## Task 4 - 240724

From a square sheet of metal with side length $a$, a cube-shaped box without a top is to be made. For the net to manufacture such a box, two variants are discussed in the image.

Both nets are arranged such that the diagonals of the given square are symmetry axes of the net.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0478.jpg?height=234&width=485&top_left_y=2230&top_left_x=1274)

Determine the size of the waste (in the image black) for both variants in dependence on $a$! If one variant results in less waste, indicate this variant!
Variant 1:

The square area can be divided into exactly 9 congruent squares with side length $\frac{a}{3}$ (Figure a); of these, 4 squares are waste, and the remaining 5 form the net for making the box. Consequently, the waste here is $\frac{4}{9} a^{2}$.

Variant 2:

The square area can be divided into exactly 32 congruent isosceles right triangles with leg length $\frac{a}{4}$ (e.g., as in Figure b); of these, 12 triangles are waste, and the remaining 20 form the net for making the box. The waste is $\frac{3}{8} a^{2}$.

Comparison: Since $4 \cdot 8 > 3 \cdot 9$, it follows that $\frac{4}{9} > \frac{3}{8}$. Therefore, the waste in Variant 2 is smaller than in Variant 1.

Solutions of the 2nd Round 1984 adopted from [5]

### 4.26.3 3rd Round 1984, Class 7