a. Euler's computation of Zeta(2), (at first with only very weak, handwaving, or no convergence arguments), as redacted in Polya or here. Obviously amazingly ingenious, and interesting for artists, connecting numbers and circles. Requires unerstanding that a polynomial is equal to the product of its first degree factors, possibly it is too well known, however. It also allows one to then revisit the convergence argument and show what mathematicians actually worry about, after the "flash of ingenuity"
b. A lovely agument I heard a long time ago on sci.math relating to Sagan's book "Contact" in which a message is encoded in the bits of Pi. Someone asked whether a deity could arrange for Pi to be a different real number. Opinions went back and forth relating to spacetime, etc. Then someone asked, in light of any of the familiar series expansions, "If Pi were different, which natural number would be missing or duplicated?duplicated, and how might that be?" Leads to a discussion of the Peano axioms. Everyone goes home wondering. :-)