Henry Ernest Dudeney/Modern Puzzles/128 - Lines and Squares/Solution

Modern Puzzles by Henry Ernest Dudeney: $128$

With how few straight lines can you make exactly one hundred squares?

Thus, in the first diagram it will be found that with nine straight lines I have made twenty squares

(twelve with sides of the length $AB$, six with sides $AC$, and two with sides of the length $AD$).

In the second diagram, although I use one more line, I only get seventeen squares.

We use $15$ straight lines as follows:

There are:

$40$ squares with sides of length $AB$

$28$ squares with sides of length $AC$

$18$ squares with sides of length $AD$

$10$ squares with sides of length $AE$

$4$ squares with sides of length $AF$

making $100$ altogether.

We could make as many as $112$ squares with $15$ straight lines, but we were asked for exactly $100$.

With $14$ straight lines we can make only $91$ squares.

With $n$ straight lines we can make as many as:

$\dfrac {\paren {n - 3} \paren {n - 1} \paren {n + 1} } {24}$ squares if $n$ is odd

$\dfrac {\paren {n - 2} \paren {n - 1} n } {24}$ squares if $n$ is even.