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A362939 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a rectangle (conjectured). 3
2, 1, 4, 3, 5, 4, 7, 4, 9, 5, 10, 7, 10, 9 (list; graph; refs; listen; history; text; internal format)
The dimensions of the rectangle can be anything you want, as long as it is a rectangle.
Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.
Apart from changing "square" to "rectangle", the rules are the same as in A110312.
I do not know which of these values have been proved to be minimal. Probably only a(3)=2 and a(4)=1.
The three related sequences A110312, A362938, A362939 are exceptions to the usual OEIS policy of requiring that all terms in sequences must be known exactly. These sequences are included because of their importance and in the hope that someone will establish the truth of some of the conjectured values.
The definitions imply that A362938(n) <= a(n) <= A110312(n).
Adam Gsellman, Another construction showing that a(5) <= 4, May 16 2023.
Adam Gsellman, First 4-piece dissection of a regular octagon to a rectangle, showing details of the dissection [Needs a very wide window to see full illustration]
Adam Gsellman, Another construction showing that a(8) <= 4, May 16 2023.
N. J. A. Sloane, Illustrating a(6) <= 3: three-piece dissection of regular hexagon to a rectangle. (Surely there is a proof that this cannot be done with only two pieces?)
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023.
Gavin Theobald, A 7-piece dissection of a 9-gon to a rectangle (See our paper "On dissecting polygons into rectangles" for details of this dissection)
Gavin Theobald, A 4-piece dissection of a 10-gon to a rectangle (See our paper "On dissecting polygons into rectangles" for details of this dissection)
See our paper "On dissecting polygons into rectangles" for illustrations of a(n) for all n <= 16 except n=13 and n=15.
Sequence in context: A244373 A107640 A030065 * A328676 A269595 A055176
N. J. A. Sloane, Aug 31 2023

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Last modified April 15 18:28 EDT 2024. Contains 371696 sequences. (Running on oeis4.)