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Three-Dimensional Guillotine Partitions

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A guillotine slice of a rectangle in the plane is a line parallel to an edge of from edge to edge, thus forming sub-rectangles. A guillotine partition is the set of sub-rectangles formed by applying guillotine slices to previously formed sub-rectangles.

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Suppose that points are inside a rectangle with no two points on a line parallel to an edge of . The Schröder numbers can be interpreted as the number of guillotine partitions of such that each point lies on exactly one slice.

An extension applies the same idea in 3D. This Demonstration shows the different ways to partition a cuboid using different numbers of guillotine slices.

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Contributed by: Robert Dickau (February 2024)
Open content licensed under CC BY-NC-SA

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Initial counts [1]: 1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, …

Reference

[1] R. Stephan. "The On-Line Encyclopedia of Integer Sequences." (Jan 18, 2024) oeis.org/A103210.

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