positions $1$ and $7$.

You can twist the puzzle along the cutout lines in each square. You can also rotate squares.

Assuming that you can only perform 90° rotations and orthogonal twists, prove whether or not two pieces of the same color can be orthogonally adjacent on a standard flexi puzzle of 12 pieces. A picture or description of such a pattern would suffice as a proof to the positive.

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I think:

No

Because:

Suppose you re-coloured each cube black and white, alternating down the line

Then:

Since each colour is 6 spaces apart, pairs of cubes that were the same colour would still be the same colour

Now:

Colour an infinite 3-D space of cubes in a checkerboard fashion, and place a configuration of the flexi puzzle in line with the cubes, and such that one of the coloured cubes in the flexi puzzle corresponds to the same colour as the cube in the space it was placed

Since:

We have an alternating coloured flexi puzzle

We can conclude:

That all the cubes of the flexi puzzle are on a similarly coloured cube in the space

But then because:

Adjacent cubes in the space are of different colour

We observe that:

The flexi puzzle cannot have two black cubes or two white cubes next to each other

I.e.:

No two cubes of the same colour can be next to each other

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