Here's a game I thought of the other day while driving in my car:
Choose four distinct points on a circle of radius 1 in such a way as to maximize your score. Your score is the area of your quadrilateral plus the area of the largest triangle that may be formed by deleting one of your points.
As a warm-up, you might show that if you choose the points to form a square, then your score is 3.
My high score is shown below. Can you beat it?
(Numerically, this is between 3.14880129427 and 3.14880129428.)