In this number pair question, you must identify the odd pair that does not follow the same rule as the others among the given options.

Introduction / Context:
This problem is from the odd one out section using number pairs. Each option consists of two numbers separated by a dash. In three of the pairs, the two numbers follow a consistent mathematical relationship, while in one pair that relationship is broken. Your goal is to find the pair that does not fit the common pattern.

Given Data / Assumptions:

• Each option shows a pair of numbers, for example 122 – 1331.
• The second number in each pair is relatively large compared to the first number.
• We suspect powers such as squares and cubes may be involved because of the size of the second numbers.
• Exactly one pair will violate an otherwise consistent rule followed by the other three pairs.

Concept / Approach:
First, we examine the larger second numbers to see whether they are perfect cubes, as values like 1331, 2197, 2744 and 4913 often appear in cube tables. After confirming this, we then relate the first number to the cube root of the second number and look for a common formula. Once that formula is identified, the pair that fails to satisfy it will be the odd one out.

Step-by-Step Solution:
Step 1: Find the cube roots of the second numbers: 1331 = 11^3, 2197 = 13^3, 2744 = 14^3 and 4913 = 17^3.Step 2: Relate each first number to the corresponding cube root n.Step 3: For 122 – 1331, with cube root n = 11, compute n^2 + 1 = 11^2 + 1 = 121 + 1 = 122. This matches the first number.Step 4: For 197 – 2744, with n = 14, compute n^2 + 1 = 14^2 + 1 = 196 + 1 = 197, again matching the first number.Step 5: For 290 – 4913, with n = 17, compute n^2 + 1 = 17^2 + 1 = 289 + 1 = 290, which matches the first number.Step 6: For 173 – 2197, with n = 13, compute n^2 + 1 = 13^2 + 1 = 169 + 1 = 170, but the first number is 173, not 170. So this pair breaks the pattern.

Verification / Alternative check:
We can summarise the relationship for the valid pairs as: second number = n^3 and first number = n^2 + 1 for n equal to 11, 14 and 17. Applying the same rule to n = 13 should yield first number 170, not 173. Therefore, 173 – 2197 does not align with the established formula, while the other three pairs do. This confirms that 173 – 2197 is the odd pair.

Why Other Options Are Wrong:
Option A, 122 – 1331, fits the rule with n = 11, because 122 = 11^2 + 1 and 1331 = 11^3. Option C, 197 – 2744, fits with n = 14, since 197 = 14^2 + 1 and 2744 = 14^3. Option D, 290 – 4913, fits with n = 17, where 290 = 17^2 + 1 and 4913 = 17^3. Because these three options follow a consistent square plus one and cube pattern, they cannot be the odd one out.

Common Pitfalls:
Some candidates may focus only on the cube nature of the second numbers and mistakenly think there is no difference between the pairs. Others may not test the relationship between the first number and the cube root of the second number. Always check for a precise numeric formula that works across multiple options instead of relying on a single observation.

Final Answer:
The pair that does not follow the pattern first number = n^2 + 1 and second number = n^3 is 173 – 2197, so option B is correct.