Question

3 11 55 6 12 24 42 4

Explanation:

Step1: Analyze the bottom row pattern

The bottom row numbers are 6, 12,?, 42. Let's check the differences: \(12 - 6 = 6\), assume the pattern is multiplying by a sequence or adding a multiple. Wait, \(6\times1 = 6\), \(6\times2 = 12\), \(6\times4 = 24\), \(6\times7 = 42\)? Wait, maybe another pattern. Let's check the first column: 11 (top) and 6 (bottom). 11 - 6 = 5? No. Wait, maybe the top and bottom rows have a relation with the left and right numbers (3 and 4). Wait, 3 and 4 are on the sides. Let's check the bottom row: 6, 12, 24, 42. Wait, 6 to 12: +6, 12 to 24: +12, 24 to 42: +18? No, 6, 12, 24, 42. The differences are 6, 12, 18, which is 6*1, 6*2, 6*3. So next difference would be 6*4=24, but 24 + 24=48? No, 42 is given. Wait, maybe the top row and bottom row have a relation with 3 and 4. Let's see: 11 (top) and 6 (bottom): 11 = 6*1 + 5? No. Wait, 3 and 4: maybe columns. Let's check column 1: 11 (top), 6 (bottom). 11 - 6 = 5. Column 2:? (top), 12 (bottom). Column 3: 55 (top), 24 (bottom). 55 - 24 = 31. Column 4:? (top), 42 (bottom). No, that's not helpful. Wait, maybe the bottom row is 6, 12, 24, 42. Let's factor them: 6=6, 12=6*2, 24=6*4, 42=6*7. The multipliers: 1, 2, 4, 7. The differences between multipliers: 1, 2, 3. Ah, 1 to 2: +1, 2 to 4: +2, 4 to 7: +3. So next would be +4, 7+4=11, 6*11=66. But top row has 11,?, 55,?. Wait, top row: 11,?, 55,?. Let's see the multipliers for top row: 11=11*1, 55=11*5. So 1, 5. The differences: 1 to 5: +4. Maybe the middle term is 11*3=33? Wait, bottom row multipliers: 1,2,4,7 (differences 1,2,3). Top row multipliers: 1,3,5,7 (differences 2,2,2)? No. Wait, bottom row: 6,12,24,42. Let's check 6*1=6, 6*2=12, 6*4=24, 6*7=42. The numbers 1,2,4,7: 1+1=2, 2+2=4, 4+3=7, 7+4=11. So top row: 11*1=11, 11*3=33, 11*5=55, 11*7=77. Let's check column 2: top 33, bottom 12. 33 - 12 = 21. Column 3: 55 - 24 = 31. No, not helpful. Wait, maybe the bottom row is 6,12,24,42. Let's check the sum of bottom row: 6+12+24+42=84. Top row: 11 + x + 55 + y = 66 + x + y. Not helpful. Wait, maybe the left number 3 and right number 4: 3 and 4 are factors. Let's see: 6 (bottom row, column 1) * 3 = 18, no. 11 (top row, column 1) * 3 = 33, no. 6 * 4 = 24, which is column 3 bottom. Oh! 6*4=24 (column 3 bottom). 12*4=48, no, column 4 bottom is 42. 11*3=33 (column 2 top). 55*3=165, no. Wait, column 3 bottom: 24. 6*4=24. Yes! 6 (column 1 bottom) * 4 = 24 (column 3 bottom). Column 1 bottom: 6, column 3 bottom: 24. 6*4=24. Column 2 bottom: 12, column 4 bottom: 42. 12*4=48, no, 42. Wait, 6*3=18, no. Wait, 6 (column 1 bottom) * 4 = 24 (column 3 bottom). 12 (column 2 bottom) * 3 = 36, no. Wait, 3 and 4: column 1 and 3 use 4? Column 2 and 4 use 3? 6*4=24 (column 3 bottom), 12*3=36, no. 55 (column 3 top) * 3 = 165, no. 11 (column 1 top) * 3 = 33 (column 2 top). 55 (column 3 top) * 3 = 165, no. Wait, 11 (column 1 top) * 3 = 33 (column 2 top), 55 (column 3 top) * 3 = 165 (column 4 top). But column 4 bottom is 42. 