True, there are many problems which cannot be solved or even vaguely approached analytically, but this isn't one of them.
There are more problems that cannot be solved by or even vaguely approached analytically by a given person than ones that cannot be solved by or even vaguely approached analytically. I think that the original poster laid out a pretty good case that he couldn't tackle this analytically, while leaving it open as to whether someone else might succeed in doing so.
Therefore your success with an analytical approach takes nothing away from the value of the non-analytical approach in this case.