Number Pattern Analysis

This problem requires us to identify a specific pattern in a sequence of numbers arranged in rows. We need to find the missing number represented by '?' in the third row by analyzing the relationship observed in the first two rows. The rule states that operations must be performed on the whole numbers, not on individual digits.

Row Data

The given data is structured in three rows:

• Row 1: 11, 12, 529
• Row 2: 13, 14, 729
• Row 3: 15, 16, ?

Pattern Logic

Let's examine the relationship between the numbers. We look at the third number in the first two rows:

• In Row 1, the third number is 529. We recognise $529$ as the square of $23$ ($23^2 = 529$).
• In Row 2, the third number is 729. We recognise $729$ as the square of $27$ ($27^2 = 729$).

Now, let's see how the first two numbers in each row relate to these square roots (23 and 27).

• Row 1: The first two numbers are 11 and 12. Their sum is $11 + 12 = 23$. This matches the square root of 529.
• Row 2: The first two numbers are 13 and 14. Their sum is $13 + 14 = 27$. This matches the square root of 729.

The pattern identified is: $(\text{First Number} + \text{Second Number})^2 = \text{Third Number}$.

Third Row Calculation

We apply this established pattern to the third row:

• The first two numbers in Row 3 are 15 and 16.
• First, calculate their sum: $15 + 16 = 31$.
• Next, square this sum to find the third number: $31^2$.
• The calculation is $31 \times 31 = 961$.

Result Confirmation

The calculation shows that the missing number is 961. This value fits the pattern observed across the rows and logically completes the sequence.