A189243 -id:A189243 - cp's OEIS frontend

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

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

"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

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

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

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.

Cf. A189243, A219861, A359146.

A219861 Number of ways to dissect a nonsquare rectangle into n rectangles of equal area up to symmetry.

Original entry on oeis.org

1, 2, 4, 11, 35, 130, 562, 2685, 13901, 76046

Offset: 1

Geoffrey H. Morley, Nov 29 2012

			There are 4 ways (up to symmetry) to form a nonsquare rectangle from 3 rectangles with the same area:
+-----+ +-+-+-+ +-----+ +-+---+
|     | | | | | |     | | |   |
+-----+ | | | | +--+--+ | |   |
|     | | | | | |  |  | | +---+
+-----+ | | | | |  |  | | |   |
|     | | | | | |  |  | | |   |
+-----+ +-+-+-+ +--+--+ +-+---+
So a(3)=4.
The eleven solutions for n=4 can be seen as a subset of the illustration of A189243(4) = 21 in that entry. - _N. J. A. Sloane_, Dec 05 2012

Cf. A108066, A189243.

a(7)-a(10) from Geoffrey H. Morley, Dec 16 2012

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs