Here is the initial board:

As our first step, let's fill in square (A,9) as the obvious 'top' ship segment, and also fill in all obvious water squares. [Ed. Using Fathom It! you do this by setting square (A,9) to the obvious 'top" ship segment and finalizing the square] Doing this we get:

We'll now show that the squares (C,1) and (C,7) must be ship segments (see the rule here). If either (C,1) or (C,7) are water, all five remaining squares will have to be ship segments (because the row tally is five). Therefore both (C,1) and (C,7) must be ship segments:

A standard technique at this point is to attempt to place the largest unfound ship, in this case the battleship. We will first show that the battleship must reside in row C.

If the battleship does NOT reside in row C, then neither (C,3) nor (C,6) can be water. This is because a water in either square would force a battleship along row C. Here's what the board would look like if both (C,3) and (C,6) are ship segments:

Notice anything special? Now there is no place to place the battleship. The forced conclusion is that the battleship must reside along row C.

So now we know the battleship is along row C and not row J. There are two ways to place the battleship along row C -- why one is correct?

Attempting to place the battleship at (C,4) - (C,7) gives the following:

A brief check shows it is impossible to place both cruisers on the board.

If a cruiser is placed anywhere along row J, (J,3) will be a ship segment. The board would then look like this:

There are no other possible locations for a cruiser. So the cruiser cannot be along row J.

The only remaining places for the cruisers are (H,1) - (H,3), (G,2) - (I,2), and (H,2) - (J,2). All three are mutually exclusive with the other.

So we have shown that the battleship must be located at (C,3) - (C6):

Let's attempt to place the two cruisers. There are three major possible locations:

- Column 2
- Row H
- Row J

It cannot be in row J, as we would arrive at the following board:

Row H is forced to be all ship segments:

But now there is nowhere to place the second cruiser. So row J does not possess a cruiser. This leaves the other two possible positions as the inevitable:

The rest is just cleaning up from this point, leading to the solution:

It's interesting to note that the Fathom It! built-in solver requires 25 moves to solve the board, while the human approach takes only 19, a savings of six moves.