How to heal a paper cut really fast?

Use super glue. Super glue was originally invented during the war as a quick heal method for soldiers cuts. Put it on before you go in the water and it will remain waterproof and keep your cut sealed.

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Super Glue is pretty effective in sealing and protecting minor cuts. It's easiest to get someone to help you, but pinch the wound closed after disinfecting it. Spread a thin layer of the glue on the cut and let it dry before touching it. If you feel it might be too thin, you can always add another layer on top, so long as it extends further out than the first layer. A nice thing is that if the glue mound sticks out too far and you're afraid you might snag it on something, you can always use a nail file to sand it smooth and flat along your finger. I used this method many times while working as a chef for small knife wounds on my hands. Never had a problem with it. Didn't really sting too much either.