What element of Smiley's character does the stranger exploit in"The Celebrated Jumping Frog of Calaveras County"?
The wording on this question seems to assume that the stranger got Smiley to challenge him with a bet and to go out to search for a frog on purpose and by intention. However the text indicates that the stranger was sincere until Smiley left him alone with Daniel Webster and went off in another overly enthusiastic spree of bet.
If you read the text this way, as though the stranger had no intent until he had "set there a good while thinking and thinking to hisself," then all he exploited was Smiley's over-enthusiasm, an over-enthusiasm that caused him to leave his prize betting jumping frog with the man with whom he had wagered a bet! So, if the stranger had no prior intent, the answer to the question, then, is that the stranger was able to trick Smiley because he ran off and left the frog with the stranger who took advantage of the situation and poured quail shot down its throat!
then he got the frog out and prized his mouth open and took a tea- spoon and filled him full of quail shot filled him pretty near up to his chin and set him on the floor.
If you prefer the reading that sees the stranger as acting with intent, then the stranger outsmarts Smiley because he feigns innocent disinterest.
says, very deliberate, "Well, I don't see no p'ints about that frog that's any better'n any other frog."
The word "deliberate" can be construed as indicating deliberate intention to find a way to swindle Smiley based upon the way Smiley's own feigned disinterestedness has exposed his yearning for a bet:
Smiley says, easy and careless, "... I should judge he can outjump any frog in Calaveras county."
The words "outjump any frog" echo a yearning for the stranger to pick up the challenge and make a bet. So, in this view, the stranger exploits (1) Smiley's naivete in leaving his frog in the care of the stranger; (2) his over-eagerness to track down resources for a bet; (3) his untempered quest for bets.
Mark Twain wrote the short story "The Notorious Jumping Frog of Calaveras County." The story is about a man named Jim Smiley who is a devout gambler. In the story it is explained that Jim will bet on just about anything. The story is a humorous account of how Jim had a horse who won a race because of it having asthma, and a dog loosing a fight because he had a penchant for biting his opponent's back leg until he fought a dog with no back legs. Then Smiley got a frog and trained it to jump real far and do what he said. A stranger came along and told Smiley he would bet against him and his frog but he had no frog of his own. Smiley was easily tricked because he could not turn a bet down. Smiley went off to fetch a frog leaving his frog with the man. The man tricked Smiley by filling his frog with shot making it very heavy. Smiley's frog lost the jumping contest.
Smiley's willingness to jump at any opportunity to gamble and his nativity made him a guidable target which led to him being swindled.