Problem Circle Solutions
In the Problem Circle from the last issue of MTCircular, we asked whether it Is possible to paint each point in the plane, using three
different colors, so that every straight line in the plane
has exactly two different colors. Here are two nicely different solutions to this problem, courtesy of Hema Gopalakrishnan (left) and Joshua Zucker (right).
 
For her solution at left, Hema Gopalakrishnan writes, "Using the XY-coordinate system to represent points in the plane, suppose the origin is colored green, points on the X-axis other than the origin are colored red and points on the Y-axis other than the origin are colored blue. Further suppose that the points in the 1st and 3rd quadrants are colored blue and points in the 2nd and 4th quadrant are colored red. Note that the only point colored green is the origin. Then every line through the origin will have two colors, blue and green or red and green. Every line not passing through the origin will have the two colors blue and red."
For his solution at right, Joshua Zucker writes, "Any line not passing through the intersection point can't be parallel to both of the intersecting lines, so it will be green and red. Any line passing through the intersection point either is one of the intersecting lines, in which case it is red and blue, or meets the intersecting
lines at only the intersection point, so it is green and blue."
|
