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Idea shamelessly stolen from this question.

I have a 4x4x4 Rubik's Cube, but all I want is a 2x2x2 Rubik's Cube.

Luckily, I also have an unlimited supply of small pieces of tape. Each piece of tape can cover the face of two neighbouring squares on my Rubik's Cube. When two pieces are taped together, they cannot be moved away from each other.

What is the minimum number of pieces of tape I must use to ensure that my 4x4x4 will always behave like a 2x2x2?

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I think:

15

Why?

To bandage a single 2x2x2 corner block of the 4x4x4, you need three pieces of tape: enter image description here
Alternatively, you can arrange the three pieces of tape like this: enter image description here or this: enter image description here

In addition, you'll need to bandage at least five corners, otherwise you would be able to group four of the 2x2x2 blocks on one side of the 2x2x2 and still make a 4x4x4 move on the other side.
The fifth corner also requires three pieces of tape, since you could orient the corner and still be able to do a 4x4x4 move if you've only used two pieces of tape. (Thanks for the correction @JulianRosen.)

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    $\begingroup$ If the fifth corner has only two pieces of tape, it should be possible to orient that corner in such a way as to make a non-2x2x2 move possible. $\endgroup$ – Julian Rosen Apr 12 '16 at 15:28
  • $\begingroup$ @JulianRosen You're indeed right, edited. $\endgroup$ – Kevin Cruijssen Apr 12 '16 at 15:46
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Answer:

5

Explanation:

If you need 5 corners immovable to keep other corresponding corners intact, you could technically use just 5 pieces of tape. Just run one piece of tape along the seam, around the edge, connecting 4 squares per corner. 2 squares on 2 faces, then none of the 3 faces could change on the corner. You might be able to do it with 4, but I would have to hold a cube to visualize it, and I don't have one. This is the minimum, but unless you have perfect tape, you would need more.

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6?
One per face of the cube. Each face has actually 2 layers that must be combined with one strip.

EDIT: wrong

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    $\begingroup$ How do you propose to tape two entire faces together with one tiny piece of tape? One piece of tape can only link two squares. $\endgroup$ – Deusovi Apr 13 '16 at 5:05
  • $\begingroup$ What I mean is that one layer cannot rotate with respect to the one below as long as one piece of tape joins them. You do not need tapping the entire face. $\endgroup$ – fffred Apr 13 '16 at 6:40
  • $\begingroup$ That still allows some faces to turn in ways other than two at a time. $\endgroup$ – Deusovi Apr 13 '16 at 6:43
  • $\begingroup$ Yes. That's why you need to put 6 pieces so that only cubes 2x2x2 remain. Another way to see this: 4x4x4 has 9 degrees of freedom (3 rotations per axis x y or z); you block 6: remain 3, just like a 2x2x2 Rubik's cube. Note that Kevin's solution starts with similar fashion as mine, but forgets to remove duplicates. $\endgroup$ – fffred Apr 13 '16 at 6:49
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    $\begingroup$ Can you show a picture of your solution? I agree that 6 pieces are necessary, but I don't think they're sufficient. $\endgroup$ – Deusovi Apr 13 '16 at 6:55

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