The reason that an atom knows what sheet of paper it belongs to is due to the fact that the atom is *bound* to a molecule. A molecule is a bunch of atoms that bond *very* strongly to one another. The atoms can be arranged in a chain, in rings, or in one of many other ways.
Paper is made up of various different molecules, most importantly [cellulose](https://en.m.wikipedia.org/wiki/Cellulose) in case you want to look it up. Many of these molecules make up fibres and many fibres make up the sheet of paper.
As I said, the bond between the atoms and the rest of their molecule is very strong. Breaking that bond would require a lot of work, that is missing when you simply push two sheets of paper together. The molecule really doesn’t want to give up the atom and there really isn’t a molecule on the other side that would like to adopt another atom.
Now when you cut a piece of paper with a pair of scissors it is very unlikely that you will break the bond between atoms. That is because, the bond between atoms inside of molecules is much stronger than the attraction between molecules. So when you cut a piece of paper you are much more likely to separate two molecules. Although, there is a chance you might also break a bond between atoms.
If you would like to know what bond I’ve been talking about look up “covalent bonding”.
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