Turn 4x4x4 Rubik's Cube into 3x3x3 Rubik's Cube

I have a 4x4x4 Rubik's Cube, but all I want is a normal 3x3x3 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 3x3x3?

Answer

Your goal is to lock out the middle-slice rotations of the 4x4. Doing this turns the puzzle into a 3x3, because the change from a 3x3 to a 4x4 is equivalent to the addition of the middle slice motion.

Another way to think about this is that without the middle slice motion of a 4x4, you can't break up the edge pairs, and you can't break up the 2x2 center blocks. Without being able to split up edges or centers (i.e., limiting the puzzle to the outside rotations only), the puzzle is a 3x3.

You can lock out the middle slices with four pieces of tape. Here's how:

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You can use two pieces of tape to lock out the middle slices through a single center. As long as the two pieces of tape are perpendicular, middle-slice motion is through that center isn't possible in either direction.

There are three middle-slice movements on the 4x4 - one for each spatial dimension. Each center intersects two of these. Therefore, your first center locks out two rotational axes. If you also tape over any adjacent center, you will also end up locking out the third rotational axis. Therefore, you need only tape over two centers.

The second center requires two pieces of tape, not one. While one piece of tape would lock out the third axis in the cube's initial configuration, if you rotate the center, that piece of tape will no longer intersect the third axis. In other words, without the second piece there, it is still possible to reach a state where you can rotate the third middle slice of the cube, even if it isn't the initial state.

Applying four pieces of tape - two pieces to two adjacent centers - locks out all middle-slice movements, reducing the puzzle to a 3x3.