SOLUTION to Problem 1 Extension

This time imagine that the plane is represented by a wall, and tap nails into each of the red points, as if you were hanging pictures. Now take a long, thin steel rod and place it against the wall in some position where it divides all the nails in half. Our goal is to twist the rod around through 180 degrees so that it returns to its starting position, all the while dividing the nails (red points) in half.

So begin twisting the rod counterclockwise until it comes up against two of the nails which block any further turning. (It cannot hit three nails simultaneously, because no three points are collinear.) Now temporarily remove the two nails in the way, turn the rod just enough so that it crosses the two nail holes, then reinsert the nails so that it now rests against the other sides of those two nails. This has the effect of moving each nail to the other side of the rod, so there will still be 25 nails on each side of the rod.

Continue this process until the rod has turned through a total of 180 degrees. Clearly it will have divided the nails in half throughout this process. Also, note that at this stage the rod will be parallel to its starting position. Furthermore, there cannot be any nails in the way to prevent us from sliding the rod so that it coincides with its original position. For otherwise the rod would no longer evenly divide the nails in half when it returned to its original position, contrary to the way we began the process.

Next reinstate the blue points, and make slight alterations to the process of turning the rod, if necessary, to ensure that it only crosses one blue point at a time. Now imagine keeping track of the number of blue points to one side of the line as it turns, just as we did before. Precisely the same argument shows that at some instant the line will divide the blue points in half also. This position of the line solves the problem.