Cutting a Pancake with an Exotic Knife

• Published in 2025
• Added on 2026-04-27

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.

Links

• https://arxiv.org/abs/2511.15864v3
• https://arxiv.org/pdf/2511.15864v3

Other information

BibTeX entry

@article{CuttingaPancakewithanExoticKnife,
	key = {CuttingaPancakewithanExoticKnife},
	type = {article},
	title = {Cutting a Pancake with an Exotic Knife},
	author = {David O. H. Cutler and Jonas Karlsson and Neil J. A. Sloane},
	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.},
	comment = {},
	date_added = {2026-04-27},
	date_published = {2025-04-27},
	urls = {https://arxiv.org/abs/2511.15864v3,https://arxiv.org/pdf/2511.15864v3},
	collections = {easily-explained,food,geometry,things-to-make-and-do},
	url = {https://arxiv.org/abs/2511.15864v3 https://arxiv.org/pdf/2511.15864v3},
	year = 2025,
	urldate = {2026-04-27},
	archivePrefix = {arXiv},
	eprint = {2511.15864},
	primaryClass = {math.CO}
}