## Task 4 - 170714

Little Uwe has cube-shaped, white-painted building blocks with an edge length of $2 \mathrm{~cm}$ and cube-shaped, red-painted building blocks with an edge length of $3 \mathrm{~cm}$. He built a larger, composite cube using only these two types of blocks.

The four vertical outer walls consisted of red building blocks, and the rest of the cube body consisted of white building blocks from bottom to top.

Determine the number of white and red building blocks used, assuming that Uwe had no more than 60 building blocks of each type available!
The height of the cube body built by Uwe can be composed of a number $r$ of red stones as well as a number $w$ of white stones. It is also equal to the width of the cube body, which is made up of the width of 2 red stones and a number $v (\geq 1)$ of white stones. Therefore, its measurement (in cm) is the number

$$
3 r=2 w=2 v+2 \cdot 3
$$

Here, $(2 v+2 \cdot 3)$ and consequently $v$ are divisible by 3. Thus, there exists a natural number $n \geq 1$ such that $v=3 n$, and from (1) it follows that $w=3 n+3$ and $r=2 n+2$.

All the white stones used form a cuboid. This consists of $w$ layers; each of them contains $v$ rows of $v$ stones each. Therefore, exactly $W=w \cdot v^2$ white stones were used.

If $n \geq 2$, then $v \geq 6, w \geq 9$, and thus $W \geq 9 \cdot 36 > 60$, which contradicts the problem statement. Therefore, $n=1, v=3, w=6, r=4$, and thus exactly $W=6 \cdot 9=54$ white stones are used. All the red stones used are arranged in $r$ layers; each of them consists of four rows of $(r-1)$ stones each. Therefore, exactly $R=r \cdot 4(r-1)=4 \cdot 4 \cdot 3=48$ red stones were used.

Solutions of the I. Round 1977 taken from [5]

### 4.19.2 II. Round 1977, Class 7