A108066 Number of distinct ways to dissect a square into n rectangles of equal area.

1, 1, 2, 6, 18, 65, 281, 1343, 6953, 38023

Offset: 1

Author

Hans Riesebos (hans.riesebos(AT)wanadoo.nl) and Herman Beeksma, Jun 03 2005

Keywords

• hard
• more
• nonn
• nice

Comments

"Distinct" here means that dissections differing only by a rotation and/or reflection are not counted as different (see A189243).

The first time the pieces can be made to all have different shapes (but the same area) is at n=7 - see Descartes (1971) and the illustration; also Wells, Weisstein. - N. J. A. Sloane, Dec 05 2012

Examples

			There are six ways to dissect a square into four rectangles of equal area, so a(4)=6:
+-+-----+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+
| |     | | | |   | | |     | | | | | | | |   | | |   |   |
| |     | | | |   | | |     | | | | | | | |   | | |   |   |
| +--+--+ | | |   | | +-----+ | | | | | | |   | | |   |   |
| |  |  | | | +---+ | |     | | | | | | | +---+ | +---+---+
| |  |  | | | |   | | |_____| | | | | | | |   | | |   |   |
| |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
| |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
+-+--+--+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+
		

References

• David Wells, Penguin Dictionary of Curious and Interesting Geometry, 1991, pp. 15-16.

Links

• Blanche Descartes, Divisions of a square into rectangles, Eureka, No. 34 (1971), 31-35.
• R. Häggkvist, P.-O. Lindberg, and B. Lindström, Dissecting a square into rectangles of equal area, Discr. Math. 47 (1983), 321-323.
• Johan Nilsson, Illustrations for terms up to a(6)..
• N. J. A. Sloane, Blanche Descartes's dissection into 7 pieces with equal areas but different shapes.
• Eric Weisstein, Blanche's Dissection.

Crossrefs

Cf. A189243, A219861, A359146.