%I
%S 4,1,6,5,7,5,9,7,10,6,11,10,11,11,12,12,15,14
%N Minimal number of polygonal pieces in a dissection of a regular ngon to a square (conjectured).
%C I do not know which of these values have been proved to be minimal.
%C Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with nonmeasurable sets.
%D G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.
%D H. Lundgren, Geometric Dissections, Van Nostrand, Princeton, 1964.
%D H. Lundgren (revised by G. N. Frederickson), Recreational Problems in Geometric Dissections and How to Solve Them, Dover, NY, 1972.
%D 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.
%H Henry Baker, <a href="/A110312/a110312_6.gif">A 5piece dissection of a hexagon to a square</a> [From HAKMEM]
%H Henry Baker, <a href="http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html">Hypertext version of HAKMEM</a>
%H Stewart T. Coffin, <a href="/A110312/a110312_3.gif">Dudeney's 1902 4piece dissection of a triangle to a square</a>, from The Puzzling World of Polyhedral Dissections.
%H Stewart T. Coffin, <a href="http://www.johnrausch.com/PuzzlingWorld/chap01e.htm">The Puzzling World of Polyhedral Dissections</a>, link to part of Chapter 1.
%H Geometry Junkyard, <a href="http://www.ics.uci.edu/~eppstein/junkyard/dissect.html">Dissection</a>
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%H N. J. A. Sloane and Vinay A. Vaishampayan, <a href="http://arxiv.org/abs/0710.3857">Generalizations of SchÃ¶bi's tetrahedral dissection</a>, Discrete and Comput. Geom., 41 (No. 2, 2009), 232248; arXiv:0710.3857.
%H Gavin Theobald, <a href="http://home.btconnect.com/GavinTheobald/HTML/Square.html">Square dissections</a>
%H Vinay Vaishampayan, <a href="/A110312/a110312_3v.jpg">Dudeney's 1902 4piece dissection of a triangle to a square</a>
%e 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.
%e 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.
%e a(4) = 1 trivially.
%e 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?
%e 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?
%e 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?
%e 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?
%e 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?
%e For n >= 10 see the Theobald web site.
%Y Cf. A110000, A110356.
%K nonn,nice,more
%O 3,1
%A _N. J. A. Sloane_, Sep 11 2005
