10.4. A point on the plane with coordinates $(x ; y)$ is allowed to be connected by a segment to a point with coordinates $(x+3 y ; y)$ or to a point with coordinates $(x ; y-2 x)$. Can points $A(19 ; 47)$ and $B(12 ; 17)$ be connected by a broken line under these conditions? Justify your answer.
Solution: It is obvious that if the coordinates of the initial point are integers, then the coordinates of all subsequent points will also be integers. The difference in the x-coordinates of the points that can be connected by a segment is either $3y$ or 0, and in both cases, it is divisible by 3. Therefore, the remainder of the division by 3 of the x-coordinates of all obtained points is the same - this is an invariant of the problem. (Another invariant is the oddness of the y-coordinate, but considering this invariant will not lead to the correct answer.) Since 19 gives a remainder of 1 when divided by 3, and 12 gives a remainder of 0, points $A$ and $B$ of the broken line cannot be connected.

Answer: No.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct justified answer | 7 points |
| The idea of considering the divisibility of the coordinates of the points by fixed numbers (in the absence of a solution) | 2 points |
| Correct answer without justification (with incorrect justification) | 0 points |
| Lack of justification for the integrity of the coordinates of all considered points | the score is not reduced |