Analyzing the Number Pattern

Let's carefully examine the given pattern of numbers to find the relationship between them. The numbers are presented in a sequence which implies they might form a grid or follow a specific rule across rows or columns.

The sequence is 12, 15, ?, 8, 7, 11, 80, 176, 240. We can arrange these numbers into a 3x3 grid:

Our goal is to find the number that replaces the question mark (?) in the third column, first row.

Identifying the Relationship in the Pattern

We need to find a rule that connects the numbers in the grid. Let's analyze the columns to see if there's a consistent relationship between the numbers in the first two rows and the number in the third row within each column.

• Column 1: Numbers are 12, 8, and 80.
• Column 2: Numbers are 15, 7, and 176.
• Column 3: Numbers are ?, 11, and 240.

Let's try various mathematical operations on the numbers in the first two rows of a column to see if we can obtain the number in the third row.

Considering Column 1 (12, 8, 80):

• Sum: \(12 + 8 = 20 \ne 80\)
• Difference: \(12 - 8 = 4 \ne 80\) or \(8 - 12 = -4 \ne 80\)
• Product: \(12 \times 8 = 96 \ne 80\)
• Let's consider squares: \(12^2 = 144\), \(8^2 = 64\). What if we subtract the square of the second number from the square of the first number? \(12^2 - 8^2 = 144 - 64 = 80\). This matches the third number in Column 1!

Let's test this potential pattern on Column 2 (15, 7, 176):

• Using the proposed pattern: \((Row 1)^2 - (Row 2)^2 = Row 3\)
• \(15^2 - 7^2 = 225 - 49 = 176\). This also matches the third number in Column 2!

The pattern seems to be confirmed: The number in the third row of each column is equal to the square of the number in the first row minus the square of the number in the second row.

Mathematically, for any given column, if \(R_1\) is the number in Row 1, \(R_2\) is the number in Row 2, and \(R_3\) is the number in Row 3, the rule is \(R_1^2 - R_2^2 = R_3\).

Calculating the Missing Number

Now we apply this discovered pattern to Column 3, which contains ?, 11, and 240. Let the missing number be $x$.

According to the pattern:

\((\text{Missing number in Row 1})^2 - (\text{Number in Row 2})^2 = \text{Number in Row 3}\)

\(x^2 - 11^2 = 240\)

First, calculate the value of \(11^2\):

\(11^2 = 11 \times 11 = 121\)

Substitute this value into the equation:

\(x^2 - 121 = 240\)

To solve for \(x^2\), add 121 to both sides of the equation:

\(x^2 = 240 + 121\)

\(x^2 = 361\)

To find the value of $x$, we need to take the square root of 361:

\(x = \sqrt{361}\)

We know that \(19 \times 19 = 361\). Therefore, the square root of 361 is 19.

$x = 19$

Verifying the Solution

The calculated missing number is 19. Let's quickly check the provided options to ensure our answer is among them and is correct based on the pattern.

• If the missing number were 18: \(18^2 - 11^2 = 324 - 121 = 203 \ne 240\).
• If the missing number were 27: \(27^2 - 11^2 = 729 - 121 = 608 \ne 240\).
• If the missing number were 22: \(22^2 - 11^2 = 484 - 121 = 363 \ne 240\).
• If the missing number were 19: \(19^2 - 11^2 = 361 - 121 = 240\). This matches the pattern exactly.

Thus, the number that replaces the question mark is 19.

Revision Table: Number Pattern Concepts

Additional Information: Strategies for Solving Pattern Problems

Solving number pattern problems often requires a combination of observation, hypothesis testing, and mathematical skills. Here are some general strategies:

• Look for Simple Relationships: Start with basic arithmetic operations (addition, subtraction, multiplication, division) between adjacent numbers or corresponding numbers in rows/columns.
• Consider Position-Based Rules: The rule might depend on the position of the number (e.g., the $n$-th term in a sequence).
• Think About Squares, Cubes, and Powers: Many patterns involve numbers raised to powers. Be familiar with common squares (\(1^2\) to \(20^2\)) and cubes (\(1^3\) to \(10^3\)).
• Look for Combinations of Operations: The pattern might involve a sequence of operations, such as multiply by 2 and add 1.
• Analyze Differences or Ratios: For sequences, calculate the difference between consecutive terms (for arithmetic patterns) or the ratio (for geometric patterns). If the first differences don't show a pattern, look at the second differences, and so on.
• Check Rows, Columns, and Diagonals in Grids: As seen in this problem, the relationship might be vertical (columns), horizontal (rows), or even diagonal.
• Break Down the Problem: If the numbers are large or complex, try to find patterns in their digits or components.
• Practice: The more number pattern problems you solve, the better you become at recognizing common patterns and developing strategies.

Recognizing patterns is a valuable skill not just in mathematics but also in logical reasoning and data analysis.