In this number-pair odd-man-out question, select the pair that does not follow the square relationship between the numbers.

Introduction / Context:
This is a number-pair classification question that focuses on the relationship between two numbers in each pair. In many reasoning tests, such pairs are used to represent operations like squaring, cubing or multiplying by a constant. Three of the pairs here show a perfect square relationship, while one pair does not. The task is to identify which pair breaks that pattern and is therefore the odd-man-out.

Given Data / Assumptions:

• Number pairs: 11 – 121, 12 – 144, 15 – 225 and 13 – 171.
• The first number in each pair is on the left, and the second number is on the right.
• We are looking for a consistent mathematical operation linking the two numbers.

Concept / Approach:
When the right-hand number is noticeably larger than the left-hand number, a natural pattern to test is whether the second number equals the square of the first. Squaring is a very common pattern in such questions because it creates nicely recognisable results. We therefore compute the square of each left-hand number and compare it with the right-hand number to see which pairs follow this relationship exactly.

Step-by-Step Solution:
Step 1: Analyse 11 – 121. Compute 11^2. We have 11^2 = 11 * 11 = 121. This matches the right-hand number exactly, so this pair follows the relation second number = square of first number. Step 2: Analyse 12 – 144. Compute 12^2. We have 12^2 = 12 * 12 = 144. Again, the right-hand number equals the square of the left-hand number. Step 3: Analyse 15 – 225. Compute 15^2. We have 15^2 = 15 * 15 = 225. This continues the same pattern. Step 4: Analyse 13 – 171. Compute 13^2. We have 13^2 = 13 * 13 = 169. However, the right-hand number given in the pair is 171, which is not equal to 169. Step 5: Summarise the pattern. Three pairs satisfy the rule second number = first number squared. One pair does not. Step 6: Conclude that 13 – 171 is the odd pair because its second number is not the correct square of the first number.

Verification / Alternative check:
We can check whether 171 has any other simple relation with 13, such as 13 * 13 + 2 or 13 * 13 + some constant. However, the other pairs match exact squares without any additional constant. Introducing an extra offset only for one pair would break the clean symmetry. Therefore, the only consistent and elegant rule is that the second number should be the exact square of the first. Under that rule, 13 – 171 fails and must be the odd-man-out.

Why Other Options Are Wrong:
The pair 11 – 121 is not odd because 121 is exactly 11^2, so it follows the main pattern. The pair 12 – 144 fits since 144 = 12^2. The pair 15 – 225 also fits because 225 = 15^2. These three pairs are therefore similar and should not be chosen as the answer. Only 13 – 171 breaks the square relationship by giving 171 instead of 169, so the others are all correct examples of the rule rather than exceptions to it.

Common Pitfalls:
One common error is simple calculation mistakes when squaring numbers, especially 13. If a learner misremembers 13^2 as 171 instead of 169, they might falsely think all four pairs follow the same rule and get confused. To avoid such mistakes, always compute squares carefully, or memorise common squares such as 11^2 = 121, 12^2 = 144, 13^2 = 169 and 15^2 = 225. Another pitfall is looking for unnecessary complex patterns when a straightforward square relationship explains most of the data.

Final Answer:
The odd number pair is 13 – 171, because in this pair the second number is not the exact square of the first number, whereas in all the other pairs the second number equals the square of the first.