1. The problem is to find the missing number represented by $??$ in the given sequence of numbers arranged in rows. 2. Let's analyze the pattern row by row and column by column to identify any relationships or sequences. 3. The numbers are: Row 1: 69, 34, 10, 77, 28, 24, 20 Row 2: 20, 71, 06, 53, 06, 74, 05 Row 3: 60, 12, 77, 53, 77, 43, 46 Row 4: 31, 49, 84, 20, 53, 53, 98 Row 5: 00, 61, 40, 32, ?? 4. Since the last row has only 5 numbers and the others have 7, it suggests the missing number is the 6th element in the last row. 5. Let's check if columns have any arithmetic pattern: - Column 1: 69, 20, 60, 31, 00 - Column 2: 34, 71, 12, 49, 61 - Column 3: 10, 06, 77, 84, 40 - Column 4: 77, 53, 53, 20, 32 - Column 5: 28, 06, 77, 53, ?? - Column 6: 24, 74, 43, 53, ? (missing) - Column 7: 20, 05, 46, 98, ? (missing) 6. Since the last row is incomplete, focus on column 5 where the missing number is. 7. Look for a pattern in column 5: 28, 06, 77, 53, ?? 8. Differences between consecutive numbers in column 5: - 06 - 28 = -22 - 77 - 06 = 71 - 53 - 77 = -24 9. The differences do not form a simple arithmetic progression. 10. Check if the sum of each row is consistent or follows a pattern: - Row 1 sum: 69+34+10+77+28+24+20 = 262 - Row 2 sum: 20+71+6+53+6+74+5 = 235 - Row 3 sum: 60+12+77+53+77+43+46 = 368 - Row 4 sum: 31+49+84+20+53+53+98 = 388 - Row 5 sum (known numbers): 0+61+40+32+?? = 133 + ?? 11. No clear pattern in sums. 12. Check if the missing number is the average of the column 5 numbers: Average of first 4 numbers in column 5: (28+6+77+53)/4 = 164/4 = 41 13. So, the missing number could be 41. 14. Verify if 41 fits any other pattern. 15. Since no other clear pattern emerges, the best estimate for $??$ is 41. Final answer: $41$