A110312 Minimum number of polygonal pieces in a dissection of a regular n-gon into a square (conjectured).

4, 1, 6, 5, 7, 5, 9, 7, 10, 6, 11, 9, 11, 10, 12, 10, 13, 11

Offset: 3

Author

N. J. A. Sloane, Sep 11 2005

Keywords

• nonn
• nice
• more
• hard
• changed

Comments

I do not know which of these values have been proved to be minimal. (Probably only a(4)!).

Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.

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) <= A362939(n) <= a(n).

a(3)=4 is proved in Demaine et al. (2025).

Examples

			a(3) <= 4 because there is a 4-piece dissection of an equilateral triangle into a square, due probably to H. Dudeney, 1902 (or possible C. W. McElroy - see Fredricksen, 1997, pp. 136-137). Surely it is known that this is minimal? See illustrations.
Coffin gives a nice description of this dissection. He notes that the points marked * are the midpoints of their respective edges and that ABC is an equilateral triangle. Suppose the square has side 1, so the triangle has side 2/3^(1/4). Locate B on the square by measuring 1/3^(1/4) from A, after which the rest is obvious. See the Sloane-Vaishampayan paper for another description of this construction, with coordinates.
a(4) = 1 trivially.
a(5) <= 6 since there is a 6-piece dissection of a regular pentagon into a square, due to R. Brodie, 1891 - see Fredricksen, 1995, p. 120. Certainly a(5) >= 5. Is it known that a(5) = 5 is impossible?
a(6) <= 5 since there is a 5-piece dissection of a regular hexagon into a square, due to P. Busschop, 1873 - see Fredricksen, 1995, p. 117. (See illustration.) Is it known that a(6) = 4 is impossible?
a(7) <= 7 since there is a 7-piece dissection of a regular heptagon into a square, due to G. Theobald, 1995 - see Fredricksen, 1995, p. 128. Is it known that a(7) = 6 is impossible?
a(8) <= 5 since there is a 5-piece dissection of a regular octagon into a square, due to G. Bennett, 1926 - see Fredricksen, 1995, p. 150. Is it known that a(8) = 4 is impossible?
a(9) <= 9 since there is a 9-piece dissection of a regular 9-gon into a square, due to G. Theobald, 1995 - see Fredricksen, 1995, p. 132. Is it known that a(9) = 8 is impossible?
For n >= 10 see the Theobald web site.
		

References

• G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.
• H. Lindgren, Geometric Dissections, Van Nostrand, Princeton, 1964.
• H. Lindgren (revised by G. N. Frederickson), Recreational Problems in Geometric Dissections and How to Solve Them, Dover, NY, 1972.
• N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Links

• Henry Baker, A 5-piece dissection of a hexagon to a square [From HAKMEM]
• Henry Baker, Hypertext version of HAKMEM
• Stewart T. Coffin, Dudeney's 1902 4-piece dissection of a triangle to a square, from The Puzzling World of Polyhedral Dissections.
• Stewart T. Coffin, The Puzzling World of Polyhedral Dissections, link to part of Chapter 1. [Broken link?]
• Stewart T. Coffin, The Puzzling World of Polyhedral Dissections [Scan of two pages from Chapter 1 that deal with the triangle-to-square dissection, annotated by _N. J. A. Sloane_, Sep 12 2019]
• Erik D. Demaine, Tonan Kamata, and Ryuhei Uehara, Dudenay's Dissection is Optimal, arXiv, 2025.
• Geometry Junkyard, Dissection
• N. J. A. Sloane, Seven Staggering Sequences.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
• 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.
• N. J. A. Sloane and Vinay A. Vaishampayan, Generalizations of Schöbi's tetrahedral dissection, Discrete and Comput. Geom., 41 (No. 2, 2009), 232-248; arXiv:0710.3857.
• Gavin Theobald, The Geometric Dissections Database
• Gavin Theobald, Illustration showing that a(20) <= 11
• Gavin Theobald, Illustration showing that a(30) <= 13
• Gavin Theobald, Illustration showing that a(50) <= 20
• Gavin Theobald, Illustration showing that a(100) <= 33
• Vinay Vaishampayan, Dudeney's 1902 4-piece dissection of a triangle to a square
• Eric Weisstein's World of Mathematics, Dissection

Crossrefs

Cf. A110000, A110356, A362938, A362939.

Extensions

New values for a(n), n = 14, 16, 18, 19, 20 from Gavin Theobald's Geometric Dissections Database. - N. J. A. Sloane, Jun 13 2023. In fact this Database gives values out to n = 30 which may be optimal or close to optimal.