The number of squares on a chessboards is :

Options

A. 102
B. 108
C. 216
D. 204

Answer

To find the total number of squares on a chessboard, we need to consider squares of different dimensions that can fit within the 8x8 grid of the chessboard.

### Step-by-Step Solution:

1. **Understanding the Chessboard Dimensions**:
A standard chessboard has 8 rows and 8 columns, making it an 8x8 grid.

2. **Counting 1x1 Squares**:
Each individual square on the chessboard is a 1x1 square. Since there are 8 rows and 8 columns, the total number of 1x1 squares is:
\[ 8 \times 8 = 64 \]

3. **Counting 2x2 Squares**:
A 2x2 square occupies 2 rows and 2 columns. The number of 2x2 squares that can fit horizontally and vertically is:
\[ (8-1) \times (8-1) = 7 \times 7 = 49 \]

4. **Counting 3x3 Squares**:
Similarly, for 3x3 squares:
\[ (8-2) \times (8-2) = 6 \times 6 = 36 \]

5. **Counting 4x4 Squares**:
For 4x4 squares:
\[ (8-3) \times (8-3) = 5 \times 5 = 25 \]

6. **Counting 5x5 Squares**:
For 5x5 squares:
\[ (8-4) \times (8-4) = 4 \times 4 = 16 \]

7. **Counting 6x6 Squares**:
For 6x6 squares:
\[ (8-5) \times (8-5) = 3 \times 3 = 9 \]

8. **Counting 7x7 Squares**:
For 7x7 squares:
\[ (8-6) \times (8-6) = 2 \times 2 = 4 \]

9. **Counting 8x8 Squares**:
Finally, for the 8x8 square (the entire chessboard):
\[ (8-7) \times (8-7) = 1 \times 1 = 1 \]

10. **Totaling All Squares**:
Now, we add all the squares together:
\[ 64 \, (1x1) + 49 \, (2x2) + 36 \, (3x3) + 25 \, (4x4) + 16 \, (5x5) + 9 \, (6x6) + 4 \, (7x7) + 1 \, (8x8) \]
\[ = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 \]

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