[bsammon]

An important detail is what kind of ship piece was pictured in the freebie square. It could be an entire-size-one-ship (a circle), the end of a ship (which would indicate what direction the rest of the ship would continue in), or the middle of a ship (which indicates a size 3 or 4 ship).

I decided to go with the assumption that it was a middle-of-a-ship square, thinking you would more likely have mentioned if it was one of the other two types. Then, based on the numbers around the outside, you can quickly determine whether the ship is horizontally or vertically oriented, and fill in two more squares as a result (and rule out a few more squares as a result of that). the rest of the solution proceeded quickly after that.

I used to do these Battleships puzzles regularly around 15 years ago, when I regularly bought/read Games magazine, which included them. My process, especially for harder puzzles, involved darkening (or otherwise tagging) the boundary between pairs of squares, as I determine that a 2-or-longer ship could not span that particular boundary. Usually based on a logic of "Well, if there was a 2 or longer ship there, there'd be a problem for the next row/column over" or just "there can't be a 2 or longer ship because I'm only allowed to darken one more square in that row/column"

This web-app version doesn't appear to have a mechanic for marking boundaries -- I frequently find electronic versions of traditional puzzles to (understandably) lack the flexibility to support the ad-hoc annotative solving process I developed over the years of solving these kind of puzzles on paper.