polyrectangle

A polyrectangle is a type of polyomino, which is a plane geometric figure formed by joining one or more equal squares edge to edge. A polyrectangle, specifically, is a polyomino that can be arranged to form a rectangle. The number of squares in a polyomino is called its order. Therefore, a polyrectangle of order n is a polyomino composed of n squares that can tile a rectangle.

The simplest polyrectangles are the monomino (1 square) and the domino (2 squares), which can form 1x1 and 1x2 rectangles respectively. All polyominoes of order 3 (trominoes) and 4 (tetrominoes) can form rectangles. For example, the straight tetromino can form a 1x4 or 2x2 rectangle. However, not all polyominoes can form rectangles. For instance, some free pentominoes, which are polyominoes of order 5, cannot tile any rectangle.

The condition for a polyomino to be a polyrectangle is that it must be able to tile at least one rectangle. Determining whether a given polyomino is a polyrectangle can be a complex problem. Some polyrectangles can tile multiple different rectangles. For example, the 2x2 square tetromino is a polyrectangle that can form a 2x2 rectangle. The study of polyrectangles is related to tiling problems and combinatorial geometry.