Perfect rectangle

A perfect rectangle is a rectangle that can be divided into squares of different sizes. If a perfect rectangle is specifically a square, it is analogously called a perfect square.

A rectangle that is not perfect is also called an imperfect rectangle.¹

Many mathematicians have been involved in the discovery of perfect rectangles and perfect squares.

Below is a selection of important discoveries in this field.

• 1925: Zbigniew Moroń decomposed a perfect smallest possible 33x32 rectangle into nine squares.
• 1939: The German mathematician Roland Sprague published a large perfect square with 55 squares.
• 1978: A. J. W. Duijvestijn dissected a perfect square into 21 squares with a total side length of 112, where 21 is the lowest possible number of subsquares of perfect squares.²

Among the numerous perfect rectangles and squares, the following selected examples are intended to highlight some special features.³

(The numbers in the squares indicate their respective side lengths.)

• 
  Smallest possible perfect rectangle (9 squares, Moroń)
• 
  Perfect rectangle with many squares (22 squares)
• 
  Almost symmetrical perfect rectangle (12 squares)
• 
  Elongated perfect rectangle (17 squares)
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  Perfect rectangle with a remarkably large side length of 7 for the smallest sub-square (10 squares)
• 
  Smallest possible simple perfect square (21 squares, Duijvestijn)