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An infinite countable family $\cal{L}$ of lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:

$\bullet$ No circle intersects infinitely many lines from the family $\cal{L}$;

$\bullet$ Let $L$ denote the union of the lines in $\cal{L}$. Each connected component of the complement of $L$ is a bounded set; the closure of each of these sets will be called a cell (observe that each cell is a convex polygon). This condition requires that the diameters of the cells formed by the cutting have a common upper bound;

Finally,

$\bullet$ All cells are of the same area.

We say that two fair cuttings are affinely equivalent if there exists an affine transformation of $\mathbb{R}^2$ sending the lines of one family onto the lines of the other. Also, we say that a fair cutting $\cal{L}$ is extra-fair, if, in at least one of its affine equivalents, all cells are congruent.

Four examples of mutually non-equivalent extra-fair cuttings are shown below.

Question 1. Is there a fair cutting whose at least one cell has more than four sides?

Question 2. Must all cells of a fair cutting have the same number of sides?

Question 3. Is every fair cutting extra-fair?

Question 4. Does there exist another example of a fair cuting, non-equivalent to any of the four pictured here?

An infinite countable family $\cal{L}$ of straight lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:

$\bullet$ No circle intersects infinitely many lines from the family $\cal{L}$;

$\bullet$ Let $L$ denote the union of the lines in $\cal{L}$. Each connected component of the complement of $L$ is a bounded set; the closure of each of these sets will be called a cell (observe that each cell is a convex polygon). This condition requires that the diameters of the cells formed by the cutting have a common upper bound;

Finally,

$\bullet$ All cells are of the same area.

We say that two fair cuttings are affinely equivalent if there exists an affine transformation of $\mathbb{R}^2$ sending the lines of one family onto the lines of the other. Also, we say that a fair cutting $\cal{L}$ is extra-fair, if, in at least one of its affine equivalents, all cells are congruent.

Four examples of mutually non-equivalent extra-fair cuttings are shown below.

Question 1. Is there a fair cutting whose at least one cell has more than four sides?

Question 2. Must all cells of a fair cutting have the same number of sides?

Question 3. Is every fair cutting extra-fair?

Question 4. Does there exist another example of a fair cuting, non-equivalent to any of the four pictured here?

An infinite countable family $\cal{L}$ of lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:

$\bullet$ No circle intersects infinitely many lines from the family $\cal{L}$;

$\bullet$ Let $L$ denote the union of the lines in $\cal{L}$. Each connected component of the complement of $L$ is a bounded set; the closure of each of these sets will be called a cell (observe that each cell is a convex polygon). This condition requires that the diameters of the cells formed by the cutting have a common upper bound;

Finally,

$\bullet$ All cells are of the same area.

We say that two fair cuttings are affinely equivalent if there exists an affine transformation of $\mathbb{R}^2$ sending the lines of one family onto the lines of the other. Also, we say that a fair cutting $\cal{L}$ is extra-fair, if, in at least one of its affine equivalents, all cells are congruent.

Four examples of mutually non-equivalent extra-fair cuttings are shown below.

**Question 1.** Is there a fair cutting whose at least one cell has more than four sides?

**Question 2.** Must all cells of a fair cutting have the same number of sides?

**Question 3.** Is every fair cutting extra-fair?

**Question 4.** Does there exist another example of a fair cuting, non-equivalent to any of the four pictured here?

An infinite countable family $\cal{L}$ of lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:

$\bullet$ No circle intersects infinitely many lines from the family $\cal{L}$;

$\bullet$ Let $L$ denote the union of the lines in $\cal{L}$. Each connected component of the complement of $L$ is a bounded set; the closure of each of these sets will be called a cell (observe that each cell is a convex polygon). This condition requires that the diameters of the cells formed by the cutting have a common upper bound;

Finally,

$\bullet$ All cells are of the same area.

We say that two fair cuttings are affinely equivalent if there exists an affine transformation of $\mathbb{R}^2$ sending the lines of one family onto the lines of the other. Also, we say that a fair cutting $\cal{L}$ is extra-fair, if, in at least one of its affine equivalents, all cells are congruent.

Four examples of mutually non-equivalent extra-fair cuttings are shown below.

Question 1. Is there a fair cutting whose at least one cell has more than four sides?

Question 2. Must all cells of a fair cutting have the same number of sides?

Question 3. Is every fair cutting extra-fair?

Question 4. Does there exist another example of a fair cuting, non-equivalent to any of the four pictured here?

An infinite countable family $\cal{L}$ of lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:

$\bullet$ No circle intersects infinitely many lines from the family $\cal{L}$;

$\bullet$ Let $L$ denote the union of the lines in $\cal{L}$. Each connected component of the complement of $L$ is a bounded set; the closure of each of these sets will be called a cell (observe that each cell is a convex polygon). This condition requires that the diameters of the cells formed by the cutting have a common upper bound;

Finally,

$\bullet$ All cells are of the same area.

We say that two fair cuttings are affinely equivalent if there exists an affine transformation of $\mathbb{R}^2$ sending the lines of one family onto the lines of the other. Also, we say that a fair cutting $\cal{L}$ is extra-fair, if, in at least one of its affine equivalents, all cells are congruent.

Four examples of mutually non-equivalent extra-fair cuttings are shown below.

Question 1. Is there a fair cutting whose at least one cell has more than four sides?

Question 2. Must all cells of a fair cutting have the same number of sides?

Question 3. Is every fair cutting extra-fair?

Question 4. Does there exist another example of a fair cuting, non-equivalent to any of the four pictured here?

An infinite countable family $\cal{L}$ of lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:

$\bullet$ No circle intersects infinitely many lines from the family $\cal{L}$;

$\bullet$ Let $L$ denote the union of the lines in $\cal{L}$. Each connected component of the complement of $L$ is a bounded set; the closure of each of these sets will be called a cell (observe that each cell is a convex polygon). This condition requires that the diameters of the cells formed by the cutting have a common upper bound;

Finally,

$\bullet$ All cells are of the same area.

We say that two fair cuttings are affinely equivalent if there exists an affine transformation of $\mathbb{R}^2$ sending the lines of one family onto the lines of the other. Also, we say that a fair cutting $\cal{L}$ is extra-fair, if, in at least one of its affine equivalents, all cells are congruent.

Four examples of mutually non-equivalent extra-fair cuttings are shown below.

**Question 1.** Is there a fair cutting whose at least one cell has more than four sides?

**Question 2.** Must all cells of a fair cutting have the same number of sides?

**Question 3.** Is every fair cutting extra-fair?

**Question 4.** Does there exist another example of a fair cuting, non-equivalent to any of the four pictured here?