We have enough amount of $2$x$2$ grid ships where you can put them on the $14$x$14$ grid battleship board. If this was a real battleship game, you could simply put $49$ of these $2$x$2$ grid ships into the board without any grid shared.
This time, you are going to put the ships one by one shown as the example below. Every time you put a ship, it cannot share more than 1 grid with other ships. That is, it must occupy at least three empty grids. In this case,
At most how many ships can you put into the board?
If this question was asked for $4$x$4$, the answer would be $5$ as shown below:
Here is the wrong way playing the game where you put the green ship, it shares two grid at the same time when you put it:
Answer
An obvious upper bound is
65, because each new ship after the first must occupy at least empty 3 squares, and $1 + \frac{196 - 4}{3} = 65$.
I couldn't do that well, however, so this might be suboptimal. My best arrangement so far gets to
64 ships:
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