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This is the only possible answer I could think of.

Spoiler

 You cut one slit in the middle of the piece of paper. The slit will almost reach the other side. When you slide the rock through the paper, the rock will pass through the half of the piece of paper because of the slit. You never said that the rock should pass through the entire sheet of paper.

If you wanted to cut a real hole through the piece of paper, then it would be impossible. It is possible to cut a square hole with one snip, if you fold the paper. But, the square hole will never pass through the cube rock. Since the dimensions of the hole would be <1 x <1, it can't go through a 1 x 1 x 1. If you turn the rock, the direct horizontal side would be from 1 to >1 across. Never <1.

So, the answer at top would be the only logical answer unless there was something you knew was possible.

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How about this...

Spoiler

     Get the 1 x 1 piece of paper and fold it in half. Then you fold it in half the other way to get a 0.5 x 0.5 square. Fold this in half diagonally now to get a right triangle. There will be an edge that is not touching the corner that is the center. Cut a line parallel to it as close as you can to the edge without overlapping it. Unfold it to half a large square hole in the middle, with a little bit left of paper surrounding it. This is only hole I can think of that will be closest to being able to fit the rock. Is this the hole that will be able to go through the cube rock?

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