This is a fairly old Puzzle that sets before the gamester a collection of Nine Points--
Whether they are mathematical points of no width or heighth,
or whether they are drawn points, tiny; such as those like a 'Period' is generally not revealed.
This distinction may be useful for solutions that require only 3 lines.
The Nine Points are Arranged like so:
. . .
. . .
. . .
While connecting The Nine Points with 5 lines is simple for most humanimals,
The Puzzle askes you to solve it by connecting all Nine Points with Four Lines...!
After solving this; i wondered--
Is there a solution for Three Lines...
Then Two Lines;
( Which is the trickiest...
Considering that for each solution,
One is looking for a Unique Approach... ! )
And Then One Line.
The One Line Solution is The Most Elegant...!
Once you have the tools, and insights from solving The One Line Solution;
Is there a Zen -- Zero Line Solution...!!! ( ? )
Finished with this;
Is there an Elegant 6, 7, 8
&/or 9 Line Solution(s...???
The Solutions: ( In Invisible Ink...!!! )
Sadly there are no accompanying illustrations, so you'll just have to read carefully...
The Three Line Solution Requires that The the top three points are connected with a line that stretches off into Infinity, As do the middle three points, and the bottom three points.
This also assumes that parallel lines meet at infinitity, which allows the lines to merge and return to the parameters of the puzzle!
If The Points have actual dimensions, then; It is possible to draw the connecting lines so that they just edge through the top of the first point, pass through the middle of the middle point, and then scratch past the last point, such that the line is at a discernable angle, and it doesn't have to extend to infinity before returning to the next line of points...!
The Two Line Solution is placed on a Inversion grid...!!! So that at the very middle of the paper is a point that represents the coordinates (0,0). Then a circle is drawn around this, it's edge represents the radial quantity of One. Such that any and all points inside the circle; such as ( .2,.6) has an Inversion Equivalency Point-- Outside the Circle. These aren't just mirror points, they are the SAME point, in Inversion Space...!!! Once this is established, The Nine Points are ploted so that the middle three points are drawn along the circles edge, while the lower three points converge at the center... meaning that the top three points are found at the outter edge of infinity...!!! The first line tranverses curved space to cut through the middle three points, then takes a tangental 90_degree turn towards the center, passes through it, which means that it simultaneously cuts through the top three points at infinity.
This is the Weakest Solution... Can you discover a better one...???
The One Line Solution:
This the most Elegant. Twist ( Slightly Offset ) and Roll the paper into a cylinder-- Then connect The Nine Points with One Line, So that the result resembles a Barber Pole!
The Zen: Zero Line Solution-- Simple fold the paper over and over upon itself, so that all The Points are occupying the same vertical ( Dimensionless ) space.