

A110312


Minimal number of polygonal pieces in a dissection of a regular ngon to a square (conjectured).


5



4, 1, 6, 5, 7, 5, 9, 7, 10, 6, 11, 10, 11, 11, 12, 12, 15, 14
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OFFSET

3,1


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 nonmeasurable sets.


REFERENCES

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. 93110.


LINKS

Table of n, a(n) for n=3..20.
Henry Baker, A 5piece dissection of a hexagon to a square [From HAKMEM]
Henry Baker, Hypertext version of HAKMEM
Stewart T. Coffin, Dudeney's 1902 4piece 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 triangletosquare dissection, annotated by N. J. A. Sloane, Sep 12 2019]
Geometry Junkyard, Dissection
N. J. A. Sloane, Seven Staggering Sequences.
N. J. A. Sloane and Vinay A. Vaishampayan, Generalizations of SchÃ¶bi's tetrahedral dissection, Discrete and Comput. Geom., 41 (No. 2, 2009), 232248; arXiv:0710.3857.
Gavin Theobald, Index to dissections
Vinay Vaishampayan, Dudeney's 1902 4piece dissection of a triangle to a square


EXAMPLE

a(3) <= 4 because there is a 4piece dissection of an equilateral triangle into a square, due probably to H. Dudeney, 1902 (or possible C. W. McElroy  see Fredricksen, 1997, pp. 136137). 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 SloaneVaishampayan paper for another description of this construction, with coordinates.
a(4) = 1 trivially.
a(5) <= 6 since there is a 6piece 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 5piece 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 7piece 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 5piece 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 9piece dissection of a regular 9gon 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.


CROSSREFS

Cf. A110000, A110356.
Sequence in context: A324056 A120422 A342033 * A011242 A008565 A205325
Adjacent sequences: A110309 A110310 A110311 * A110313 A110314 A110315


KEYWORD

nonn,nice,more,hard


AUTHOR

N. J. A. Sloane, Sep 11 2005


STATUS

approved



