10.5. On a plane, there are eight different points. A numbering of these points with numbers from 1 to 8 is called good if the following condition is met:

there exists a line such that all points lie on one side of it and at different distances from it, and the distances from the points to this line increase with the increase in the number. That is, the closest point is number 1, the next in distance is number 2, and so on.

What is the maximum number of different good numberings that a given set of eight points can have?
Answer. $2 C_{8}^{2}=56$.

Solution. Note that when numbering points relative to a line, we are only interested in its direction and which side of it the points are on. All this information can be uniquely restored from a unit normal to it, directed towards the points. Therefore, the numbering of the octet is derived from a vector of unit length. Let's parameterize such vectors by points on a circle.

For a moment, consider only two points $A$ and $B$ from the octet. These two points can be associated with two opposite points on the circle: the two normals to the line $A B$. The circle is divided into two arcs. It is clear that the numbering of $A$ and $B$ on one arc is the same, and on the other arc, it is the remaining one.

Next, note that the numbering can be uniquely restored based on the order of each separate pair. Therefore, if we mark all the resulting $C_{8}^{2}$ pairs of opposite points on the circle, the numbering on each of the resulting arcs will be fixed. Consequently, for any octet of points, the number of numberings will not exceed the number of arcs, i.e., not more than $2 C_{8}^{2}$. On the other hand, we can imagine a sufficiently general octet of points for which no pairs of opposite points coincide. Then the circle will be divided into exactly $2 C_{8}^{2}$ arcs. Finally, any two arcs can be separated by some pair of opposite points, meaning that the numbering on any two different arcs is different. Therefore, $2 C_{8}^{2}$ is achievable.

Criteria.

(-.) Correctly found and justified the number of numberings for some non-maximal configuration.

(-/+) Correct reasoning, not leading to the correct answer due to a minor error.

(+/2) Correct answer + significant progress in justification.

(+/-) Fully proven that 56 is achievable, $\underline{and}$ and that more than 56 is not possible.

(+) Fully proven that 56 is achievable, and more than 56 is not possible.