Board Trick

Danila the Magician and Mia the Assistant are showing a trick with a board.

First, Danila secludes himself, while Mia asks the audience to fill the $8 \times 8$ square table drawn on the board: each cell can contain either zero or one.

Next, another member of the audience chooses the secret number: an integer from $1$ to $64$. The number is announced to Mia but not to Danila.

After that, Mia comes to the board and changes exactly one number on it: either a zero into one, or a one into zero.

Finally, Danila returns, looks at the board for a while, and happily announces the secret number he did not know.

How do they perform the trick? Help Danila and Mia to prepare and then show it.