Finding the Odd Number Triads Pattern

The question asks us to identify the number triad that is different from the others. Three of the given triads follow the same rule, while one does not. We need to find the underlying pattern that connects the three numbers in each triad.

Let's look closely at the given number triads:

• 4 ∶ 16 ∶ 12
• 15 ∶ 225 ∶ 210
• 11 ∶ 121 ∶ 110
• 7 ∶ 49 ∶ 40

Analyzing Number Triads to Find the Rule

Let the first number in the triad be \(n\), the second number be \(n_2\), and the third number be \(n_3\). We are told that \(n_2\) and \(n_3\) are formed by operations on \(n\).

Let's analyze the first triad: 4 ∶ 16 ∶ 12

• The first number \(n\) is 4.
• The second number \(n_2\) is 16. Notice that \(16 = 4^2\). This suggests the second number might be the square of the first number.
• The third number \(n_3\) is 12. How is 12 related to 4 and 16? We can see that \(16 - 4 = 12\). This suggests the third number might be the second number minus the first number.

Based on this observation, let's hypothesize a rule: For a triad \(n : n_2 : n_3\), the rule is \(n_2 = n^2\) and \(n_3 = n_2 - n\).

Testing the Pattern on Other Triads

Now we will apply the hypothesized rule \(n : n^2 : n^2 - n\) to the remaining triads to see if they follow this pattern.

Triad 2: 15 ∶ 225 ∶ 210

• First number \(n = 15\).
• Expected second number: \(n^2 = 15^2 = 225\). This matches the given second number.
• Expected third number: \(n^2 - n = 225 - 15 = 210\). This matches the given third number.

This triad follows the pattern.

Triad 3: 11 ∶ 121 ∶ 110

• First number \(n = 11\).
• Expected second number: \(n^2 = 11^2 = 121\). This matches the given second number.
• Expected third number: \(n^2 - n = 121 - 11 = 110\). This matches the given third number.

This triad also follows the pattern.

Triad 4: 7 ∶ 49 ∶ 40

• First number \(n = 7\).
• Expected second number: \(n^2 = 7^2 = 49\). This matches the given second number.
• Expected third number: \(n^2 - n = 49 - 7 = 42\).
• The given third number is 40. This does not match the expected third number (42).

This triad does not follow the pattern.

Identifying the Odd Number Triad

We found that the first three triads (4:16:12, 15:225:210, and 11:121:110) all adhere to the rule where the second number is the square of the first number, and the third number is the result of subtracting the first number from the second number (\(n : n^2 : n^2 - n\)). The fourth triad (7:49:40) follows the rule for the second number (\(7^2 = 49\)), but the third number (40) does not match the result of subtracting the first number from the second number (\(49 - 7 = 42\)).

Therefore, the number triad 7 ∶ 49 ∶ 40 is the one that is different or "odd" compared to the others.

Summary Table of Triad Analysis

Revision Table: Number Pattern Types

Recognizing different types of numerical patterns is crucial for solving logical reasoning problems like the odd number triads. Here are some common pattern types:

Additional Information: Approaching Pattern Recognition Questions

When faced with pattern recognition questions involving numbers, especially in formats like triads or sequences, adopt a systematic approach:

• Initial Scan: Look at the numbers quickly. Do they seem to be increasing or decreasing rapidly (suggesting multiplication or powers) or slowly (suggesting addition or subtraction)?
• Check Basic Operations: Test for simple relationships like addition, subtraction, multiplication, or division between consecutive numbers or between the first number and the others in a set.
• Consider Powers: Check if numbers are squares, cubes, or close to them. Look for relationships involving \(n^2\), \(n^3\), etc.
• Look at Differences/Ratios: Calculate the differences or ratios between consecutive numbers. Do these differences or ratios form a new, simpler pattern?
• Combine Operations: If simple operations don't work, look for rules that combine operations, like multiplying by a number and then adding/subtracting a constant, or the pattern found in this problem (\(n : n^2 : n^2 - n\)).
• Apply and Verify: Once you think you've found a pattern, apply it to all the given examples to ensure it consistently holds true for most, identifying the one that deviates.
• Eliminate Options: If working with multiple-choice options, sometimes testing the rule on the options can help confirm or reject your suspected pattern.

Practice is key to developing intuition for recognizing common numerical patterns quickly.