Entrance Slip - How many squares in a chessboard ?

When thinking of how many squares are in a chessboard, I begin by thinking of a regular 8x8 chessboard.

In order to see all of the different types of squares that can be formed from this board, I must begin by thinking of the smallest squares that can be found, and work my way up to an 8x8 square.

To begin, I think of a 1x1 chessboard, seeing that it just has 1 square.

Moving to a 2x2 chessboard, one would see 5 squares.

* There is one large 2x2 square, and 4 1x1 squares.

In a 3x3 chessboard, one would see 14 squares

* There are 9 1x1 squares, 4 2x2 squares, and 1 3x3 square.

From these you can see that a pattern emerges:

For a 1x1 chessboard, 1^2 =1

For a 2x2 chessboard, 1^2 + 2^2 = 5

For a 3x3 chessboard, 1^2 + 2^2 + 3^2 = 14

etc.

When applying this pattern to an 8x8 chessboard, we see that 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 1+4+9+16+25+36+49+64=204.

From this we would see that there are 204 squares in a chessboard.

Initially, I entered this problem thinking of an obvious answer of 64 squares. I quickly realized there was more to this problem than I thought. I needed to find a way of calculating the number of squares in a quicker way than just counting. I began with the smallest size of squares and worked my way up to 3, before finding a pattern. I realized that through finding out the each number of squares, they were all "perfect squares", meaning they have a square root. Knowing perfect squares helps me, as well as deductive reasoning and seeing a pattern emerging. This problem could be extended by asking for the number of squares of a larger chessboard and checking understanding.