Plot five points at random at the intersections of a coordinate grid. Between each pair of points a line segment can be drawn.
Prove that the midpoint of at least one of these segments occurs at an intersection of grid lines.
With 5 points there must be either 3 all with even x coordinates or 3 all with odd x coordinates. Choose the three points that fit this criterion.
Of the three chosen, at least two must have the same ycoordinate parity. These two points have the same ycoordinate parity and therefore differ by an even number in their y coordinates, and also have the same true of their x coordinates. Therefore taking the mean in either dimension results in an integer, for the midpoint.

Posted by Charlie
on 20160518 19:49:08 