3,1

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.

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.

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.

Henry Baker, A 5-piece dissection of a hexagon to a square [From HAKMEM]

Henry Baker, Hypertext version of HAKMEM

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

N. J. A. Sloane, Seven Staggering Sequences.

Gavin Theobald, Square dissections

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

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.

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.

a(4) = 1 trivially.

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?

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?

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?

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?

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?

For n >= 10 see the Theobald web site.

Cf. A110000, A110356.

Sequence in context: A115607 A076717 A120422 this_sequence A011242 A008565 A021100

Adjacent sequences: A110309 A110310 A110311 this_sequence A110313 A110314 A110315

nonn

N. J. A. Sloane (njas(AT)research.att.com), Sep 11 2005