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Cutting a Pancake with an Exotic Knife

David O. H. Cutler, Neil J. A. Sloane

TL;DR

This work extends the classical pancake-cutting problem to a broad family of polygonal and exotic cookie-cutters S by modeling n copies as a planar graph G_S(n) and proving that, for most shapes, maximizing crossings maximizes the number of regions. The authors derive explicit formulas for a wide range of affine- and similarity-type shapes (including hatpins, k-armed Vs, k-chains, long-legged letters, finite polygons, circles, and lollipops) and establish numerous exact results, equivalences, and conjectures. They reveal several surprising connections, such as the equivalence in growth between long-legged A, 3-armed Vs, and 3-chains, and they frame many results in terms of Euler-type region counts and crossing bounds. The paper also ties these geometric dissections to OEIS sequences, enumerative geometry, and geometrical configurations, and it identifies several open problems and conjectures for future work. Overall, it provides a coherent method to compute a_S(n) for a rich collection of shapes and highlights both rigorous results and fertile directions for further research.

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, H, L, M, T, W, or X, a convex polygon, a circle, a phi, 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. "Constrained" versions of the long-legged letters A and T are also considered, for which we have only conjectural formulas.

Cutting a Pancake with an Exotic Knife

TL;DR

This work extends the classical pancake-cutting problem to a broad family of polygonal and exotic cookie-cutters S by modeling n copies as a planar graph G_S(n) and proving that, for most shapes, maximizing crossings maximizes the number of regions. The authors derive explicit formulas for a wide range of affine- and similarity-type shapes (including hatpins, k-armed Vs, k-chains, long-legged letters, finite polygons, circles, and lollipops) and establish numerous exact results, equivalences, and conjectures. They reveal several surprising connections, such as the equivalence in growth between long-legged A, 3-armed Vs, and 3-chains, and they frame many results in terms of Euler-type region counts and crossing bounds. The paper also ties these geometric dissections to OEIS sequences, enumerative geometry, and geometrical configurations, and it identifies several open problems and conjectures for future work. Overall, it provides a coherent method to compute a_S(n) for a rich collection of shapes and highlights both rigorous results and fertile directions for further research.

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, H, L, M, T, W, or X, a convex polygon, a circle, a phi, 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. "Constrained" versions of the long-legged letters A and T are also considered, for which we have only conjectural formulas.

Paper Structure

This paper contains 24 sections, 1 theorem, 30 equations, 32 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The planar graph and the maximum number of regions
  3. The hatpin
  4. The k-armed V
  5. The k-chain
  6. The long-legged A
  7. The long-legged Z, W, and M
  8. The constrained long-legged L, X, H, and phi
  9. The constrained long-legged L
  10. The constrained long-legged X
  11. The constrained long-legged H and phi
  12. The constrained long-legged T and A
  13. The constrained long-legged T
  14. The constrained long-legged T
  15. The constrained long-legged A
  16. ...and 9 more sections

Key Result

Theorem 1

The maximum number of regions that can be formed in the plane by drawing $n$ long-legged 3-armed V's, or $n$ 3-chains, or $n$ long-legged ${\sf A}$'s, is $(9n^2-5n+2)/2$ (https://oeis.org/A140064).

Figures (32)

  • Figure 1: Cutting an infinite pancake with 1, 2, 3, 4, or 5 cuts using an infinite straight knife produces a maximum of 2, 4, 7, 11, or 16 pieces. Arrowheads indicate infinite lines.
  • Figure 2: A $6$-set Venn diagram (with $64$ regions) constructed from six copies of a sausage shape $S$, from Grünbaum Gru75. (A circle is a very simple sausage.) The construction easily generalizes to show that at least $2^n$ regions can be achieved using $n$ copies of a simple Jordan curve.
  • Figure 3: A single twisted sausage shape can produce arbitrarily many regions.
  • Figure 4: An optimal pancake graph $G_K(7)$ defined by \ref{['EqPancake']} with $n=6$ and $\theta = 15^{\circ}$. The $t=0$ line is the $x$-axis, and the $t=6$ line is $x=6$.
  • Figure 5: A graph $G_{3{\sf V}}(2)$ with 14 regions, formed from two 3-armed V's (or Wu's).
  • ...and 27 more figures

Theorems & Definitions (1)

  • Theorem 1