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Runner on second. Batter hits into right center, rounds first and starts to head to second and the runner on second rounds third and heads home. Right fielder makes the throw home to try and stop the run, which scores despite a decent throw. Meanwhile the batter-runner has safely advanced to third. Is the batter rewarded a double or triple? That is, it probably would have been a double without the runner on second and the throw home. Thx.

I would score as double based on your description. Then advance the runner to third "on the throw." you would have better first hand knowledge whether the batter would have made it to third without the throw to home.