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, 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.
Cite
@article{arxiv.2511.15864,
title = {Cutting a Pancake with an Exotic Knife},
author = {David O. H. Cutler and Jonas Karlsson and Neil J. A. Sloane},
journal= {arXiv preprint arXiv:2511.15864},
year = {2026}
}
Comments
47 pages, 8 tables, 47 figures. Added a third author, new sections, some proofs tightened