Cutting a Pancake with an Exotic Knife

TL;DR

This paper explores the maximum number of pancake pieces obtainable with various exotic-shaped knives, extending classical results to complex shapes and constrained versions.

Contribution

It introduces new maximum piece counts for diverse exotic knife shapes, expanding the classical pancake-cutting problem.

Findings

Maximum pieces for various shapes determined in most cases.

Bounds provided for constrained A and lollipop shapes.

Extended classical pancake-cutting results to complex knife geometries.

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, long-legged versions of the letters A, E, H, L, M, T, W, or X, a convex polygon, a circle, a phi, a figure 8, a pentagram, a hexagram, or a lollipop (or qoppa). We also consider "constrained" versions of the long-legged letters A, H, L, T, and X. In most cases we are able to determine the maximum number of pieces, although for the constrained A and the lollipop we can only give bounds.