Since in each case it is a square pyramid with a base side length of $a$, each lateral face of each of the pyramids is an isosceles triangle with a base length of $a$, and the triangles are all congruent to each other.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0893.jpg?height=537&width=437&top_left_y=2147&top_left_x=815)

We calculate the size of the angle between two triangular faces that share the same edge of the cube. The angle is composed of:

1. the angle between a triangular face and the base of the corresponding pyramid.
2. the right angle between two adjacent side faces of the cube, and
3. the angle between the other triangular face and the base of the corresponding pyramid. Its size is thus $45^{\circ}+90^{\circ}+45^{\circ}=180^{\circ}$.

The two adjacent triangles lie in the same plane, and because they are congruent and isosceles and share a common side, their legs form the sides of a rhombus. Since the cube has 12 edges, the composite body has 12 rhombic faces as side faces; because no two rhombi belonging to different edges lie in the same plane. The rhombi are congruent to each other.

We also show that the rhombi are not squares. The heights of the pyramids placed on top, due to the angle between the lateral and base faces of each pyramid being $45^{\circ}$, have a length of $\frac{a}{2}$. The heights of the triangular faces forming the lateral faces have, by the Pythagorean theorem, a length of $\sqrt{2\left(\frac{a}{2}\right)^{2}}$, i.e., $\frac{a}{2} \sqrt{2}$.

Thus, the diagonals of the rhombi have different lengths, namely $a$ and $a \sqrt{2}$.

Each of the pyramids placed on top has a volume of $\frac{1}{3}\left(a^{2} \frac{a}{2}\right)$, i.e., $\frac{a^{3}}{6}$.

The six pyramids placed on top thus have a total volume of $2 a^{3}$ together with the cube.

Solutions of the III. Round 1964 adopted from [5] and edited

### 6.7 V. Olympiad 1965

### 6.7.1 I. Round 1965, Class 9
## Task 6 - 040936

Pyramids are placed on the faces of a cube, whose bases are congruent to the faces of the cube and whose side faces form an angle of $45^{\circ}$ with the base.

1. How many faces does the new solid have, and what is the shape of these faces?
2. Give the volume of the composite solid as a function of the cube edge $a$!