Sauron was really "stupid." The nature of
the Ring was such that it was literally inconceivable to him that anyone would attempt to destroy it.
But he has so many pointers: He knows that a
Halfling is carrying the Ring (first from
Gollum and then certainly from the
Nazgûl after they attack the
Hobbits at
Weathertop), he knows that the
Halflings live really far away from
Mordor, and he must know, that at least one of them has entered his
Realm. After all, the
Mouth of Sauron even presents
Frodo's mithril-coat and
sword to
Gandalf and describes the
Hobbit’s stature. Sauron must be really stupid to not even consider the possibility that this Halfling could be
the one carrying the Ring and that he intends to destroy it.
A discussion of "motives in
The Silmarillion" (from "
Morgoth's Ring," History of
Middle-earth Vol. X) has an interesting sentence that may shed some light on this subject. Discussing Sauron's perception of
Saruman and Gandalf, he says that Sauron supposed that
Manwë's motives in sending the
Istari were similar to his own, i.e., a power play. "His cynicism.... seemed fully justified in Saruman. Gandalf he did not understand. But certainly he had already become
evil, and therefore stupid, enough to imagine that his different behaviour was due simply to weaker intelligence and lack of a firm masterful purpose." The phrase "evil, and therefore stupid" speaks volumes for
Tolkien's characterization of Sauron. Sauron had become literally unable to understand Good.