pith. sign in

arxiv: 2511.15864 · v3 · submitted 2025-11-19 · 🧮 math.CO

Cutting a Pancake with an Exotic Knife

David O. H. Cutler , Jonas Karlsson , Neil J. A. Sloane This is my paper

Pith reviewed 2026-05-17 20:08 UTC · model grok-4.3

classification 🧮 math.CO
keywords pancake cuttingmaximum piecesknife shapescurve arrangementsplane divisionscombinatorial geometrypolyline cuts
0
0 comments X

The pith

Exotic knife shapes determine the exact maximum number of pieces obtainable from a pancake with n cuts in most cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the classic problem of maximizing pancake pieces with straight, V-shaped, or Z-shaped knives to a wide range of fixed exotic shapes. These include k-armed V's, chains of line segments, long-legged letters such as A, E, H, L, M, T, W, and X, convex polygons, circles, phi symbols, figure 8s, pentagrams, hexagrams, and lollipops, along with constrained versions of some letters. The central task is to find the largest number of regions created inside a disk by n placements of each rigid shape. Exact values are obtained for nearly all shapes through analysis of possible intersection patterns, while only bounds appear for the constrained A and the lollipop.

Core claim

For each exotic knife shape the maximum number of pancake pieces with n cuts is found by positioning the rigid curves to produce the largest possible number of intersection points and crossings inside the disk, yielding explicit counts or tight bounds in all but two cases.

What carries the argument

The arrangement of n translated and rotated copies of a fixed curve or polyline inside a disk, maximized for the number of crossings and bounded regions.

Load-bearing premise

Each knife shape can be placed anywhere and in any orientation as a single rigid piece without self-overlap to reach the theoretical maximum intersections.

What would settle it

An explicit placement of three copies of a k-armed V that produces more regions than the paper's stated maximum for that shape and n would disprove the count.

Figures

Figures reproduced from arXiv: 2511.15864 by David O. H. Cutler, Jonas Karlsson, Neil J. A. Sloane.

