Yashchenko I.V.

On a grid sheet of paper, several cells were colored in such a way that the resulting figure had no axes of symmetry. Vanya colored one more cell. Could the resulting figure have four axes of symmetry?
To construct an example, one needs to take a figure with four axes of symmetry and remove a cell that does not lie on any of these axes. For example, a square has four axes of symmetry (see the left image). This results in the example in the center.

There are other examples as well (for example, on the right image).
![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-05.jpg?height=446&width=1288&top_left_y=2335&top_left_x=386)

\section*{[ Polyhedra and Polygons (Miscellaneous) $]$

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-06.jpg?height=108&width=1295&top_left_y=206&top_left_x=10)  Difficulty: 3 Grades: 10,11}

Author: $\underline{\text { Folklore }}$

What is the maximum number of triangular faces a pentahedron can have?

## Solution

A quadrilateral pyramid has four triangular faces.

We will prove that all five faces of a pentahedron cannot be triangles. Indeed, suppose such a pentahedron exists. Each of its edges is shared by two faces, so it should have $5 \cdot 3$ : 2 edges, which is impossible.

## Answer

4 faces.