Consider the leftmost column and lowest row of table, we color all these cells. We can see that for all ways to put the number of the sub square $7 \times 7$ that not colored, we can choose the number for the correspondent colored position at same column or row. Indeed, if sum of the $7$ numbers is odd, then we put $1$ on the remain cell; otherwise, we put $2$.

Finally, the number for the cell at corner can choose base on the parity of the sum of all number in square $7 \times 7$. These mean that the way to fill in the square $7 \times 7$ uniquely define the numbers on the rest cells.

Since we can fill each cell among $49$ cells of the square $7 \times 7$ by $1$ or $2$ in any way then the number of way is $2^{49}$.