What is the most surprising result that you have personally discovered?

I was very happy to find out that if we look at a notebook with a magnifying glass, then the lines become curves; and the fact that they are parallel is remained (especially if you keep them at the focal point of the magnification).

However the curves all meet at the "edge" of the glass. So we can have a sense of geometry where parallel lines meet at infinity.

I remember telling about this to my brother who was an engineering student (I was merely 16), and he said that it's impossible. Some years later I learned that this was already known as non-Euclidean geometry and played an important part of Einstein's relativity.

Some time ago I saw that the record for Broccard's Problem ( http://en.wikipedia.org/wiki/Brocard%27s_problem ) was pretty low($10^{9}$) because of the scarce attention it has, so I coded a program to prove it up to $10^{11}$. Then I presented it at my school's mathfest. The other presentations were just informative about popular things like fibonacci or pascal, and it really surprised the jury :). I don't think it was a big deal, after all neither my code or my computer were too fast, and it only took 2 days. Any decent investigator can easily do way better than that in a couple of days.

But it was a very nice experience discovering something new for the first time on my own.

This was the most surprising to me, I suppose, because it was one of my first:

One day I considered the alternating harmonic series, and realized that if you replace the $-1$ with the imaginary unit, $i$, then the resulting series still converges. (This can be shown in several ways.)

I asked a professor whether this was true for any root of unity, and we quickly decided not only that it is, but that this is true using any complex number $z \neq 1$ such that $|z| = 1$. This was via a hand-wavy geometric argument, but it subsequently became clear to me that this result is known and easily proved with the Dirichlet Convergence Test.

Years later, I posted on MO to ask about a rigorous geometric proof for the general case, and someone nicely provided one. But: It was pretty cool to think it up the first time, especially since it is tough to guess at a first glance that the harmonic series diverges; turns out any fixed wobble (rotation) after each step would, indeed, give convergence.

Edit: On second thought, I once explored the following problem: Suppose a bag contains $m$ black marbles, $n$ white marbles, and $m \leq n$. Remove a marble, note its color, and put it back in the bag. What is the chance that the total number of black marbles counted - at some point - exceeds the total number of white marbles counted at that same point?

I figured this would be a tractable problem, and maybe even that it would always be $1$, i.e., at some point there would be enough black marbles picked in a row to exceed the number of white ones picked. However, this did not turn out to be the case, though there ends up being a very simple formula: $m/n$. (Surprise!)

Of course, this number syncs up with a few reality checks (e.g., when we have $m = n$). I should also note that there are surprising ways of tackling this problem: Random walks, Catalan numbers, recurrence relations, and the Gambler's Ruin can each be used to provide "different" proofs that the formula provided above holds.