Let each side of smallest square of chess board in one unit in length. Find the sum of area of all possible squares whose side parallel to side of chess board.

Answer

To find the sum of the areas of all possible squares whose sides are parallel to the sides of a chessboard, we can follow these steps:

### Step 1: Understand the Chessboard Structure
A standard chessboard consists of 8 rows and 8 columns of unit squares, which means it has a total of 64 unit squares (1x1).

### Step 2: Count the Squares of Each Size
We will count the number of squares of each possible size that can be formed on the chessboard.

1. **1x1 Squares:**
- There are 8 rows and 8 columns.
- Total number of 1x1 squares = \(8 \times 8 = 64\)
- Area of each 1x1 square = \(1^2 = 1\)
- Total area contributed by 1x1 squares = \(64 \times 1 = 64\)

2. **2x2 Squares:**
- There are 7 rows and 7 columns where a 2x2 square can fit.
- Total number of 2x2 squares = \(7 \times 7 = 49\)
- Area of each 2x2 square = \(2^2 = 4\)
- Total area contributed by 2x2 squares = \(49 \times 4 = 196\)

3. **3x3 Squares:**
- There are 6 rows and 6 columns where a 3x3 square can fit.
- Total number of 3x3 squares = \(6 \times 6 = 36\)
- Area of each 3x3 square = \(3^2 = 9\)
- Total area contributed by 3x3 squares = \(36 \times 9 = 324\)

4. **4x4 Squares:**
- There are 5 rows and 5 columns where a 4x4 square can fit.
- Total number of 4x4 squares = \(5 \times 5 = 25\)
- Area of each 4x4 square = \(4^2 = 16\)
- Total area contributed by 4x4 squares = \(25 \times 16 = 400\)

5. **5x5 Squares:**
- There are 4 rows and 4 columns where a 5x5 square can fit.
- Total number of 5x5 squares = \(4 \times 4 = 16\)
- Area of each 5x5 square = \(5^2 = 25\)
- Total area contributed by 5x5 squares = \(16 \times 25 = 400\)

6. **6x6 Squares:**
- There are 3 rows and 3 columns where a 6x6 square can fit.
- Total number of 6x6 squares = \(3 \times 3 = 9\)
- Area of each 6x6 square = \(6^2 = 36\)
- Total area contributed by 6x6 squares = \(9 \times 36 = 324\)

7. **7x7 Squares:**
- There are 2 rows and 2 columns where a 7x7 square can fit.
- Total number of 7x7 squares = \(2 \times 2 = 4\)
- Area of each 7x7 square = \(7^2 = 49\)
- Total area contributed by 7x7 squares = \(4 \times 49 = 196\)

8. **8x8 Squares:**
- There is only 1 square that covers the entire chessboard.
- Total number of 8x8 squares = \(1 \times 1 = 1\)
- Area of the 8x8 square = \(8^2 = 64\)
- Total area contributed by 8x8 squares = \(1 \times 64 = 64\)

### Step 3: Sum the Areas
Now, we sum the total areas contributed by all the squares:

\[ \text{Total Area} = 64 + 196 + 324 + 400 + 400 + 324 + 196 + 64 \]

Calculating this gives:

\[ \text{Total Area} = 64 + 196 + 324 + 400 + 400 + 324 + 196 + 64 = 1968 \]

### Final Answer
The sum of the area of all possible squares whose sides are parallel to the sides of the chessboard is **1968**.

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