A362939 -id:A362939 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 11 2005

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).

			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.

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.

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

Cf. A110000, A110356, A362938, A362939.

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.

A362938 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, i.e. a polygonal tile which tiles the plane (conjectured).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6

Offset: 3

N. J. A. Sloane, Aug 29 2023

I do not know which of these values have been proved to be minimal. Probably only a(n) for n = 3, 4, 5, 6, 8, and 10.
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.
More formally, a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, that is, a prototile for a monohedral tiling of the plane.
Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.
The definitions imply that a(n) <= A362939(n) <= A110312(n).
On Aug 31 2023 Gavin Theobald sent me two different solutions for a(9) = 3 and one solution for a(11) = 4 (see links). He reports that he found these dissections in the 1990's. In his email and in a later email (Sep 04 2023) he also gives the values a(13) = 5, a(14) = 3, a(15) = 5 (with one piece turned over), a(16) = a(18) = 4, a(20) = 5. He conjectures that a(2t) = floor(t/2) for all t >= 2.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. [The sequence is defined in Section 2.6, pp. 91-95.]
Harry Lindgren, Geometric Dissections, Van Nostrand, Princeton, NJ, 1964. Plates B6, B7, B8, B9, B10, and B12 illustrate n = 6, 7, 8, 9, 10, and 12, respectively. One would expect that plates B11 and B13 would refer to n = 11 and 13, but in fact they appear to give alternative solutions for n = 10 and 12, respectively.
Harry Lindgren, Recreational Problems in Geometric Dissections and How to Solve Them, Revised and enlarged by Greg Frederickson, Dover Publications, NY, 1972.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. Annotated copy of Fig. 2.6.1, illustrating a(5), a(7), a(8), a(10), and a(12). (Their value for a(9) is out-of-date.)
Harry Lindgren, Geometric Dissections, Annotated scan of Plate B12, showing tiling of plane arising from the conjectured a(12) = 3.
N. J. A. Sloane, Illustration for a(5) = 2, after Grunbaum and Shephard, Fig. 2.6.1. Left: The 2-piece dissection of the pentagon. Right: Shows how the hexagonal tile made from those two pieces tiles the plane.
N. J. A. Sloane, An illustration for a(12) = 3, based on Lindgren's plate B12.
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, Illustration for a(5)
Gavin Theobald, Illustration for a(7)
Gavin Theobald, Another illustration for a(7) <= 3, after Lindgren.
Gavin Theobald, Illustration for a(8)
Gavin Theobald, A 9-piece dissection of a 9-gon into a monotile, with the monotile outlined in red, illustrating a(9) = 3.
Gavin Theobald, A 9-piece dissection of a 9-gon into a monotile, showing how the monotile is obtained from the 9-gon.
Gavin Theobald, An alternative illustration for a(9) = 3.
Gavin Theobald, Yet another illustration for a(9)
Gavin Theobald, Illustration for a(10)
Gavin Theobald, Illustration for a(11)
Gavin Theobald, Illustration for a(12)
Gavin Theobald, Illustration for a(13) <= 4
Gavin Theobald, Illustration for a(14)
Gavin Theobald, Illustration for a(15) (5 pieces)
Gavin Theobald, Another illustration for a(15) <= 5
Gavin Theobald, Illustration for a(17) (The piece marked X must be turned over)
Gavin Theobald, Illustration for a(19) <= 7
Gavin Theobald, The Geometric Dissections Database

Cf. A110312, A362939.

a(9) = 3, a(11) = 4, a(13) = 5, a(14) = 3, a(16) = 4 from Gavin Theobald, Aug 31 2023 - Sep 11 2023.

Updated with many further illustrations from Gavin Theobald. - N. J. A. Sloane, Sep 19 2023

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

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