Solution. The square marked with the number 1 has a side length of 9. Let the side length of square 2 be $x$. Then the side length of square 3 is $x+9$, the side length of square 4 is $x-9$, the side length of square 5 is $x+18$, the side length of square 6 is $2 x-9$, the side length of square 7 is $2 x+27$, and the side length of square 8 is $3 x-18$.

If the side of square 9 is $y$, then $y+(x+18)+9=(3 x-18)+(x-9), \quad$ from which $y=3 x-54$. Thus, the side of square 10 is $6 x-72$, and the side of square 11 is $9 x-126$.

The top side of the given rectangle is the sum of the sides of squares 10 and 11, i.e., it has a length

$$
(6 x-72)+(9 x-126)=15 x-198
$$

and its bottom side is the sum of the sides of squares 7, 3, 2, and 6, i.e., it has a length

![](https://cdn.mathpix.com/cropped/2024_06_05_d0c212a3407f50323a50g-03.jpg?height=486&width=522&top_left_y=1437&top_left_x=960)

$$
(2 x+27)+(x+9)+x+(2 x-9)=6 x+27
$$

Equating the obtained expressions, we get $x=25$. Therefore, the sides of the squares marked with numbers from 1 to 11 are $9,25,34,16,43,41,77,57,21,78,99$, respectively. Thus, the length of the given rectangle is 177, and the width is 176.
1. The rectangle drawn in the diagram is divided into 11 squares with different side lengths. The smallest square has a side length of 9. What are the dimensions of the rectangle?

![](https://cdn.mathpix.com/cropped/2024_06_05_d0c212a3407f50323a50g-03.jpg?height=276&width=295&top_left_y=910&top_left_x=1184)