

A110312


Minimum number of polygonal pieces in a dissection of a regular ngon into a square (conjectured).


7



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


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. 93110.


LINKS

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.)


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



KEYWORD

nonn,nice,more,hard


AUTHOR



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.


STATUS

approved



