%I A110312
%S A110312 4,1,6,5,7,5,9,7,10,6,11,10,11,11,12,12,15,14
%N A110312 Minimal number of polygonal pieces in a dissection of a regular n-gon 
               to a square (conjectured).
%C A110312 I do not know which of these values have been proved to be minimal.
%C A110312 Turning over is allowed. The pieces must be bounded by simple curves 
               to avoid difficulties with non-measurable sets.
%D A110312 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.
%D A110312 G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.
%D A110312 H. Lundgren, Geometric Dissections, Van Nostrand, Princeton, 1964.
%D A110312 H. Lundgren (revised by G. N. Frederickson), Recreational Problems in 
               Geometric Dissections and How to Solve Them, Dover, NY, 1972.
%H A110312 Henry Baker, <a href="a110312_6.gif">A 5-piece dissection of a hexagon 
               to a square</a> [From HAKMEM]
%H A110312 Henry Baker, <a href="http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html">
               Hypertext version of HAKMEM</a>
%H A110312 Stewart T. Coffin, <a href="a110312_3.gif">Dudeney's 1902 4-piece dissection 
               of a triangle to a square</a>, from The Puzzling World of Polyhedral 
               Dissections.
%H A110312 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 A110312 Geometry Junkyard, <a href="http://www.ics.uci.edu/~eppstein/junkyard/
               dissect.html">Dissection</a>
%H A110312 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">
               Seven Staggering Sequences</a>.
%H A110312 Gavin Theobald, <a href="http://home.btconnect.com/GavinTheobald/HTML/
               Square.html">Square dissections</a>
%H A110312 Vinay Vaishampayan, <a href="a110312_3v.jpg">Dudeney's 1902 4-piece dissection 
               of a triangle to a square</a>
%e A110312 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.
%e A110312 Coffin gives a nice description of this dissection. He notes that the 
               points marked * are the mid-points 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.
%e A110312 a(4) = 1 trivially.
%e A110312 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?
%e A110312 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?
%e A110312 a(7) <= 7 since there is a 7-piece dissection of a regular 7-gon into 
               a square, due to G. Theobald, 1995 - see Fredricksen, 1995, p. 128). 
               Is it known that a(7) = 6 is impossible?
%e A110312 a(8) <= 5 since there is a 5-piece dissection of a regular 8-gon into 
               a square, due to G. Bennett, 1926 - see Fredricksen, 1995, p. 150). 
               Is it known that a(8) = 4 is impossible?
%e A110312 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?
%e A110312 For n >= 10 see the Theobald web site.
%Y A110312 Cf. A110000, A110356.
%Y A110312 Sequence in context: A115607 A076717 A120422 this_sequence A011242 A008565 
               A021100
%Y A110312 Adjacent sequences: A110309 A110310 A110311 this_sequence A110313 A110314 
               A110315
%K A110312 nonn
%O A110312 3,1
%A A110312 N. J. A. Sloane (njas(AT)research.att.com), Sep 11 2005