## Task 4 - 330814

Ria builds a pyramid-like body from cubes with an edge length of $2 \mathrm{~cm}$. It consists of layers that each completely cover a square area. The illustration shows the principle of its construction. The total height of this body is $10 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0825.jpg?height=471&width=734&top_left_y=458&top_left_x=661)

The "visible surface" of a body composed of cubes is understood to be the total of all partial surfaces of the body that are visible from above or from the sides. This "visible surface" therefore does not include surfaces that, like the base of Ria's body, are only accessible from below.

a) How many cubes does this body consist of?

b) Ria paints the "visible surface" of this body. In how many of the cubes are then six, five, four, three, two, one, or no faces painted?

c) Beate removes a number of the partial cubes that have at least one painted face. She chooses these partial cubes such that the remaining body has the same "visible surface" as the original body. Under this condition, Beate chooses the number of partial cubes to be removed as large as possible. How large is this number?

d) On the body left by Beate, Ria paints its "visible surface" again. Then Beate removes, under the same conditions as in c), the largest possible number of partial cubes with at least one colored face. How large is this number?

Hint: For b), c), d), only descriptions (including sketches if necessary) are required, but no justifications.
a) Due to the total height of $10 \, \text{cm}$ and the edge length of $2 \, \text{cm}$ of the cubes, the body consists of five layers. The sides of the square faces covered by the layers are formed by $1, 3, 5, 7, 9$ cubes, respectively.

Thus, the body consists of $1 + 9 + 25 + 49 + 81 = 165$ cubes.

b) In no cube is the number of colored faces 6, 4, or 1.

In exactly 1 cube, namely the top one, it is 5.

In the $4 \cdot 4 = 16$ corner cubes of the other four layers, it is 3.

In the remaining $4 \cdot (1 + 3 + 5 + 7) = 64$ edge cubes of these layers, it is 2.

The remaining 84 cubes have no colored face.

c) In the four upper layers, only those corner and edge cubes that are located at the midpoints of the edges are to remain. To be removed from these layers are the other corner and edge cubes. These are, listed according to the layers from top to bottom, $4 \cdot (0 + 1 + 3 + 5) = 36$ cubes.

d) In the second to fourth layers, counted from the top, there are now, in addition to the cubes described as "to remain" in c), partial layers that each cover a square face again.

From these, the corner and edge cubes are to be removed, except for the cubes at the midpoints of the edges, which are $4 \cdot (0 + 1 + 3) = 16$ cubes.

Solutions of the 1st Round 1993 adopted from [5]

### 5.35.2 2nd Round 1993, Class 8