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What's My Line?


Two million points are randomly scattered in a circle (all distinct).

Can you always draw a straight line that passes through the circle so that the line has a million points on each side?

Explain your answer/reasoning.

 


Hint: Try it with a random number of points (even). Is it always possible? Think about your strategies for drawing the line.

 


Solution Commentary: Rather than offer my own solution, I suggest you consider the solution provided by Dr. Math via the MathForum. Do you agree with this solution?

Also, M.N. (Norway) sent this solution...is it different: "Construct lines through every possible pair of points. Place an anchor point, A, somewhere outside of the cluster of 1 000 000 points so it is not lying on any line. Now you can place a second point, B, somewhere on the other side of the cluster and construct line AB. By moving point B, line AB will only cross over one point at a time, because point A is not colinear with more than one point in the cluster. Thus, we can find a position for point B where there are the same number of points on each side of AB....Thank you to the late Herb Wills, one of my doctoral advisors, for this elegant solution. It is one of the many excellent problems he shared with me over the years."