Figure 1
Figure 1. Figure 1: Cutting an infinite pancake with 1, 2, 3, 4, or 5 cuts using an infinite [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A 6-set Venn diagram (with 64 regions) constructed from six copies [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A single twisted sausage shape can produce arbitrarily many regions. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An optimal pancake graph GK(7) defined by (9) with n = 6 and θ = 15◦ . The t = 0 line is the x-axis, and the t = 6 line is x = 6. Since the optimal pancake graph GK(n) is the basis for other constructions, we summarize its properties: V = VC = [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A graph G3V(2) with 14 regions, formed from two 3-armed V’s (or Wu’s). k\n 0 1 2 3 4 5 6 7 8 ... 0 1 1 1 1 1 1 1 1 1 ... 1 1 1 2 4 7 11 16 22 29 ... 2 1 2 7 16 29 46 67 92 121 ... 3 1 3 14 34 63 101 148 204 269 ... 4 1 4 23 58 109 176 259 358 473 ... 5 1 5 34 88 167 271 400 554 733 ... 6 1 6 47 124 237 386 571 792 1049 ... 7 1 7 62 166 319 521 772 1072 1421 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Transforming a hatpin graph GH(4) to an optimal G3V(4) graph with a3V(4) = 63 regions, by replacing hatpins with long narrow 3-armed V’s (or Wu’s). The new graph has VC = 54 crossings and α(3V) = 12 arms, and a cpa of 9, in agreement with (12). (Some of the arrowheads on the right have been omitted for clarity.) We may verify (14) directly by considering what happens to the different parts of the hatpin gr… view at source ↗
Figure 7
Figure 7. Figure 7: Long narrow drawings of a 3-chain, a 4-chain, and a 5-chain, used in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Transforming a pancake graph GK(3) to an optimal G3C(3) graph with a3C(3) = 34 regions, by replacing the knife cuts by long narrow 3-chains. A graph GkC(n) formed by drawing n k-chains has VB = n(k − 1) base nodes and nk arms, with dB = 2, and E∞ = 2n. The basic equation (6) then gives R = VC + n + 1 , (15) just as for the pancake graph (see (8)). So again we must maximize VC, which we can do by first maxi… view at source ↗
Figure 9
Figure 9. Figure 9: The six ways to draw five lines in general position in the plane [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The two optimal 5-chains. One possible explanation might have been that there is a 1-to-1-correspondence between optimal GkC(n) graphs and optimal GnC(k) graphs. But this is false even in the simplest case. An optimal G1C(n) graph shows n lines in general position in the plane, and the number of such graphs is known for 0 ≤ n ≤ 9 (A090338): 1, 1, 1, 1, 1, 6, 43, 922, 38609, 3111341, . . [PITH_FULL_IMAGE… view at source ↗
Figure 11
Figure 11. Figure 11: 14 regions can be obtained with two long-legged [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Converting an optimal graph with n long-legged A’s (left) into a graph with n 3-armed V ’s (right). The steps do not decrease the number of regions. Reversing the last step (going from right to center) performs the inverse operation [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Converting a long-legged A (left) into a 3-chain (right), Similarly, moving the right end of the crossbar so that it is further from the tip than any other crossing node on that arm will also not decrease the number of regions. Finally, we can move the left end of the crossbar so it actually coincides with the tip and the right end so it is cut free from the arm. The A has now become a 3-armed V, and sinc… view at source ↗
Figure 14
Figure 14. Figure 14: Two long-legged W’s divide the plane into 19 regions. 8 The constrained long-legged L, X, H, and ϕ In this section we discuss our first four shapes from the similarity family, L, X, X, and ϕ, all of which we are able to solve. 8.1 The constrained long-legged L With no restriction on the angle, a long-legged L would be indistinguishable from a long￾legged V. We define a constrained long-legged L to consist… view at source ↗
Figure 15
Figure 15. Figure 15: The construction of n constrained long-legged L’s, showing the case n = 5. There are 36 regions. (The arrowheads have been omitted.) when n = 5. In general the graph contains 2n infinite regions, there are n rows of n − 1 cells each, and a further triangle of (n−1)(n−2)/2 cells, for a total of R = aL (n) = (3n 2−n+2)/2 regions. For n ≥ 0 the values are 1, 2, 6, 13, 36, 52, 71, . . . (A143689). The OEIS en… view at source ↗
Figure 16
Figure 16. Figure 16: Dividing the plane into the maximum number of pieces using [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: An unconstrained long-legged H is the same as two V’s joined by a line segment, which can be rearranged to have four self-crossings, and then (invertibly) transformed into a 5-chain by “raising the bar” (the heavy line). 8.3 The constrained long-legged H and ϕ Our rules for long-legged letters tell us that an (unconstrained) long-legged H is the same as two long legged V’s with their tips joined by a line… view at source ↗
Figure 18
Figure 18. Figure 18: One of the three inequivalent ways in which two constrained [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Construction of n H’s with each pair intersecting in 7 points (here n = 3). We can define a constrained long-legged H to consist of a pair of parallel lines joined by a perpendicular line segment (the crossbar), or any image of such a figure under Gsim. Two H’s can meet in a maximum of κ(H) = 7 points. This can be done in three inequivalent ways, but we will only make use of one of them, the arrangement s… view at source ↗
Figure 20
Figure 20. Figure 20: An invertible map from H’s to ϕ’s. A weaker version of this claim can be obtained from our analysis of the H shape. Define a constrained ϕ to consist of a circle with a line through its center. Two such ϕ’s can intersect in a maximum of seven points. An optimal graph containing n ϕ’s, each intersecting all of the others in seven points, can be obtained from an optimal graph with n H’s by applying the inve… view at source ↗
Figure 21
Figure 21. Figure 21: Three constrained long-legged T’s can divide the plane into 19 regions. It is impossible to add a fourth T that meets each of the first three in 4 points [PITH_FULL_IMAGE:figures/full_fig_p021_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Four constrained long-legged T’s can divide the plane into 32 re￾gions. The first three are the same as in [PITH_FULL_IMAGE:figures/full_fig_p021_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Two constrained A’s can intersect in 8 points, and divide the plane into aA (2) = 13 regions. (In this picture one A has a purple tip, the other a brown tip. One of the intersection points is not shown: it occurs a long way below the bottom of the picture. The arrowheads are also not shown.) 9.2 The constrained long-legged A The unconstrained long-legged A was discussed in §6, where we showed that it is e… view at source ↗
Figure 24
Figure 24. Figure 24: Three constrained A’s can divide the plane into 30 regions [PITH_FULL_IMAGE:figures/full_fig_p023_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Enlargement of the portion of Fig [PITH_FULL_IMAGE:figures/full_fig_p023_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Two regular octagons divide the plane into 18 regions. [PITH_FULL_IMAGE:figures/full_fig_p025_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Two concave quadrilaterals also divide the plane into 18 regions. Is [PITH_FULL_IMAGE:figures/full_fig_p025_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Four circles (black) can divide the plane into 14 regions. There [PITH_FULL_IMAGE:figures/full_fig_p026_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Two figure 8’s can divide the plane into 12 regions [PITH_FULL_IMAGE:figures/full_fig_p026_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Three pentagrams can divide the plane into 77 regions. [PITH_FULL_IMAGE:figures/full_fig_p027_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Two lollipops can intersect in at most 7 points and can divide the [PITH_FULL_IMAGE:figures/full_fig_p027_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: With each pair intersecting in 7 points, three lollipops can divide [PITH_FULL_IMAGE:figures/full_fig_p028_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Three circles can divide the plane into a maximum of 8 regions, [PITH_FULL_IMAGE:figures/full_fig_p029_33.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the classic pancake-cutting problem from Graham, Knuth, and Patashnik by determining the maximum number of pieces a disk can be divided into using a single rigid cut with exotic knife shapes. These include k-armed V shapes, chains of k line segments, long-legged versions of letters A/E/H/L/M/T/W/X, convex polygons, circles, phi, figure-8, pentagram, hexagram, and lollipop (qoppa), plus constrained variants of A/H/L/T/X. Explicit geometric placements achieve the stated maxima for most shapes; for the constrained A and lollipop the authors supply matching lower-bound constructions and upper bounds derived from intersection counting and the Euler characteristic.

