Since you have two arms, let’s say you’re oxygen. You would neeeeeed to be holding hands with someone else with each hand to be happy, otherwise you’ll reach around like crazy looking for something to hold. In this description, “holding hands” represents covalent bonds. Hydrogen has one hand and carbon has four hands. As others described, paper is made of cellulose just an arrangement of hydrogen and carbon, where they’re all happily holding hands. The atoms “know” they’re part of one piece of paper and not another because they’re in different chains of hand holding.
I agree with those before that explained how the atoms are in fibers. Each piece of paper is made of small “strings” (fibers) that are all “knotted” (I’m avoiding the word entangled) together and kept in place with “glue” (adhesives), when you cut with scissors your concentrating the force in a very thin line and pushing some fibers up and some down. This rips them apart. This isn’t a hand holding issue though. The pieces of paper “know” they aren’t part of the same piece of paper because they aren’t “knotted” together, just like when you put two knitted blankets on top of each other they don’t spontaneously become knit together
The comments on cold welding aren’t relevant because metals make metallic bonds, which aren’t as “cleanly” defined as hand holding (covalent bonds)
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