[oh8eleven]

One was Trinity, one was Little Boy, one was Fat Man and the fourth was unused.
IIRC, Trinity was the very first A-bomb that was named for the Trinity test site in NM where it was detonated. I still believe the two we used in Japan were it, but could be wrong.

[Harmless Teddy Bear]

No, there was a fourth.

And as pointed out the pipeline would have been churning out more of them just as fast as they could. All the components were in place and they could turn out about two a month.