How many total squares are there in a standard chess board (or check board)?

Options

A. 65
B. 204
C. 256
D. 206

Answer

To find the total number of squares in a standard chessboard, we need to consider all possible squares of different sizes that can be formed on the board. A standard chessboard has 8 rows and 8 columns.

### Step-by-Step Solution:

1. **Identify the Sizes of Squares**:
A square can be of size 1x1, 2x2, 3x3, up to 8x8. We will count the number of squares for each size.

2. **Count the Number of Squares for Each Size**:
- **1x1 Squares**: There are 8 rows and 8 columns, so the number of 1x1 squares is:
\[ 8 \times 8 = 64 \]
- **2x2 Squares**: A 2x2 square can start from any cell in the first 7 rows and the first 7 columns (since it occupies 2 rows and 2 columns), so the number of 2x2 squares is:
\[ 7 \times 7 = 49 \]
- **3x3 Squares**: A 3x3 square can start from any cell in the first 6 rows and the first 6 columns, so the number of 3x3 squares is:
\[ 6 \times 6 = 36 \]
- **4x4 Squares**: A 4x4 square can start from any cell in the first 5 rows and the first 5 columns, so the number of 4x4 squares is:
\[ 5 \times 5 = 25 \]
- **5x5 Squares**: A 5x5 square can start from any cell in the first 4 rows and the first 4 columns, so the number of 5x5 squares is:
\[ 4 \times 4 = 16 \]
- **6x6 Squares**: A 6x6 square can start from any cell in the first 3 rows and the first 3 columns, so the number of 6x6 squares is:
\[ 3 \times 3 = 9 \]
- **7x7 Squares**: A 7x7 square can start from any cell in the first 2 rows and the first 2 columns, so the number of 7x7 squares is:
\[ 2 \times 2 = 4 \]
- **8x8 Squares**: There is only one 8x8 square, which is the entire chessboard:
\[ 1 \times 1 = 1 \]

3. **Sum All the Squares**:
Now, we add all the squares together:
\[ 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 \]

### Final Answer:
The total number of squares in a standard chessboard is **204**.