[ Regular Polyhedra. Duality and Relationships ]  [ Distance between_two points. Equation of a sphere ]

Authors: Rabzimiotskyl., Giadkih A.

Can an octahedron be inscribed in a cube such that the vertices of the octahedron lie on the edges of the cube?

#
In the figure, an octahedron is inscribed in a cube with an edge length of 4; the vertices of the octahedron divide the edges of the cube in the ratio $1: 3$. The square of the length of each edge of the octahedron is 18 (either $3^{2}+3^{2}$, or $4^{2}+1^{2}+$ $\left.1^{2}\right)$, that is, all its edges are equal.

![](https://cdn.mathpix.com/cropped/2024_05_06_6fdbe810ffc70cc86b90g-01.jpg?height=389&width=394&top_left_y=1692&top_left_x=838)

## Answer

It is possible.

| Problem 66183 Topics: | [ Chessboards and chess pieces ] | Difficulty: 3+  Grades: 7,8,9,10 |
| :---: | :---: | :---: |
|  | [ Examples and counterexamples. Constructions ] |  |
|  | Dirichlet's principle (other). |  |
|  | Parity and oddity |  |

![](https://cdn.mathpix.com/cropped/2024_05_06_6fdbe810ffc70cc86b90g-01.jpg?height=42&width=292&top_left_y=2513&top_left_x=0)

What is the minimum number of rooks needed to place on an $8 \times 8$ chessboard so that all white cells are under attack by these rooks? (A cell is considered under attack by a rook if it is in the same row or column as the rook.)

## Solution

Estimate. Each rook attacks no more than two cells of the white diagonal.

Example. Place rooks on the fields a1, c3, e5, and g7.

| Problem 66186 Topics: | $\left[\begin{array}{l}\text { [ Chessboards and chess pieces ] }] \text { ] } \\ \text { Examples and counterexamples. Constructions }]\end{array}\right.$ | Difficulty: 3+ |
| :---: | :---: | :---: |
|  | Dirichlet's principle (other). | Grades: 8,9,10,11 |
|  | Parity and oddity |  |

Author: Kenooarov. P.

The cells of a $9 \times 9$ board are colored in a checkerboard pattern in black and white (corner cells are white). What is the minimum number of rooks needed to place on this board so that all white cells are under attack by these rooks? (A cell is considered under attack by a rook if it is in the same row or column as the rook.)

## Solution

Estimate. A rook attacks no more than two cells of the white diagonal, so four rooks are insufficient.

Example. Place rooks on the fields indicated in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_6fdbe810ffc70cc86b90g-02.jpg?height=494&width=481&top_left_y=848&top_left_x=800)

## Answer

5 rooks.