How to Solve Boat-Placement Puzzles: Find Every Hidden Boat

To solve a boat-placement puzzle, use the row and column numbers to decide which cells are water and which hold part of a boat, and remember that boats never touch, not even at a corner. Start with the lines whose numbers force an answer, then place the longest boat, because it has the fewest places to go.

This guide covers the rules, the marks to use, a step-by-step method, a full worked puzzle and a practice puzzle with its answer.

What is a boat-placement puzzle?

A boat-placement puzzle, also called a hidden boat grid, is a square grid with a hidden set of boats. Each boat is a straight line of cells, lying either across a row or down a column. The puzzle gives you a boat list, such as one three-cell boat, two two-cell boats and three one-cell boats.

Beside each row and under each column is a number. It counts how many cells in that line are covered by boats. Every other cell is water.

There is one more rule. No two boats may touch, whether side by side, end to end or diagonally. Every boat is surrounded by a ring of water.

A well-made puzzle has exactly one answer, and you can reach it by reasoning alone.

What marks should you use?

You need two marks:

• A filled square for a boat cell.
• A dot for water.

Some solvers also draw a small rounded end when they know a cell is the end of a boat, but plain squares are enough. Mark a cell only when you can say why.

What do the row and column numbers tell you?

The numbers are your main source of facts. Read them in three ways.

• A zero means the whole line is water. Dot it at once.
• Cells still needed equal to the unmarked cells left means every unmarked cell in that line is a boat cell. The cells still needed are the number minus the cells already filled.
• A number already reached means every other cell in that line is water.

The totals also check each other. The row numbers and the column numbers must both add up to the number of boat cells on the boat list. If your marks ever break that, something has gone wrong.

Why can boats never touch?

The no-touching rule turns every boat cell into a source of water. As soon as you know a cell holds part of a boat, its four diagonal neighbours must be water. The same boat cannot reach them, because boats are straight, and no other boat may touch it.

The side neighbours follow once you know which way the boat runs. A boat lying across a row has water directly above and below every one of its cells. A boat running down a column has water directly to the left and right of every cell.

That is handy in crowded corners. Two filled cells that meet only at a corner are always a mistake, and a filled cell can never have filled neighbours both beside it and above or below it.

What is the step-by-step method?

1. Clear the zeros. Dot every row and column whose number is zero.
2. Fill the full lines. Where the cells still needed equal the unmarked cells left, fill them all.
3. Ring every boat cell. Dot its diagonal neighbours. When a boat is complete, dot every cell around it.
4. Place the longest boat. List the places where it fits, using the numbers, and rule them out one by one.
5. Count off the boat list. When a run of filled cells is closed by water at both ends, you know its length. Count off one boat of that length from the list as you place it. A list may hold several boats of the same length.
6. Rule out finished lengths. Only when every boat of one length on the list is counted off can no more runs of that length appear. If a line could then only be filled by such a run, split its cells into shorter boats with water between them.
7. Check at the end. Every row and column count, every boat length and every gap.

Which clue patterns give quick answers?

How does the method solve a six by six puzzle?

Here is an original puzzle. The boat list is one three-cell boat, two two-cell boats and three one-cell boats. The last column gives the row numbers and the bottom row gives the column numbers.

Try it first, then read on.

Step 1: clear the zeros. Row 2, column 1 and column 5 each have a count of zero, so every cell in those three lines is water.

Step 2: fill row 1. Row 1 needs four boat cells. Columns 1 and 5 are water, so only columns 2, 3, 4 and 6 are open, and all four must be filled.

Step 3: name the boats in row 1. Columns 2 to 4 form a run of three, with water at both ends and water below in row 2. That is the three-cell boat. Row 1, column 6 has water to its left and below, so it is a one-cell boat.

Step 4: narrow the lower rows. Column 3 needed only one cell, and row 1 has it, so the rest of column 3 is water. Rows 3 to 6 can now use only columns 2, 4 and 6.

Step 5: fill row 5. Row 5 needs three cells, and only three columns are open, so row 5 columns 2, 4 and 6 are all filled. Column 4 is now complete, with its cells in rows 1 and 5. Row 5, column 4 has water on every side, so it is the second one-cell boat.

Step 6: work down column 2. Column 2 needs three more cells in rows 3 to 6, and row 5 is one of them. Rows 3, 4 and 5 together would make a second three-cell boat, and so would rows 4, 5 and 6. The boat list has only one, so column 2 uses rows 3, 5 and 6.

Row 3, column 2 is the third one-cell boat, and rows 5 and 6 form a two-cell boat. Count it off: one two-cell boat is still on the list.

Step 7: finish column 6. Rows 3 and 6 are now full. Column 6 still needs one more cell, and row 4 is the only open row, so rows 4 and 5 of column 6 form the last two-cell boat.

Check. Every row and column count matches. The boat list is complete: one three-cell boat, two two-cell boats and three one-cell boats. No two boats touch, even at a corner. For this guide, we checked by computer every possible placement of these boats, and this is the only answer.

How do these puzzles relate to tiling and packing?

Each boat is a polyomino, a shape made of unit squares joined edge to edge. A paper published on 1 January 2003 recalls a simple measure for such shapes, called the order: the smallest number of copies of a shape that fit together into a rectangle ^[1]. The same paper describes as totally open the question of whether any polyomino has an odd order of five or more ^[1].

A straight boat has no such trouble, because it is already a rectangle on its own. A hidden boat grid asks a gentler question. The boats do not have to fill the grid, and they must keep apart rather than fit tightly together.

A paper published on 25 September 2018 studied whether the squares from one by one up to 24 by 24 could be packed into a 70 by 70 square, which their areas would fill exactly, and reports earlier work showing that no such packing exists ^[2]. In our own comparison, boat grids share one habit with that kind of problem: every cell is either covered or not, and the totals must agree, though matching totals alone never prove a fit.

Which mistakes should you watch for?

• Forgetting diagonal water. Two boats meeting at a corner break the rules.
• Bending a boat. Boats are straight. A filled cell with neighbours both to its side and below cannot be right.
• Losing track of the boat list. Count off each boat as soon as it is closed at both ends. A length can appear more than once.
• Filling before checking the column. A row may allow a cell that the column number has already used up.
• Skipping the final count. Add the filled cells in every row and column once more.

Can you solve this practice puzzle?

The boat list is one three-cell boat, one two-cell boat and two one-cell boats.

A hint if you need one: ask where the three-cell boat can fit.

Answer: the three-cell boat lies across row 2 in columns 1 to 3, the two-cell boat runs down column 1 in rows 4 and 5, and the one-cell boats sit in column 5 at rows 1 and 3.

Column 4 is water. Columns 2 and 3 hold one cell each, so the three-cell boat either lies across columns 1 to 3 or runs down column 1.

Suppose it runs down column 1. Any cell of column 2 beside or diagonal to the boat is water, so row 2 can reach three only with the boat cell plus columns 3 and 5. The boat must then cover row 2, in rows 1 to 3 or rows 2 to 4.

In rows 2 to 4, every cell of column 2 touches the boat, yet column 2 needs one cell. In rows 1 to 3, columns 1 and 3 are full, so the two-cell boat must run down column 5 through row 2. That puts a second cell in row 1 or row 3, whose count is one. So the boat lies across, in row 2, the only row with a count of three.

Rows 1 and 3 touch that boat in column 1, so column 1 takes its last two cells in rows 4 and 5. Column 5 then takes rows 1 and 3.

When you have finished, try this live riddle. It is about fitting a shape into a gap, which is the same kind of space reasoning in a different form.