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A110000
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Minimal number of polygonal pieces in a dissection of a regular n-gon to an equilateral triangle (conjectured).
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2
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OFFSET
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3,2
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COMMENTS
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I do not know which of these values have been proved to be minimal.
Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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a(3) = 1 trivially.
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.
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.
For n >= 5 see the Theobald web site.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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