Journal of Integer Sequences, Vol. 29 (2026), Article 26.4.5

Cutting a Pancake with an Exotic Knife


David O. H. Cutler
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109
USA

Jonas Karlsson
Boden, 961 42
Sweden

Neil J. A. Sloane
The OEIS Foundation Inc.
Highland Park, NJ 08904
USA
and
Visiting Scholar, Math. Dept.
Rutgers University
Piscataway, NJ 08854
USA

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 ${\sf Z}$. 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 ${\sf A}$, ${\sf E}$, ${\sf H}$, ${\sf L}$, ${\sf M}$, ${\sf T}$, ${\sf W}$, or ${\sf X}$, 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 ${\sf A}$, ${\sf H}$, ${\sf L}$, ${\sf T}$, and ${\sf X}$. In most cases we are able to determine the maximum number of pieces, although for the constrained ${\sf A}$ and the lollipop we can only give bounds.


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(Concerned with sequences A000124 A000125 A000326 A008865 A014206 A034856 A046127 A051890 A054554 A058331 A069894 A077588 A077591 A080856 A084849 A090338 A117625 A125201 A130883 A140063 A140064 A143689 A152948 A241600 A250001 A272906 A288554 A383464 A383465 A383466 A386477 A386478 A386479 A386480 A386481 A386485 A386486 A387525 A389608 A389614 A389624 A393441 A393442 A393448 A397182.)


Received December 20 2025; revised versions received April 19 2026; July 16 2026. Published in Journal of Integer Sequences, July 28 2026.


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