6 equidistant vertical lines are drawn on a board. 6 equidistant horizontal lines are also drawn on the board cutting the 6 vertical lines, and the distance between any two consecutive horizontal lines is equal to that between any two consecutive vertical lines. What is the maximum number of squares thus formed?

Options

A. A) 37
B. B) 55
C. C) 91
D. D) 225

Answer

To solve the problem of finding the maximum number of squares formed by 6 equidistant vertical lines and 6 equidistant horizontal lines, we can follow these steps:

### Step 1: Understand the Grid Formation
We have 6 vertical lines and 6 horizontal lines. The intersection of these lines forms a grid. Each square is formed by selecting two vertical lines and two horizontal lines.

### Step 2: Count the Squares of Different Sizes
We can form squares of various sizes within this grid. The sizes of squares we can form are:
- 1x1 squares
- 2x2 squares
- 3x3 squares
- 4x4 squares
- 5x5 squares

### Step 3: Calculate the Number of Each Size of Square

1. **1x1 Squares**:
- The number of 1x1 squares is determined by the number of ways to choose 2 vertical lines and 2 horizontal lines from the 6 lines.
- The number of 1x1 squares = (6 - 1) * (6 - 1) = 5 * 5 = 25.

2. **2x2 Squares**:
- The number of 2x2 squares = (6 - 2) * (6 - 2) = 4 * 4 = 16.

3. **3x3 Squares**:
- The number of 3x3 squares = (6 - 3) * (6 - 3) = 3 * 3 = 9.

4. **4x4 Squares**:
- The number of 4x4 squares = (6 - 4) * (6 - 4) = 2 * 2 = 4.

5. **5x5 Squares**:
- The number of 5x5 squares = (6 - 5) * (6 - 5) = 1 * 1 = 1.

### Step 4: Sum All the Squares
Now, we sum all the squares calculated:
- Total squares = 25 (1x1) + 16 (2x2) + 9 (3x3) + 4 (4x4) + 1 (5x5) = 55.

### Conclusion
The maximum number of squares that can be formed is **55**.