Question 60
Exactly one integer number from $1$ to $49$ is written in each cell of the $7 \times 7$ square table (see the figure). Per move it is allowed to choose any cell then simultaneously to increase (to decrease) the number in this cell by $1$ and to decrease (to increase respectively) the numbers in some two adjacent cells by $1$ (two cells are adjacent if they have a common side).
|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|26|27|28|
|29|30|31|32|33|34|35|
|36|37|39|39|40|41|42|
|43|44|45|46|47|48|49|
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 $2013$ in all its cells after the finite number of these moves?
|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|26|27|28|
|29|30|31|32|33|34|35|
|36|37|39|39|40|41|42|
|43|44|45|46|47|48|49|
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 $2013$ in all its cells after the finite number of these moves?