Cutting a Pancake with an Exotic Knife

Abstract: In the first chapter of their classic book "Concrete Mathematics", Graham, Knuth, and Patashnik consider the maximum number of pieces that can be obtained from a pancake by making n cuts with a knife blade that is straight, or bent into a V, or bent twice into a Z. We extend their work by considering knives, or "cookie-cutters", of even more exotic shapes, including a k-armed V, a chain of k connected line segments, a long-legged version of one of the letters A, E, L, M, T, W, or X, a convex polygon, a circle, a figure 8, a pentagram, a hexagram, or a lollipop. In many cases a counting argument combined with Euler's formula produces an explicit expression for the maximum number of pieces. ``Strict'' versions of the letters A and T are also considered, for which we have only conjectural formulas.