Alice has a cardboard sheet measuring $60 \mathrm{~cm}$ by $25 \mathrm{~cm}$. She wants to cut the sheet to build a cube. What is the largest cube she can construct?
Let $a$ be the edge of the cube we want to build. Since the lateral area of the cube is $6a^2$, we must have $6a^2 \leq 25 \times 60$, which means $a^2 \leq 250$ and thus $a < 16$. With $a = 15$, we have $4 = 60 \div 15$ squares with a side of $15 \text{ cm}$ and a remaining rectangle of $60 \text{ cm}$ by $10 \text{ cm}$.

We can cut a rectangle of $60 \text{ cm}$ by $2.5 \text{ cm}$ and the pieces marked with $\circledast$ with dimensions $15 \text{ cm}$ by $7.5 \text{ cm}$. Thus, in the figure, the dotted line indicates a fold and the solid line indicates a cut, and the cardboard pieces marked with $\circledast$ form the lid.

![](https://cdn.mathpix.com/cropped/2024_05_01_2b4d243432ecfddb5212g-5.jpg?height=223&width=537&top_left_y=1162&top_left_x=768)