A362939 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a rectangle (conjectured).

2, 1, 4, 3, 5, 4, 7, 4, 9, 5, 10, 7, 10, 9

Offset: 3

Author

N. J. A. Sloane, Aug 31 2023

Keywords

• nonn
• more
• hard

Comments

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).

Examples

			See our paper "On dissecting polygons into rectangles" for illustrations of a(n) for all n <= 16 except n=13 and n=15.
		

Links

• Adam Gsellman, Illustration for a(5) <= 4, a 4-piece dissection of a regular pentagon to a rectangle, May 16 2023.
• Adam Gsellman, Another construction showing that a(5) <= 4, May 16 2023.
• Adam Gsellman, Illustration for r(8) <= 4, a 4-piece dissection of a regular octagon to a rectangle, 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, Another 4-piece dissection of a regular pentagon to a rectangle, showing a(5) <= 4, Jun 08 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, Illustration 12gonA for a(12) <= 5, a 5-piece dissection of a regular dodecagon to a rectangle, May 18 2023.
• N. J. A. Sloane, Illustration 12gonB2 for a(12) <= 5, showing the rearranged pieces.
• N. J. A. Sloane, Illustration 12gonC for a(12) <= 5, showing vertex and edge labels.
• N. J. A. Sloane, Illustration 12gonD for a(12) <= 5, giving proof of correctness.
• 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)
• Gavin Theobald, The Geometric Dissections Database

Crossrefs

Cf. A110312, A362938.