## Task 2 - 130512

Karl is supposed to divide a cube with an edge length of $4 \mathrm{~cm}$ made of modeling clay into smaller cubes, each with an edge length of $1 \mathrm{~cm}$, without deforming them.

a) Determine the number of cubes (of the required type) that will result from this!

b) Determine how many cuts Karl has to make in total if a cut must never divide more than one of the previously existing bodies, i.e., if cutting "in bulk" is not allowed!
a) The volume of the large cube is $4 \mathrm{~cm} \cdot 4 \mathrm{~cm} \cdot 4 \mathrm{~cm}=64 \mathrm{~cm}^{3}$, and the volume of a small cube is $1 \mathrm{~cm}^{3}$. This results in 64 small cubes.
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0061.jpg?height=272&width=438&top_left_y=2388&top_left_x=313)

b) First, the cube is cut along the blue lines, resulting in 4 cuboids with dimensions $1 \mathrm{~cm} \times 4 \mathrm{~cm} \times 4 \mathrm{~cm}$ after three cuts.

Then, each of these cuboids is cut along the red lines with 3 cuts each, totaling 12 cuts, resulting in 16 cuboids with dimensions $1 \mathrm{~cm} \times 4 \mathrm{~cm} \times 1 \mathrm{~cm}$.

Finally, each of these 16 cuboids is cut into 4 cubes of the desired size with 3 cuts each, totaling 48 cuts.

In total, this requires $3+12+48=63$ cuts.