Imagine taking a sheet of clear plastic, drawing a line down the middle, and then painting half of the sheet (to one side of the line) black. Now attach the painted sheet of plastic to our plane with the 101 points by pushing a thumbtack through the line on the plastic (which divides the clear and dark halves) and affixing it to point Now position the plastic sheet so that the dividing line is horizontal and we can only see the points above the line. If there happen to be exactly 50 visible points, then we are done. Suppose instead that there are more than 50 points in view. (And hence fewer than 50 out of sight below the line.) Begin slowly rotating the sheet of plastic in a counterclockwise direction through a total of 180 degrees, keeping track along the way of the number of visible points. We make two basic observations. First, because no three points are collinear, points come into view or disappear behind the dark half one at a time. Next, although we began with more than 50 visible points, we finish with less than 50, because after 180 degrees of rotation we will be looking at exactly those points which were originally hidden. Since the number of visible points only ever increases or decreases by one point at a time, there must have been exactly 50 visible points somewhere along the way. And this position of the dividing line solves the problem. Of course, the same argument works if there were initially fewer than 50 points in view, or if the initial position of the dividing line were something other than horizontal, for that matter. So we have settled the matter of dividing the points in half. |