In the following version of the puzzle, two squares in the center 3-by-3 region have ones on corners across from each other diagonally. Show that it is impossible to fill in the blank squares on this puzzle satisfying conditions (i) and (ii).

Sudoku-style puzzle

8. For the following sudoku-style puzzle, you are given the following 9-by-9 grid,  and you need to fill it in with zeros and ones satisfying the following conditions:

• (i) Each row, each column, and each red or blue 3-by-3 box must contain exactly two ones and seven zeros.
• (ii) No two ones can be in squares that touch horizontally, vertically, or diagonally. For example, the following is a solution to this puzzle:

However, the following is not a solution. Although it satisfies condition (i), the highlighted ones are in squares that touch diagonally, so it does not satisfy condition (ii).

• a. In the following version of the puzzle, two squares in the center 3-by-3 region have ones on corners across from each other diagonally. Show that it is impossible to fill in the blank squares on this puzzle satisfying conditions (i) and (ii).
• b. In the following version of the puzzle there are ones on either end of the middle row of the center 3-by-3 region. How many possible solutions are there for completing the blank squares on this puzzle satisfying conditions (i) and (ii)?
• c. Starting with all blank squares, what is the total number of solutions to this puzzle satisfying conditions (i) and (ii)?