Ship of Theseus

"Theseus owns a sailing ship. One day he sets sail on a long voyage.

Things are going to get confusing quickly, so to help keep things clear let's call the original ship that Theseus starts his voyage with "Ship A".

Ship A = Theseus' original ship.

During this voyage, the ship needs various repairs. Theseus anticipated this and he brought with him a complete supply of new parts to make repairs on his voyage. As repairs ar needed, Theseus throws the old parts overboard and replaces them with new parts.
The voyage is very long and eventually Theseus completely rebuildsthe ship. That is, he eventually replaces every part of the ship with new parts.
He returns home to Greece (or wherever it was) after defeating the Minotaur (or whatever he did) in this completely new ship. Let's call this ship, "Ship B".

Ship B = The ship Theseus returns in.

We already have the elements for the basic puzzle. The question is:

Does Ship A = Ship B?

But to give the puzzle some more bite, consider the following additional details to the story. Let's suppose that Theseus is followed on his voyage by another ship. The captain of this ship collects all of the old parts that Theseus throws overboard. As we said, Theseus eventually throws every part of his original ship overboard, so the captain of the following ship eventually collects every part of the original ship. Then this captain returns to Greece and takes the collected parts to make a ship which is exactly like Theseus' original ship (exactly like it, not just in that it's a faithful replica, it even has all the same parts, put together in the same way as ShipA). Let's call this rebuilt ship, "Ship C".

Ship C = The ship that is built from the original parts that Theseus threw overboard.

The question is:

Which of these ships is identical to the original ship?

It looks like there are four possible answers:

Ship A = Ship B (and not C)
Ship A = Ship C (and not B)
Ship A = Both Ship B and C.
Ship A = Neither Ship B or C"

All of the above by Ben Theis located at http://www.unc.edu/~theis/phil20/theseus.html