How many squares are there in a 4 by 4 grid?

There are 30 squares in a 4 by 4 grid. Here's how it breaks down.

There are 16 of the 1 x 1 squares.

There are 9 of the 2 x 2 squares.

There are 4 of the 3 x 3 squares.

There is 1 of the 4 x 4 squares.

It's fairly easy to see the individual 1 x 1 squares and count them. Same with the 4 x 4 square, as it's the perimeter of the whole array. With the 2 x 2 squares, start with a 2 x 2 square in, say, the upper left corner and "move" the square one space to the right. Then one more to "complete" the top row. Then move it one square "down" and continue until arriving at the place you started. There will be 9 different "positions" for the 2 x 2 square.