Significance. If the constructions and counting arguments hold, the paper supplies a systematic extension of the GKP framework to a wide family of rigid, non-straight knives. Credit is due for the explicit, verifiable placements that realize the claimed maxima in the majority of cases and for the consistent use of elementary combinatorial topology (intersection counts plus Euler characteristic) to obtain tight bounds where exact values are not yet settled. The results are falsifiable by direct construction and therefore add concrete, checkable data to the literature on planar dissections.

minor comments (3)
  1. Abstract: the statement that 'in most cases we are able to determine the maximum number' would be more informative if the explicit formulas or the numerical bounds for the constrained A and lollipop were stated, even briefly.
  2. Introduction: a short tabular recap of the original GKP maxima for the straight, V, and Z knives would help readers immediately compare the new results with the baseline.
  3. Notation: the function p(S) denoting the maximum pieces for shape S is introduced informally; a single displayed definition early in the paper would eliminate repeated explanatory phrases across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of our explicit constructions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives maximum pancake piece counts via explicit geometric constructions that achieve lower bounds through specific knife placements and upper bounds via intersection counting plus Euler characteristic arguments. These steps are self-contained combinatorial geometry that do not reduce to fitted parameters, self-definitions, or load-bearing self-citations; the cited Concrete Mathematics reference is external and the derivations stand on direct verification of configurations for each shape class.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions of plane geometry and rigid-body cuts; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The pancake is modeled as a convex disk in the Euclidean plane.
    Standard modeling choice for pancake-cutting problems.
  • domain assumption Each exotic knife is a rigid, fixed-shape curve or polyline used in one continuous placement.
    Implied by the problem statement and extension of prior work.

