Question

Lal divides his garden into several identical squares and places posts at all the corners of all the squares. He then plants one tree per square. If a rectangular garden uses 36 posts in all, find the maximum number of trees that he could have planted.

• A. 25
• B. 36
• C. 16
• D. 49

Hint:

When maximizing the number of interior rectangles/squares with fixed posts, choose dimensions so that post counts along each side are as close as possible.

Answer 1

The Correct Option is A

Step 1: Relation between posts and squares

If the garden is divided into \( m \) rows and \( n \) columns of squares, there will be \( (m+1) \) posts along the length and \( (n+1) \) posts along the breadth. The total number of posts is given by: \[ (m+1)(n+1) = 36 \]

Step 2: Maximizing the number of squares (trees)

The number of squares is: \[ mn = (m+1)(n+1) - m - n - 1 \] From \((m+1)(n+1) = 36\), we have: \[ mn = 36 - m - n - 1 = 35 - (m + n) \] To maximize \( mn \), we must minimize \( m + n \).

Step 3: Minimizing \( m + n \)

For a fixed product \((m+1)(n+1) = 36\), the sum \( m+1 + n+1 \) is minimized when the factors are closest in value. Factors of \( 36 \) closest together are \( (6, 6) \), giving: \[ m = 5, \quad n = 5 \]

Step 4: Maximum number of squares

\[ \text{Squares} = m \times n = 5 \times 5 = 25 \]

Final Answer:

\[ \boxed{25} \]