165 - 42 = 123, no. Wait, 6 (column 1 bottom) * 2 = 12 (column 2 bottom), 12 (column 2 bottom) * 2 = 24 (column 3 bottom), 24 (column 3 bottom) * 1.75 = 42 (column 4 bottom). No, that's not a pattern. Wait, the bottom row: 6, 12, 24, 42. Let's check the differences between terms: 12-6=6, 24-12=12, 42-24=18. So the differences are 6, 12, 18, which is an arithmetic sequence with common difference 6. So next difference would be 24, so next term would be 42+24=66, but that's not in the table. Wait, maybe the top row: 11, 33, 55, 77. The differences: 33-11=22, 55-33=22, 77-55=22. So that's an arithmetic sequence with common difference 22. Let's check the bottom row: 6, 12, 24, 42. If top row is 11,33,55,77 (difference 22), bottom row: 6,12,24,42. Let's see the ratio of top to bottom: 11/6 ≈1.83, 33/12=2.75, 55/24≈2.29, 77/42≈1.83. No, not consistent. Wait, another approach: the table has two rows, four columns. Let's denote the top row as \(a, b, 55, d\) and bottom row as \(6, 12, 24, 42\). Let's check the sum of each column: \(a + 6\), \(b + 12\), \(55 + 24 = 79\), \(d + 42\). Not helpful. Wait, 3 and 4 are on the left and right. Maybe 3 is a factor for the first two columns and 4 for the last two. So column 1: 11 (top) = 6 (bottom) * 1 + 5? No. Column 1: 6 * 3 = 18, 11 is less. Column 2: 12 * 3 = 36, so top column 2 is 33? 11, 33, 55, 77: that's an arithmetic sequence with difference 22. Yes! 11 + 22 = 33, 33 + 22 = 55, 55 + 22 = 77. So top row: 11, 33, 55, 77. Now bottom row: 6, 12, 24, 42. Let's check the difference between top and bottom: 11 - 6 = 5, 33 - 12 = 21, 55 - 24 = 31, 77 - 42 = 35. No. Wait, bottom row: 6, 12, 24, 42. Let's see the pattern of 6, 12, 24, 42. The differences between terms: 6, 12, 18. Which is 6*1, 6*2, 6*3. So the next difference would be 6*4=24, but 42 + 24=66, not in the table. Wait, but 24 is given as the third term (bottom row, third column). So bottom row: 6, 12, 24, 42. Let's check the multiples of 6: 6*1=6, 6*2=12, 6*4=24, 6*7=42. The exponents or multipliers: 1, 2, 4, 7. The differences between multipliers: 1, 2, 3. So that's 1 (start), +1=2, +2=4, +3=7, +4=11. So next multiplier would be 11, 6*11=66. But top row has 11, 33, 55, 77 (multipliers 1, 3, 5, 7 of 11). Ah! Top row: 11*1=11, 11*3=33, 11*5=55, 11*7=77. Bottom row: 6*1=6, 6*2=12, 6*4=24, 6*7=42. The multipliers for top: 1,3,5,7 (odd numbers, difference 2). Multipliers for bottom: 1,2,4,7 (differences 1,2,3). So that's a possible pattern. So top row second column: 11*3=33, fourth column: 11*7=77. Bottom row third column: 6*4=24 (which is given). So the missing top row second column is 33, top row fourth column is 77, bottom row third column is 24 (already given). Wait, the problem has some boxes: top row second and fourth, bottom row third. So bottom row third is 24 (as per the handwritten 24), top row second is 33, top row fourth is 77. Let's verify: top row: 11, 33, 55, 77 (difference 22). Bottom row: 6, 12, 24, 42 (differences 6,12,18). Yes, that works.

Step2: Determine the missing values

- Top row second column: \(11 + 22 = 33\) (since 11 to 55 is +44, so half is +22)
- Bottom row third column: \(12 + 12 = 24\) (or 6*4=24)
- Top row fourth column: \(55 + 22 = 77\)

Answer:

Top row second box: 33, Bottom row third box: 24, Top row fourth box: 77