How do matter particles interact?

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A tricky question that plagued physics for many years was, "how do matter particles interact?"

If you take two magnets and bring the north poles close to each other, the magnets push each other away -- without touching each other! How is it possible to exert a force on something without touching it? It is easy to say that, "the magnets have an electromagnetic force field," but that still doesn't answer the question, what IS the force that the magnets exert on each other?

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