Problem 4. The cake is packed in a box with a square base. The height of the box is half the side of this square. A ribbon of length 156 cm can be used to tie the box and make a bow on top (as shown in the left image). To tie it with the same bow on the side (as shown in the right image), a ribbon of length 178 cm is needed. Find the dimensions of the box.

(I. V. Raskina)

![](https://cdn.mathpix.com/cropped/2024_05_06_18371eb476ec0d24f876g-2.jpg?height=236&width=331&top_left_y=1339&top_left_x=367)
Answer: 22 cm $\times$ 22 cm $\times$ 11 cm.

Solution: In the first method of tying, the ribbon encircles the box twice along the length, twice along the width, and four times along the height, meaning its length is equal to six sides of the base plus the bow. In the second method of tying, the ribbon encircles the box twice along the length, four times along the width, and twice along the height, meaning its length is equal to seven sides of the base plus the bow. Now it is clear that the difference in the lengths of the ribbons, $178-156=22$ cm, is exactly equal to the side of the base of the box. Therefore, the dimensions of the box are 22 cm $\times$ 22 cm $\times$ 11 cm. Let's check: in the first case, 132 cm is used for wrapping the box ($22 \times 6=132$ cm), and 24 cm is used for the bow. In the second case, 154 cm is used for wrapping the box ($22 \times 7=154$ cm), and the same 24 cm is used for the bow.