## Solution to "Battleships 4" Board with Incomplete Tallies

### Solution by: Moshe Rubin

In GAMES Magazine's soft-covered "Brain Twisters from The World Puzzle Championships (volume 2)", you'll find two fascinating Battleship boards on pages 38 and 39. What is unique about them is that the reader is challenged to solve the boards, even though row and column tallies are missing! To quote GAMES Magazine, "Believe it or not, this is enough information to solve [the board]".

Here is the initial board, as found on page 39

As you can see, there are row and column tallies missing. Amazingly, this is all the information we need to solve the board.

Let's fill in all known water squares. Do this by finalizing any obvious water square, for example F1. We should see:

Let's turn our attention to Row B. With a row tally of 8, we should be able to make an initial break. We'll be making a lot of use of the fact that row B has exactly two water squares. Keep this is mind during the next few steps.

Row B already has one water square at B8. Where can we place the second water along row B? Certainly not in columns 1, 2, 6, 7, 9 and 10. Placing a water in any of these squares would produce a ship in row B with more than four segments (the longest ship allowed for this board is a battleship). Here's what we'd see if B1 were water:

Therefore, we can finalize columns 1, 2, 6, 7, 9 and 10 in row B as ship segments:

Column 3 (with a tally of 7) already has three finalized water squares, so the remaining squares in the column are ship segments:

Since the one battleship has now been placed at G3-J3, we know that B1-B3 cannot be part of a battleship, and must be a cruiser:

Row E needs one more water square (we have three already). That square cannot be in columns 1, 9 or 10, as this would force a third cruiser (too many!), so they are ship segments:

What we just gained is that the last destroyer was found at E9-E10, and we have only the submarines left to place.

Where is the last water square along row E? It cannot be in columns 5 or 7, because this would force a fourth destroyer (too many!) along row E. Finalizing E5 and E7 as ship segments gives us:

Since the only ships left to locate are submarines, the ship segments at E1, E5 and E7 must be submarines:

Where do we place the last submarine? Row I and column 7 both have a remaining ship segment left to place. The intersection between them, I7, must be a ship segment:

We've located every ship, so all remaining squares must be water. Filling them in leads to the solution: