1. The problem shows a 3x3 grid with numbers and a missing number represented by a question mark in the bottom left cell. 2. The grid rows are: - Top: 1, 85, 19 - Middle: 46, 4, 64 - Bottom: ?, 13, 31 3. To find the missing number, we look for a relationship in rows or columns. Let's examine column-wise patterns. 4. Column 2 has 85 (top), 4 (middle), 13 (bottom). 4 and 13 are related as 13 is roughly 3 times 4 plus 1. 5. Column 3 has 19 (top), 64 (middle), 31 (bottom). 64 is roughly $19^2$ or a different pattern; no clear arithmetic progression. 6. Check row sums: - Top row sum: $1 + 85 + 19 = 105$ - Middle row sum: $46 + 4 + 64 = 114$ - Bottom row sum: $? + 13 + 31 = ? + 44$ 7. No simple arithmetic progression across rows sums (105,114,?+44). 8. Check differences between columns in each row: - Top row: $85 - 1=84$, $19 - 85 = -66$ - Middle row: $4 - 46 = -42$, $64 - 4 = 60$ - Bottom row: $13 - ? = ?$, $31 - 13 = 18$ 9. No clear pattern emerges in differences. 10. Now check multiplication or division patterns column wise: Looking at the second column: - $85 o 4 o 13$ - 85 reduced to 4 suggests a division or modulo. 11. Now consider the possibility each column elements add up to a constant: - Column 1: $1 + 46 + ? = ? + 47$ - Column 2: $85 + 4 + 13 = 102$ - Column 3: $19 + 64 + 31 = 114$ 12. Columns 2 and 3 sums are 102 and 114; column 1 sum could be 114 (to match column 3 sum). 13. Set $? + 47 = 114$, so $? = 114 - 47 = 67$. 14. Therefore, the missing number is 67. Final answer: The missing number is **67**.