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%I #27 Sep 01 2023 11:18:32
%S 1,4,6,5,8,7,8,7
%N Minimal number of polygonal pieces in a dissection of a regular n-gon to an equilateral triangle (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 non-measurable sets.
%D G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.
%D H. Lindgren, Geometric Dissections, Van Nostrand, Princeton, 1964.
%D H. Lindgren (revised by G. N. Frederickson), Recreational Problems in Geometric Dissections and How to Solve Them, Dover, NY, 1972.
%H Stewart T. Coffin, <a href="/A110312/a110312_3.gif">Dudeney's 1902 4-piece dissection of a triangle to a square</a>, from The Puzzling World of Polyhedral Dissections.
%H Stewart T. Coffin, <a href="https://johnrausch.com/PuzzlingWorld/chap01.htm#p5">The Puzzling World of Polyhedral Dissections</a>, Chapter 1. (See section "Geometrical Dissections".)
%H Geometry Junkyard, <a href="http://www.ics.uci.edu/~eppstein/junkyard/dissect.html">Dissection</a>
%H Gavin Theobald, <a href="http://www.gavin-theobald.uk/HTML/Triangle.html">Triangle dissections</a>
%H Vinay Vaishampayan, <a href="/A110312/a110312_3v.jpg">Dudeney's 1902 4-piece dissection of a triangle to a square</a>
%e a(3) = 1 trivially.
%e a(4) <= 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 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.
%e For n >= 5 see the Theobald web site.
%Y Cf. A110312, A110356.
%K nonn
%O 3,2
%A _N. J. A. Sloane_, Sep 11 2005