1. **Stating the problem:** Dominik has a 3×3 grid with a 2×2 shaded block in the bottom right corner. He wants to place a dark and a light stone so that they are not in the same row or column. The dark stone must be placed on one of the shaded squares. We need to find the number of different ways to place both stones under these conditions. 2. **Understanding the grid and constraints:** The grid has 3 rows and 3 columns. The shaded squares form a 2×2 block in the bottom right, so these shaded squares are at positions (2,2), (2,3), (3,2), and (3,3) if we number rows and columns from 1 to 3. 3. **Step 1: Place the dark stone:** It must be on one of the 4 shaded squares. So, there are 4 choices for the dark stone. 4. **Step 2: Place the light stone:** It cannot be in the same row or column as the dark stone. - If the dark stone is at (2,2), the light stone cannot be in row 2 or column 2. - The remaining available squares are those not in row 2 or column 2. 5. **Counting available squares for the light stone:** - Total squares: 9 - Remove row 2 (3 squares) and column 2 (3 squares), but the intersection (2,2) is counted twice, so total removed is $3 + 3 - 1 = 5$ squares. - Remaining squares: $9 - 5 = 4$ squares. 6. **Check for each dark stone position:** - For each dark stone position, the number of available squares for the light stone is 4. 7. **Calculate total ways:** - Number of ways = number of dark stone positions × number of light stone positions per dark stone - $$4 \times 4 = 16$$ 8. **But the problem states the dark stone must be on shaded squares only (4 options), and the light stone can be on any unshaded square not in the same row or column. The unshaded squares are the 5 remaining squares outside the shaded 2×2 block. We must verify which unshaded squares are allowed for the light stone for each dark stone position.** 9. **List unshaded squares:** - Positions: (1,1), (1,2), (1,3), (2,1), (3,1) 10. **For each dark stone position, count allowed light stone positions:** - Dark stone at (2,2): light stone cannot be in row 2 or column 2 - Unshaded squares: - (1,1): row 1, col 1 → allowed - (1,2): col 2 → not allowed - (1,3): col 3 → allowed - (2,1): row 2 → not allowed - (3,1): row 3, col 1 → allowed - Allowed: 3 squares - Dark stone at (2,3): light stone cannot be in row 2 or column 3 - Unshaded squares: - (1,1): allowed - (1,2): allowed - (1,3): col 3 → no - (2,1): row 2 → no - (3,1): allowed - Allowed: 3 squares - Dark stone at (3,2): light stone cannot be in row 3 or column 2 - Unshaded squares: - (1,1): allowed - (1,2): col 2 → no - (1,3): allowed - (2,1): allowed - (3,1): row 3 → no - Allowed: 3 squares - Dark stone at (3,3): light stone cannot be in row 3 or column 3 - Unshaded squares: - (1,1): allowed - (1,2): allowed - (1,3): col 3 → no - (2,1): allowed - (3,1): row 3 → no - Allowed: 3 squares 11. **Total ways:** - For each of the 4 dark stone positions, 3 allowed light stone positions - $$4 \times 3 = 12$$ 12. **Final answer:** 12 ways. **Answer: D) 12**