Recall that given two points in the plane, an ellipse can be defined as the collection of points whose sum of distances to the given points is constant. But here in the Prisoner’s Dilemma, we know there are more operations than just addition. (Remember Dilemma 4?)
So suppose you are given points and a distance two units apart in the plane. We write just for the distance between points and in the plane (so for example, ). Describe the collection of all points in the plane such that As a start, figure 2 shows five of the points in this collection (four of which also happen to lie on the orange ellipse defined by ). You might also want to consider how the shape of this collection changes when you use a constant other than one; in other words, what does the collection of points such that look like for other values of ?
This problem originally appeared in the Prisoner’s Dilemma in the 2022 Fall issue of the PMP Newsletter. Solutions are no longer being accepted for this Dilemma.