You should certainly look at the two books by Ross Honsberger, "Mathematical Gems I" and "Mathematical Gems II". A favourite example of mine the proof due to Conway that there are configurations of checkers below the half plane on an infinite board that allow you to move a checker 4 rows into the upper half plane, but not five rows.
The negative result is an ingenious argument using nothing more than the quadratic formula, but provides a great example of to apply mathematics in unexpected contexts.
