33. A square tiling is a partition of a square into a finite number of rectangles whose sides are parallel to the sides of the square. A tiling is called primitive if it is not a subdivision of a larger tiling. For which $n$ does there exist a primitive tiling of a square into $n$ rectangles?
66.33. Answer. For $n=5$ and for $n \geqslant 7$. In Fig. 32, a, the division into an arbitrary odd number of rectangles greater than 4 is illustrated - an example for $n=9$. In Fig. 32, b, the left half of the square is divided according to Fig. 32, a, while the division of the right half into four rectangles is fixed.
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_55ce716f238acb5de3adg-184.jpg?height=665&width=612&top_left_y=250&top_left_x=859)

Fig. 32