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The way I've seen it explained is like this: it all has to do with Newton's third law. However, the "equal and opposite" reaction predicted by the third law acts on the paddle - which is not physically a part of the boat - and I can't find a good explanation that involves the boat itself.

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Well the paddle is physically part of the boat. The person holding the paddle is sitting in the boat making contact with it. If you were sitting say on friction-less raft (say with ice on its surface) you will propel yourself in the water and the raft will not move.

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