Katka cut out 20 rectangles from paper. The first one had dimensions $1 \mathrm{~cm} \times 2 \mathrm{~cm}$, the second $2 \mathrm{~cm} \times 3 \mathrm{~cm}$, the third $3 \mathrm{~cm} \times 4 \mathrm{~cm}$, and so on. Then she started folding them together with the same length sides, ensuring they did not overlap and no rectangle touched more than two other rectangles.

a) Draw the polygon that Katka formed from the first six of her rectangles.

b) Determine the perimeter of Katka's polygon formed from all the cut-out rectangles.
a) The solution is in Fig. 1. (2 points)

![](https://cdn.mathpix.com/cropped/2024_04_17_3fba1832e0e68708cfa9g-4.jpg?height=721&width=550&top_left_y=141&top_left_x=771)

Fig. 1

b) For the perimeter of Katčina's polygon, the following holds:

$$
\begin{gathered}
2 \cdot 1 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4 + \cdots + 2 \cdot 21 = \\
\quad = 2 \cdot (1 + 2 + 3 + \cdots + 21) = 462
\end{gathered}
$$

The perimeter of Katčina's polygon is $462 \mathrm{~cm}$.

(Full solution 6 points)