Find the total number of squares when 8 vertical lines intersect 15 horizontal lines such that all the vertical lines are parallel and equidistant to each other, and all the horizontal lines are also parallel and equidistant to each other.

Options

A. 336
B. 456
C. 1096
D. 376

Answer

To find the total number of squares formed by 8 vertical lines and 15 horizontal lines, we can follow these steps:

### Step 1: Understand the arrangement of lines
We have 8 vertical lines and 15 horizontal lines. The vertical lines are parallel and equidistant from each other, as are the horizontal lines.

### Step 2: Determine the number of rows and columns
- The number of rows formed by the vertical lines is \(8 - 1 = 7\) (since the squares are formed between the lines).
- The number of columns formed by the horizontal lines is \(15 - 1 = 14\).

### Step 3: Count the squares of different sizes
To find the total number of squares, we need to count squares of all possible sizes that can be formed:

1. **1x1 squares**: The number of 1x1 squares is given by the product of the number of rows and columns:
\[ 1x1 \text{ squares} = 7 \times 14 = 98 \]

2. **2x2 squares**: The number of 2x2 squares is given by:
\[ 2x2 \text{ squares} = (7 - 1) \times (14 - 1) = 6 \times 13 = 78 \]

3. **3x3 squares**: The number of 3x3 squares is given by:
\[ 3x3 \text{ squares} = (7 - 2) \times (14 - 2) = 5 \times 12 = 60 \]

4. **4x4 squares**: The number of 4x4 squares is given by:
\[ 4x4 \text{ squares} = (7 - 3) \times (14 - 3) = 4 \times 11 = 44 \]

5. **5x5 squares**: The number of 5x5 squares is given by:
\[ 5x5 \text{ squares} = (7 - 4) \times (14 - 4) = 3 \times 10 = 30 \]

6. **6x6 squares**: The number of 6x6 squares is given by:
\[ 6x6 \text{ squares} = (7 - 5) \times (14 - 5) = 2 \times 9 = 18 \]

7. **7x7 squares**: The number of 7x7 squares is given by:
\[ 7x7 \text{ squares} = (7 - 6) \times (14 - 6) = 1 \times 8 = 8 \]

### Step 4: Add all the squares together
Now, we will sum all the squares calculated:
\[ \text{Total squares} = 98 + 78 + 60 + 44 + 30 + 18 + 8 \]

Calculating the total:
\[ \text{Total squares} = 98 + 78 = 176 \]
\[ 176 + 60 = 236 \]
\[ 236 + 44 = 280 \]
\[ 280 + 30 = 310 \]
\[ 310 + 18 = 328 \]
\[ 328 + 8 = 336 \]

### Final Answer
The total number of squares formed is **336**.

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