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You really need to take a fecal sample to your vet and have them analyze it. Different wormers work for different worms. It sounds like you are describing hook worms or roundworms but you cannot be sure. Since different wormers work for different worms, you will need to see the vet, have them analyze some fecal matter and give you a dewormer that will WORK. Over the counter wormers are not terribly effective.
Let me know if you still have questions.
You can't tell just by looking at the poop, unfortunately. If you see worms in the poop, it's because they are dead. Worms live high up in the intestinal tract so you normally only see the eggs of the worms in the poop and you can't even see those with the naked eye.
As soon as you worm your puppy, you will see dead worms in the stool within 24 hours.
It sounds like you are describing a regular tape worm. Especially if it was white and resembled a grain of rice. In this case, you need to get some Droncit or Drontal Plus from your vet. It's the only thing that will treat tapeworms. Over the counter wormers will not work.
You should also keep in mind that tapeworms come from fleas so you need to treat any flea problems you have and get the dog on a prescription flea remedy. The worm is the flea larvae.
I hope this helps more!