1. **Problem statement:** We have a 4x4 wall (16 squares) painted with two colours: yellow and another colour. At least one diagonal is painted yellow, and the rest of the squares are painted equally by the two colours. 2. **Understanding the problem:** - Total squares: 16 - Two colours: yellow (Y) and another colour (N) - At least one diagonal is fully yellow - The rest of the squares are painted so that the total number of yellow and non-yellow squares are equal (8 each) 3. **Step 1: Identify the diagonals** - Main diagonal squares: 4 (positions (1,1), (2,2), (3,3), (4,4)) - Secondary diagonal squares: 4 (positions (1,4), (2,3), (3,2), (4,1)) - Intersection of diagonals: 1 square (center square (2,2) or (3,3) depending on indexing, but here both diagonals share the center square at (2,2) and (3,3) are different, so no overlap in 4x4, so no intersection) 4. **Step 2: Cases for yellow diagonals** - Case A: Main diagonal is yellow - Case B: Secondary diagonal is yellow - Case C: Both diagonals are yellow 5. **Step 3: Count yellow squares in each case** - Case A: Main diagonal yellow = 4 yellow squares - Case B: Secondary diagonal yellow = 4 yellow squares - Case C: Both diagonals yellow = 8 yellow squares (4 + 4) 6. **Step 4: Since total yellow squares must be 8 (equal division), analyze each case:** - Case A: 4 yellow on main diagonal, need 4 more yellow squares from the remaining 12 squares - Case B: 4 yellow on secondary diagonal, need 4 more yellow squares from the remaining 12 squares - Case C: 8 yellow on both diagonals, no more yellow squares needed 7. **Step 5: Calculate number of ways for each case** - Case A: Choose 4 yellow squares from 12 non-diagonal squares: $\binom{12}{4} = 495$ - Case B: Same as Case A: $495$ - Case C: Both diagonals yellow, no choice for yellow squares outside diagonals 8. **Step 6: Calculate number of ways to paint non-yellow squares** - Non-yellow squares are the remaining 8 squares in all cases - Since only two colours (yellow and one other), non-yellow squares are fixed as the other colour 9. **Step 7: Combine cases considering overlap** - Cases A and B overlap in Case C (both diagonals yellow) - Use inclusion-exclusion principle: Total ways = Case A + Case B - Case C = 495 + 495 - 1 = 989 10. **Final answer:** $$\boxed{989}$$ ways to paint the wall under the given conditions.