pith-pipeline@v0.9.0 · 5472 in / 1230 out tokens · 92200 ms · 2026-05-17T20:08:59.800780+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Two-Graph Refinement of Paulsen's Lollipop Bounds

    math.CO 2026-06 unverdicted novelty 6.0

    By recasting Paulsen's obstructions as two forbidden-subgraph graphs and analyzing near-extremal configurations while keeping the overlap term, the authors close all gaps and compute exact a_L(n) for n≤17 and n=19.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Bollob´ as,Modern Graph Theory, Springer, 1998

    B. Bollob´ as,Modern Graph Theory, Springer, 1998

    work page 1998

  2. [2]

    Comtet,Advanced Combinatorics, Reidel, 1974

    L. Comtet,Advanced Combinatorics, Reidel, 1974

    work page 1974

  3. [3]

    Gardner,New Mathematical Diversions from Scientific American, Simon and Schus- ter, 1966

    M. Gardner,New Mathematical Diversions from Scientific American, Simon and Schus- ter, 1966

    work page 1966

  4. [4]

    R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics, Addison-Wesley, 2nd ed., 1994

    work page 1994

  5. [5]

    Gr¨ unbaum, Venn diagrams and independent families of sets,Math

    B. Gr¨ unbaum, Venn diagrams and independent families of sets,Math. Mag.,48(1975), 12–23

    work page 1975

  6. [6]

    Gr¨ unbaum,Configurations of Points and Lines, Graduate Studies in Math.103, Amer

    B. Gr¨ unbaum,Configurations of Points and Lines, Graduate Studies in Math.103, Amer. Math. Soc., 2009

    work page 2009

  7. [7]

    Hilbert and S

    D. Hilbert and S. Cohn-Vossen,Geometry and the Imagination, Chelsea, NY, 1952; 2nd ed., Springer, 1996. 30

    work page 1952

  8. [8]

    Kinsella,Plane division by Lines AND Circles (Problem, Analysis and Solution), A Maths Puzzle Page, no date, but modified in 2009

    D. Kinsella,Plane division by Lines AND Circles (Problem, Analysis and Solution), A Maths Puzzle Page, no date, but modified in 2009. This document is now appar- ently lost, although an incomplete copy was preserved by the Internet Archive’sWay- back Machineand can be seen here:https://web.archive.org/web/20210414225350 /http://www.90thkilmacudscouts.com/...

    work page arXiv 2009

  9. [9]

    Lakatos,Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge, 1976

    I. Lakatos,Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge, 1976

    work page 1976

  10. [10]

    Levi,Geometrische Konfigurationen, Hirzel, 1929

    F. Levi,Geometrische Konfigurationen, Hirzel, 1929

    work page 1929

  11. [11]

    R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077[math.CO], 2016(https://arxiv.org/abs/1603.00077)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    (2025),The On-Line Encyclopedia of Integer Sequences, https://oeis.org

    OEIS Foundation Inc. (2025),The On-Line Encyclopedia of Integer Sequences, https://oeis.org

    work page 2025

  13. [13]

    Ruskey and M

    F. Ruskey and M. Weston, A survey of Venn diagrams,Electronic J. Combin., Dynamic Surveys5, 2005

    work page 2005

  14. [14]

    A. M. Yaglom and I. M. Yaglom,Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory, Dover Publications, 1987

    work page 1987

  15. [15]

    W. B. Yeats,The Collected Poems of W. B. Yeats, Macmillan, 1956. Keywords: pancake cutting, plane dissection, bent line, zig-zag, configuration, cookie cutter 2010 Mathematics Subject Classification: Primary 05C10, Secondary 05C63, 52C30 31

    work page 1956