An eccentric, if you will, twist: maximizing the number of pieces when cutting an infinite pancake.
Two mathematicians recovered a classic puzzles combinatorial geometry with an eccentric approach: maximizing the number of “pieces” obtained when cutting an infinite pancake.
It’s a new approach to the infinite pancake problemas described by .
Neil JA Sloane, founder of the On-Line Encyclopedia of Integer Sequences (OEIS), and David OH Cutler, a graduate student at Tufts University, set out to undertake this task not just with a straight knife, but with “knives” of unusual shapessome with infinitely long arms.
The original problem, known as , asks how many regions can be obtained with a certain number of straight cuts in a plane.
In this new approach, the plan is a pancake that it extends endlessly in all directions and the “cuts” can be curves and composite figures.
The base strategy maintains a intuitive principle: each new cut should ideally intersect as many previous cuts as possible to increase the number of regions.
A difficulty arises when knives stop being simple lines and start to include “legs”, “arms” and infinite segments that need to be positioned with precision to produce maximum divisions.
Among the instruments proposed is a “lollipop”-shaped knife, with an infinite handle, and a set of letter-shaped knives, highlights the .
The most demanding, according to the authors, was the “embarrassed”: a capital A with strict geometric requirements, including a fixed horizontal bar that, together with the top vertex, defines an isosceles triangle.
With a single “constraint”, the pancake is divided into three regions; with two, in an optimal configuration, you reach 13 regions. From there, the number of pieces grows quickly: computational results suggest 30 regions with three knives, 53 with four and 83 with five — generating the sequence 1, 3, 13, 30, 53, 83… (also counting the case of zero cuts, in which the pancake remains whole).
To explore settings possible, Cutler developed, with the help of colleagues, a program that starts with random arrangements of the “knives” and adjusts small variations (“jiggling and wiggling”) to look for maximums.
A central mathematical ingredient is the relationship of Euler for polyhedra, here used as a counting formula in flat configurations: the number of regions can be derived from the number of vertices and edges created by the intersections of the cuts.
The study also covers other knives: “hatpins” (semi-infinite), a V and a “three-armed V” nicknamed “Wu” (reference to a Unicode character), as well as a shape called “nunchucks”.
Experimental mathematics, as Zeilberger emphasizes, does not need a traditional laboratory: the computer functions as a test bench.
One of the most curious results comes from a “A” elongated and less restricted (with tilt bar): three of these A’s produce 34 regions — the same total obtained with three “Wu” and three “nunchucks”.
That “triple coincidence” corresponds to the same sequence registered in the OEIS (1, 3, 14, 34, 63, 101…) and ended up motivating a theorem in the work itself.
After A, the authors extend the exploration to other elongated letters, such as E, H, L, M, T, W and X, suggesting that the geometry of unlikely utensils can continue to generate new links between designs, algorithms and entire sequences.
