# Solution to Fathom It! Hard Board #9

Here is the initial board:

The first step (although Fathom It! purists/masochists will disagree <g>) is to fill in all obvious water squares.  [Ed. Using Fathom It! you do this by left-clicking an obvious water square to show water, then right-clicking it to "finalize" the square.  All squares around a submarine are water, so we'll finalize square (C,4) as water.]  Doing this we get:

## Where does the battleship go?

A good first move is to place the largest ships, in this case the battleship.  Where is the battleship positioned?  Checking the board, we see the battleship can only be positioned vertically in two columns: column 8 and column 10.  Don't be put off by the fact that it can go anywhere along column 10; we will use clever logic to show the battleship must be placed in (B,8)-(E,8)

Here are two different but equivalent methods, one using indirect logic and the other direct logic, for placing the battleship.

### (1) Prove the battleship cannot be placed in column 10

We will now use indirect logic to show that the battleship cannot be anywhere in column 10.

Let's assume the battleship belongs somewhere vertically in column 10.  Where can we place the two cruisers?  There are three possible positions:

1. Somewhere in the four (4) squares (B,8) - (E,8)
2. (E,1) - (E,3)
3. (F,1) - (F,3)

Remember: given our assumption that the battleship is in column 10, no cruiser can be placed in that column.

Possible positions 2 and 3 above are mutually exclusive -- they cannot both contain cruisers at the same time because the cruisers would be adjacent to each other.  So they can (and actually must) contain one and only one cruiser between them.  That means that the remaining cruiser must exist in position #1: in the sequence of squares (B,8) - (E,8).  Wherever this cruiser is positioned, squares (C,8) and (D,8) are ship segments:

But now, lo and behold, there are no positions to place the remaining cruiser.  Placing a cruiser in (E,1) - (E,3) would "close off" row F, leaving too few empty squares to place the required four (4) segments.:

Similarly, placing the remaining cruiser in (F,1) - (F,3) would close off row E:

The only logical conclusion is that our assumption that the battleship is in column 10 is false.  Therefore, it must be placed in (B,8) - (E,8):

### (2) Place the battleship by first placing all cruisers

In this method, pointed out to me by Shmuel Siegel, we will show the battleship must be in column 8.  The method differs from the previous one in that it uses only positive assertions, not indirect logic.

Once again, let's look at the current board:

Rather than place the battleship first, let's try to place the cruisers.

There are four (4) potential areas to place the cruisers:

• (E,1) - (E,3)
• (F,1) - (F,3)
• Column 8
• Column 10

#### A cruiser cannot be placed in (E,1) - (E,3)

This is not a valid placement, because it would "close off" row F:

#### (F,1) - (F,3) and column 8 are mutually exclusive

It is not possible to have a cruiser in both row F and column 8 at the same time.  If we place a cruiser in row E this forces a destroyer in (E,7) - (E,8) which excludes placing a cruiser in column 8:

So row E and column 8 can take at most one cruiser.  This means that the remaining cruiser must be in column 10.  And this means the battleship can only be placed in column 8:

## Solving the rest of the board

After placing the battleship, the rest of the solution is relatively straightforward.

Filling in the remaining squares in column 7 with segments, we get:

### Place the two cruisers

There are only three places to put a cruiser, all of them vertically in column 10:

1. (A,10) - (C,10)
2. (E,10) - (G,10)
3. (F,10) - (H,10)

Positions 2 and 3 are mutually exclusive (they overlap each other) so a cruiser must go in (A,10) - (C,10):

The last cruiser, as mentioned a moment ago, must be placed within the sequence of squares (E,10) - (H,10).  In any case, the squares (F,10) and (G,10) are ship segments:

Let's try to place the two remaining destroyers.  Take some time to convince yourself that only way to place both destroyers on the board is in (E,1) - (F,1) and (E,3) - (F,3).  Remember, the two segments in (F,10) and (G,10) belong to a cruiser, not to a destroyer.

Filling in rows F, A, and H in turn with ship segments leads to the solution: