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:
0 0 0 0 0 1 0 1 0 0 0 0 1 0 o 0 0 E a 0 1 a 0 0 0 1 0 1 0 0 0 1 0 0 a a 0 0 1 0 0 1 0 E 1 1 0 0 1 1 .1, 1 0 0 1 0 0 0 1 0 0 0 1 0 E. 1 0 0 1 0 0 1
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).
of 0 0 0 1 0 1 0 0 0 1=.: 1 Ci a 0 0 0 a 0 a C I 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 K, 0 0 1 0 0 1 0 0 1 0 0 1 a 0 0 0 0 0 a 1 0 1 0 1 of cl 0 0 1 0 0 0 1 a 0
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).
1
111
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)?
1
1
=
c. Starting with all blank squares, what is the total number of solutions to this puzzle satisfying conditions (i) and (ii)?
II
