The problem involves identifying the missing number in a sequence. Let's analyze it step by step:

Step 1: Observing the given pattern

The structure consists of three rows:

1. First Row: $16, 64, 256$
2. Second Row: $10, 25, ?$
3. Third Row: $4, 16, 64$

Step 2: Relations between rows

Relationship between the First Row and the Third Row:

• $\sqrt{16} = 4$
• $\sqrt{64} = 16$
• $\sqrt{256} = 64$

This shows that the third row is the square root of the first row.

Relationship between the First Row and the Second Row:

Let us try to understand the relationship with the middle row $10, 25, ?$:

• For the first column: $\sqrt{16} + 6 = 10$
• For the second column: $\sqrt{64} + 9 = 25$
• For the third column (missing number): $\sqrt{256} + 12 = ?$

Step 3: Calculate the missing value

• $\sqrt{256} = 16$
• $16 + 12 = 42$

Thus, the missing number is 42.

Final Answer:

42

Would you like further explanation or related examples?

Here are five related questions for deeper practice:

1. What is the relationship between square numbers and arithmetic sequences?
2. Can this pattern be extended to higher numbers?
3. What if we subtract instead of adding in the middle row?
4. How can this problem be visualized in a geometric context?
5. Are there other mathematical operations that fit this sequence?

Tip: Always check for patterns between rows systematically—start with addition, subtraction, multiplication, division, and square roots!