Question 27
Exactly one integer number from $1$ to $25$ is written in each cell of the $5 \times 5$ square table (see the figure). Per move it is allowed:
| 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 |
| 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 |
1) to choose any two cells with a common side then to increase (to decrease) the number in one of the cells by $2$ and to decrease (to increase respectively) the number in the other cell by $1$;
2) to choose any two cells with a common vertex then to increase the number in one of the cells by $1$ and to decrease the number in the other cell by $1$;
a) Is it possible to obtain the table with the same numbers in all its cells after the finite number of these moves?
b) Is it possible to obtain the table with $2$ in all its cells after the finite number of these moves?
Clarification for this practice version (AI editorial addition): In move 2, choose two cells meeting only at a vertex, not sharing a side; thus they are diagonally adjacent. The original wording is retained above.
| 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 |
| 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 |
1) to choose any two cells with a common side then to increase (to decrease) the number in one of the cells by $2$ and to decrease (to increase respectively) the number in the other cell by $1$;
2) to choose any two cells with a common vertex then to increase the number in one of the cells by $1$ and to decrease the number in the other cell by $1$;
a) Is it possible to obtain the table with the same numbers in all its cells after the finite number of these moves?
b) Is it possible to obtain the table with $2$ in all its cells after the finite number of these moves?
Clarification for this practice version (AI editorial addition): In move 2, choose two cells meeting only at a vertex, not sharing a side; thus they are diagonally adjacent. The original wording is retained above.