A *squared rectangle* is a rectangle dissected into squares. It
is *perfect* if no two of the squares are the same size, and
*imperfect* otherwise. It is *simple* if it contains
no smaller squared rectangle, other than its component squares; if
it is not simple it is *compound*.
The number of squares a squared rectangle is divided into is called its
*order*.

A squared rectangle might also be a square.
The first perfect squared
square is due to Sprague (*Mathematische Zeitschrift*, 45
(1939), pp. 607-608; a translation of the text of the article
is available in TeX (3K),
DVI (5K),
Postscript (43K), or
PDF (49K))
and was made by piecing squared rectangles together. The perfect
squared square with smallest order, 21,
was found by Duijvestijn and his
computer; it is simple (Simple perfect squared square of lowest order, *J.
Combinatorial Theory, series B*, 25 (1978), pp. 240-243.)

A squared rectangle can be described by its *Bouwkamp code*.
This code is obtained by looking at a list of the horizontal segments found
in the squared rectangle, ordered from top to bottom and starting with the top
edge, and making, for each segment, a parenthesized list of the side-lengths
of the squares immediately below the segment, going from left to right. For
example, the squared square above has squares of lengths 50, 35, and 27
along its top edge, and its code runs:

(50,35,27)(8,19)(15,17,11)(6,24)(29,25,9,2)(7,18)(16)(42)(4,37)(33).

Sprague's squared square has Bouwkamp code

(1885,429,299,312,615,665)(221,78)(65,247)(143)(195,234)(325,39)(286) (565,50)(104,91)(52,182)(375,340)(13,130)(117)(1040)(35,305)(140,270) (705)(575)(696,551,638,2320)(145,319,87)(725)(667,174)(493)(957,522,406) (116,290)(435,203)(29,261)(232).

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David Moews ( dmoews@fastmail.fm )

Last updated 29-IV-2007