In the pairs 12 : 144, 20 : 400, 15 : 225 and 17 : 285, find the odd pair based on the square relationship between the numbers.

Introduction / Context:
This question tests your understanding of basic number patterns, specifically the relationship between a number and its square. In many quantitative aptitude exams, number pairs are constructed such that the second number is a square, cube or some other power of the first. Your task is to recognize this pattern and then pick the pair that does not follow it, which is the odd one out.

Given Data / Assumptions:
The given pairs are 12 : 144, 20 : 400, 15 : 225 and 17 : 285. Each pair is written in the form first number : second number. We suspect that the second number may be related to the square of the first number. We work with ordinary integer squares, that is n^2 = n * n.

Concept / Approach:
When examining number pairs, a natural first step is to check whether the second number equals the square of the first. Squaring is common because it is easy to compute for small integers and gives memorable results. If three out of four pairs clearly fit the relationship second number = (first number)^2 and one does not, then the pair that fails this rule is the odd one out. This direct approach avoids overcomplicating the puzzle.

Step-by-Step Solution:
Step 1: For 12 : 144, compute the square of 12. We have 12^2 = 12 * 12 = 144. So this pair matches the rule second number = first number squared. Step 2: For 20 : 400, compute the square of 20. We have 20^2 = 20 * 20 = 400. This also fits the same rule. Step 3: For 15 : 225, compute the square of 15. We have 15^2 = 15 * 15 = 225, so this pair again satisfies second number = first number squared. Step 4: For 17 : 285, compute the square of 17. We have 17^2 = 17 * 17 = 289, not 285. Therefore, the second number in this pair is not equal to the square of the first number. Step 5: Since the first three pairs strictly follow the square relationship and the fourth pair does not, 17 : 285 is the odd one out.

Verification / Alternative check:
Recalculate each square to avoid mistakes: 12^2 is 144, 20^2 is 400, 15^2 is 225 and 17^2 is 289. Compare each result with the given second number in the pair. For the first three pairs the values match exactly, confirming the pattern. Only in the pair with 17, the second number 285 is four less than 289, which shows that it has been chosen deliberately to break the rule and serve as the odd option.

Why Other Options Are Wrong:
12 : 144 is not the odd pair because 144 is exactly 12^2. 20 : 400 is not the odd pair because 400 is exactly 20^2. 15 : 225 is not the odd pair because 225 is exactly 15^2. 17 : 285 is the odd pair because 285 is not equal to 17^2, which should be 289.

Common Pitfalls:
Some learners may focus on differences or ratios between the numbers instead of checking the square relationship, which can hide the simple logic. Under exam pressure, it is easy to miscalculate 17^2 and accept 285 as correct without verifying that 17 * 17 actually equals 289. To avoid such errors, always compute squares carefully and compare exactly with the given numbers, especially when the mismatch is small.

Final Answer:
The pair that does not follow the rule second number = first number squared and is therefore the odd one out is 17 : 285.