256. Lines and Squares. Here is one simple question. What is the smallest number of straight lines needed to construct exactly 100 squares? In the figure on the left, nine lines are used to construct 20 squares (12 with side length $A B$, 6 with side length $A C$, and 2 with side length $A D$). In the figure on the right, there is one more line, and the number of squares has increased to 17. Thus, what matters is not the total number of lines, but how they are drawn. Remember, the goal is to get exactly 100 squares (no more and no less).

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-074.jpg?height=492&width=484&top_left_y=328&top_left_x=792)
256. If 15 lines are drawn as shown in the figure, exactly 100 squares will be formed. Forty of them have a side equal to \(A B\), twenty have a side equal to \(-A C\), eighteen have a side equal to \(-A D\), ten have a side equal to \(-A E\), and four have a side equal to \(-A F\). With 15 lines, it is even possible to form 112 squares, but we are required to form exactly 100. With 14 lines, you cannot form more than 91 squares.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-232.jpg?height=589&width=368&top_left_y=902&top_left_x=841)

In general, with \(n\) lines, \(\frac{(n-3)(n-1)(n+1)}{24}\) squares can be formed if \(n\) is odd, and \(\frac{(n-2) n(n-1)}{24}\) squares if \(n\) is even.

If we have 15 lines perpendicular to another \(n\) lines, where \(m\) is less than \(n\), the number of squares is

\[
\frac{m(m-1)(3 n-m-1)}{6}
\]