In the following question, a matrix of numbers is given. These numbers follow a specific pattern row wise or column wise. Study the matrix carefully and choose the missing pair of numbers accordingly. The matrix is: Row 1: 1 4 9 ?; Row 2: 1 2 3 4; Row 3: 2 4 6 ?. Which pair of numbers should replace the two question marks in Row 1 and Row 3 respectively?

Introduction / Context:
This puzzle presents a 3 by 4 matrix where two entries are missing, one in the first row and one in the third row. The second row is fully known and provides a base sequence of simple increasing numbers. By examining how the first and third rows are related to this middle row, we can deduce the rules that generate them and then fill in the missing numbers.

Given Data / Assumptions:

• Row 1: 1, 4, 9, ?.
• Row 2: 1, 2, 3, 4.
• Row 3: 2, 4, 6, ?.
• The missing numbers are the fourth entry of row 1 and the fourth entry of row 3.
• The second row looks like a simple sequence 1, 2, 3, 4.

Concept / Approach:
The middle row forms the base: 1, 2, 3, 4. It is common in such puzzles that one row may contain squares of these numbers while another row may contain some simple multiple, such as double. We test whether the first row entries correspond to squares of the second row, and whether the third row entries correspond to twice the second row. If these relationships hold for the known columns, we can confidently extend them to the unknown entries.

Step-by-Step Solution:
Step 1: Examine row 2, which is 1, 2, 3, 4. This is a simple increasing sequence of natural numbers. Step 2: Compare row 1 with row 2 column wise. Row 1 entries for the first three columns are 1, 4, 9 while row 2 entries are 1, 2, 3. Step 3: Notice that 1 is 1 squared, 4 is 2 squared, and 9 is 3 squared. Thus row 1 appears to be the squares of row 2: n^2 for each corresponding n. Step 4: If this pattern continues, the fourth entry of row 1 should be 4 squared, which is 16. Step 5: Now compare row 3 with row 2. Row 3 entries for the first three columns are 2, 4, 6, while row 2 entries are 1, 2, 3. Step 6: We see that 2 is 2 * 1, 4 is 2 * 2, and 6 is 2 * 3. So each entry in row 3 is double the corresponding entry in row 2. Step 7: Therefore, the missing fourth entry of row 3 should be 2 times the fourth entry of row 2, that is 2 * 4 = 8.

Verification / Alternative check:
With the missing values filled, the matrix becomes: Row 1: 1, 4, 9, 16. Row 2: 1, 2, 3, 4. Row 3: 2, 4, 6, 8. Row 1 is clearly 1^2, 2^2, 3^2, 4^2. Row 3 is clearly 2 * 1, 2 * 2, 2 * 3, 2 * 4. Since the relationships hold across all columns, the pattern is neat and internally consistent, confirming the correctness of 16 and 8 as the missing numbers.

Why Other Options Are Wrong:
If we choose 49, 7 then the first row would suddenly jump from 9 to 49, which is 7 squared, breaking the clear square pattern of 1^2, 2^2, 3^2, 4^2. The options 36, 4 and 25, 5 similarly do not align the first row with squares of the second row and the third row with doubles of the second row. Therefore, they disrupt the simple and elegant pattern observed across multiple columns.

Common Pitfalls:
Some learners focus only row wise or only on differences between numbers and may miss the square and double relationships. Others search for an overly complicated rule when a simple square and doubling pattern fits perfectly. Always check for basic arithmetic and algebraic patterns like squares, cubes, and multiples before trying more complex ideas.

Final Answer:
The two missing numbers are 16 in the first row and 8 in the third row, so the correct pair is 16, 8.