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A110312 Minimal number of polygonal pieces in a dissection of a regular n-gon to a square (conjectured). 5
4, 1, 6, 5, 7, 5, 9, 7, 10, 6, 11, 10, 11, 11, 12, 12, 15, 14 (list; graph; refs; listen; history; text; internal format)



I do not know which of these values have been proved to be minimal.

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


G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.

H. Lundgren, Geometric Dissections, Van Nostrand, Princeton, 1964.

H. Lundgren (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.


Table of n, a(n) for n=3..20.

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.

Geometry Junkyard, Dissection

N. J. A. Sloane, Seven Staggering Sequences.

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, Square dissections

Vinay Vaishampayan, Dudeney's 1902 4-piece dissection of a triangle to a square


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.


Cf. A110000, A110356.

Sequence in context: A200360 A228451 A120422 * A011242 A008565 A205325

Adjacent sequences:  A110309 A110310 A110311 * A110313 A110314 A110315




N. J. A. Sloane, Sep 11 2005



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Last modified August 17 03:29 EDT 2018. Contains 313810 sequences. (Running on oeis4.)