Sharygin I.F.

A rectangle is composed of six squares (see the right figure). Find the side of the largest square if the side of the smallest one is 1.

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-08.jpg?height=346&width=417&top_left_y=2354&top_left_x=838)

The side of the largest square is equal to the sum of the sides of two squares: the next one in the clockwise direction and the smallest one.

#
Notice that the side of the largest square is equal to the sum of the sides of two squares: the one following it clockwise and the smallest one. Denoting the side of the largest square as \( x \), we can sequentially express the sides of the other squares: \( x-1, x-2, x-3, x-3 \) (see the figure). Now notice that the length of the upper side of the rectangle is \( x+(x-1) \), and the length of the lower side is \( (x-2)+(x-3)+(x-3) \). But the opposite sides of the rectangle are equal. We get the equation

\[
x+(x-1)=(x-2)+(x-3)+(x-3)
\]

From this, \( 2x-1=3x-8 \) and, therefore, \( x=7 \).

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-09.jpg?height=343&width=412&top_left_y=592&top_left_x=823)

## Answer

7.00