How many squares are there in a chessboard of size $8 imes 8$?

Question

How many squares are there in a chessboard of size $8 	imes 8$?

Prepp Team · Accepted Answer

Solving the Chessboard Squares Puzzle ($8 \times 8$)

Understanding the Chessboard Problem

A standard chessboard is an $8 \times 8$ grid. While it's easy to see the smallest $1 \times 1$ squares, the challenge involves counting squares of all possible sizes within this grid. These include $1 \times 1$, $2 \times 2$, $3 \times 3$, and so on, up to the single large $8 \times 8$ square that represents the entire board.

Step-by-Step Calculation of Squares

To find the total number of squares, we need to count how many squares of each size can fit onto the $8 \times 8$ grid.

Counting $1 \times 1$ Squares:

The most obvious squares are the individual cells. On an $8 \times 8$ board, there are $8$ rows and $8$ columns, giving us a total of $8 \times 8 = 64$ squares of size $1 \times 1$.

Counting $2 \times 2$ Squares:

Now, consider squares with side length 2. Imagine the top-left corner of a $2 \times 2$ square. It can be placed in any of the first 7 rows and any of the first 7 columns. Therefore, there are $7 \times 7 = 49$ possible positions for a $2 \times 2$ square.

Counting Larger Squares:

We continue this pattern for squares of increasing size:

For $3 \times 3$ squares, the top-left corner can be placed in the first 6 rows and 6 columns, resulting in $6 \times 6 = 36$ squares.
For $4 \times 4$ squares, there are $5 \times 5 = 25$ squares.
For $5 \times 5$ squares, there are $4 \times 4 = 16$ squares.
For $6 \times 6$ squares, there are $3 \times 3 = 9$ squares.
For $7 \times 7$ squares, there are $2 \times 2 = 4$ squares.
For the single $8 \times 8$ square (the entire board), there is $1 \times 1 = 1$ square.

Summary Table:

Size of Square	Number of Squares Possible
$1 \times 1$	$8 \times 8 = 64$
$2 \times 2$	$7 \times 7 = 49$
$3 \times 3$	$6 \times 6 = 36$
$4 \times 4$	$5 \times 5 = 25$
$5 \times 5$	$4 \times 4 = 16$
$6 \times 6$	$3 \times 3 = 9$
$7 \times 7$	$2 \times 2 = 4$
$8 \times 8$	$1 \times 1 = 1$

Counting squares of different sizes on an 8x8 chessboard.

Using the Sum of Squares Formula

The total number of squares is the sum of the counts for each size:

Total Squares = $8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2$

This is the sum of the squares of the first 8 positive integers. There's a direct formula for the sum of the first $n$ squares:

$ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} $

In our case, $n=8$. Plugging this into the formula:

$ \text{Total Squares} = \frac{8(8+1)(2 \times 8 + 1)}{6} $ $ \text{Total Squares} = \frac{8 \times 9 \times (16 + 1)}{6} $ $ \text{Total Squares} = \frac{8 \times 9 \times 17}{6} $ $ \text{Total Squares} = \frac{72 \times 17}{6} $ $ \text{Total Squares} = 12 \times 17 $ $ \text{Total Squares} = 204 $

Final Result

By summing the counts of squares of all possible sizes ($1 \times 1$ up to $8 \times 8$) or by using the sum of squares formula, we find that there are exactly 204 squares on an $8 \times 8$ chessboard.