Understanding the Number Pattern

This question asks us to find a hidden pattern within the given set of numbers arranged in rows and columns. We are given three rows, each with three numbers. The goal is to figure out the rule that connects the first two numbers to the third number in each row and then use that rule to find the missing number in the third row.

The question specifies that operations must be performed on the whole numbers themselves, not on their individual digits. This is a crucial constraint to keep in mind while looking for the pattern.

Analyzing the Given Rows

Let's look at the numbers in each row:

• Row 1: 15, 3, 252
• Row 2: 11, 5, 246
• Row 3: 9, 6, ?

We need to find a mathematical operation or combination of operations involving the first number (let's call it A) and the second number (let's call it B) that results in the third number (let's call it C) for Rows 1 and 2. Once we find this pattern, we can apply it to Row 3 to find the missing number.

Discovering the Pattern Rule

Let's try some common operations or combinations using the numbers from the first row (A=15, B=3, C=252):

• A + B = $15 + 3 = 18$ (Not 252)
• A × B = $15 \times 3 = 45$ (Not 252)
• $A^2 = 15^2 = 225$
• $B^2 = 3^2 = 9$
• $B^3 = 3^3 = 3 \times 3 \times 3 = 27$

Let's see if combining powers of A and B works:

• $A^2 + B = 225 + 3 = 228$ (Close, but not 252)
• $A^2 + B^2 = 225 + 9 = 234$ (Closer)
• $A^2 + B^3 = 225 + 27 = 252$ (This matches the third number in Row 1!)

So, the potential pattern is $A^2 + B^3 = C$. Let's verify this pattern with the second row (A=11, B=5, C=246).

• $A^2 = 11^2 = 121$
• $B^3 = 5^3 = 5 \times 5 \times 5 = 125$
• $A^2 + B^3 = 121 + 125 = 246$ (This matches the third number in Row 2!)

The pattern $A^2 + B^3 = C$ is consistent for both Row 1 and Row 2.

Applying the Pattern to Find the Missing Number

Now we apply the discovered pattern to the third row (A=9, B=6, C=?).

Using the rule $C = A^2 + B^3$:

• $A^2 = 9^2 = 81$
• $B^3 = 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$
• $C = 81 + 216 = 297$

The missing number in the third row is 297.

Summary of the Pattern

The pattern is that the third number in each row is the sum of the square of the first number and the cube of the second number.

Revision Table: Number Pattern Analysis

Reviewing the steps helps reinforce the process of solving number pattern questions:

• Examine the relationship between the given numbers in the completed rows.
• Consider basic arithmetic operations, powers, or combinations.
• Test the potential pattern on all available complete rows.
• Apply the confirmed pattern to the row with the missing number.
• Ensure operations are performed on whole numbers as specified.

Additional Information: Powers and Cubes

Understanding squares and cubes is essential for solving patterns like this. A number squared ($n^2$) means the number multiplied by itself ($n \times n$). A number cubed ($n^3$) means the number multiplied by itself three times ($n \times n \times n$).

• Examples of squares: $1^2=1$, $2^2=4$, $3^2=9$, ..., $10^2=100$, $11^2=121$, $15^2=225$, etc.
• Examples of cubes: $1^3=1$, $2^3=8$, $3^3=27$, ..., $5^3=125$, $6^3=216$, etc.

Being familiar with common squares and cubes can speed up the process of identifying patterns in number series and grids.