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A110000 Minimal number of polygonal pieces in a dissection of a regular n-gon to an equilateral triangle (conjectured). 2
1, 4, 6, 5, 8, 7, 8, 7 (list; graph; refs; listen; history; text; internal format)



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.


Mohammad K. Azarian, A Trigonometric Characterization of  Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96.  Solution published in Vol. 32, No. 1, Winter 1998, pp. 84-85.

Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337.  Solution published in Vol. 29, No. 3, Fall 1995, pp. 324-325.

G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.

H. Lundgren, Geometric Dissections, Van Nostrand, Princeton, 1964.

H. Lundgren (revised by G. N. Frederickson), Recreational Problems in Geometric Dissections and How to Solve Them, Dover, NY, 1972.


Table of n, a(n) for n=3..10.

Stewart T. Coffin, Dudeney's 1902 4-piece dissection of a triangle to a square, from The Puzzling World of Polyhedral Dissections.

Stewart T. Coffin, The Puzzling World of Polyhedral Dissections, link to part of Chapter 1.

Geometry Junkyard, Dissection

Gavin Theobald, Triangle dissections

Vinay Vaishampayan, Dudeney's 1902 4-piece dissection of a triangle to a square


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.


Cf. A110312, A110356.

Sequence in context: A140243 A199289 A114602 * A073922 A201945 A258827

Adjacent sequences:  A109997 A109998 A109999 * A110001 A110002 A110003




N. J. A. Sloane, Sep 11 2005



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