## Task 3 - 300723

a) A cube of wood, colored on all its surfaces, is to be cut into smaller identical cubes. It is required that at least 40 of these smaller cubes are completely uncolored.

Determine the smallest possible number of smaller cubes into which the entire wooden cube can be divided to meet this requirement!

b) A rectangular prism (without any hollow spaces inside) is to be constructed from 40 such uncolored smaller cubes; all 40 of these smaller cubes must be used.

Determine the volume of this rectangular prism, given that the original wooden cube had a volume of $27 \mathrm{dm}^{3}$.
a) A cube can be divided into smaller, equally sized sub-cubes by choosing a natural number \( n > 1 \), dividing three edges emanating from a corner into \( n \) equal parts, and laying planes through the division points that are parallel to the boundary surfaces of the cube.

This divides the cube into \( n \) layers, each layer into \( n \cdot n \) sub-cubes. The total number \( A \) of resulting sub-cubes is thus \( A = n^3 \).

The completely uncolored sub-cubes form a smaller cube contained within the larger cube. For the number of its sub-cubes to be \( A = 40 \), \( n \geq 4 \) must hold. For the entire wooden cube, this \( n \) is replaced by \( n + 2 \).

The smallest possible number of sub-cubes that meets the given requirement is thus the number \( A = 216 \) corresponding to \( n = 6 \).

b) Since \( 3^3 = 27 \), the edge length of the original wooden cube is \( 3 \, \text{dm} \). From a) it follows that the edge length of one of the 216 sub-cubes is \( 3 \, \text{dm} : 6 = 0.5 \, \text{dm} \). The cuboid consists of 40 cubes of this edge length.

Its volume is therefore \( 40 \cdot (0.5 \, \text{dm})^3 = 5 \, \text{dm}^3 \) (regardless of how this cuboid is composed of the cubes).