The easiest way to understand how to solve the puzzles is to look at a completed puzzle.
The image to the right shows a completed puzzle. As you can see by looking at it, the number clues tell how many filled squares are in each row and column. Each number represents a group of filled squares in that particular row or column. There is always at least 1 empty square separating the groups of filled squares. 

When solving a puzzle, I first look for any rows or columns that have 0 filled squares or have every square filled.
The example to the right shows a puzzle grid that is 9 squares wide by 12 squares high. The ideal place to begin is with the rows marked with arrows. The number clue for each of these rows is 9; meaning that each of these rows will contain a block of 9 filled squares. Since the puzzle is 9 squares wide, then every square must be filled, as shown. 

Now that a few squares have been filled in, we have more information to work with.
Each of the columns I marked with arrows has only 1 number clue. This means that that column has only 1 group of filled squares. Since we have two filled squares that are separated by empty squares, we know that the squares in between must be filled! 

Looking at the marked columns, we know that each column contains a group 6 filled squares. We know the location of 4 of those.
The other 2 filled squares in the column can only be the 2 squares directly above OR the 2 squares directly below the group of 4. The other squares in the column must be empty! I have marked those empty squares with a dot. 

Knowing that there are 2 squares that must remain empty (marked with dots), we can solve the marked row. The marked row contains a group of 7 filled squares, which includes every remaining square in this row.
Puzzles are solved by working through the clues, alternating between rows and columns to figure out which squares are filled and which must remain empty. As each puzzle is solved, a picture will appear in the grid.
Let's continue solving this puzzle. 

The row I have marked here is not as simple to solve. The row contains a group of 5 filled squares, but there are 7 possible squares in the row.
To help in solving this row, we will use the overlapping method.
If you begin at the first possible square on the left and count out 5 squares, you will find that if these squares were filled, they would span the squares I marked with "x". (fig.a) 

If you begin at the first possible square on the right and count out 5 squares, you will find that if these squares were filled, they would span the squares I marked with "o". (fig.b)
So, whether you begin the group of 5 at the left or right, there is an overlap of 3 squares. No matter where the group of 5 is located, these 3 squares must be filled. (fig. c) 

Take a look at the marked column in this grid. We know that this column will contain a group of 8 and a group of 2 filled squares, in order.
At the bottom of the column, you can see that the group of 2 filled squares has already been determined. We know that groups are separated by at least 1 empty square, so we know that the square directly above the 2 filled squares must be empty! I will mark the empty square with a dot. 

We also know that we can fill in the 2 empty squares between the filled squares, since we only have one group of 8 filled squares remaining in this column.
Using the overlapping method on this column helps us to determine the location of more of the filled squares in this column.
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