In this numerical odd-man-out question, the numbers 484, 529, 625 and 566 are given. Three of these numbers are perfect squares of integers, while one number is not a perfect square. Which number is the odd one out on the basis of the perfect square property?

Introduction / Context:
This question is an odd-man-out problem from the number reasoning section. The options are 484, 529, 625 and 566. Many aptitude questions use the concept of perfect squares and cubes to create patterns. Here, three of the given numbers are perfect squares of integers, while one is not. The task is to identify which number is not a perfect square and therefore does not belong to the main group.

Given Data / Assumptions:
The numbers given are 484, 529, 625 and 566.
A perfect square is any number that can be written as n^2 for some integer n.
We assume standard integer arithmetic with no approximations.
Exactly one number in the list is not a perfect square.

Concept / Approach:
To solve this, we recall or compute the squares of nearby integers. For example, 22^2 = 484, 23^2 = 529, 24^2 = 576 and 25^2 = 625. By comparing each option with known square values, we can check which numbers match exactly and which one does not. The number that does not equal any n^2 for integer n is the odd one out. This type of question emphasizes the importance of remembering square values for competitive exams.

Step-by-Step Solution:
Step 1: Check 484. We know that 22^2 = 22 * 22 = 484, so 484 is a perfect square. Step 2: Check 529. We know that 23^2 = 23 * 23 = 529, so 529 is a perfect square. Step 3: Check 625. We know that 25^2 = 25 * 25 = 625, so 625 is a perfect square. Step 4: Check 566. It lies between 23^2 = 529 and 24^2 = 576 and does not match any standard square value. Step 5: Conclude that 484, 529 and 625 are perfect squares, while 566 is not, making 566 the odd one out.
Verification / Alternative check:
As an alternative, calculate approximate square roots. The square root of 484 is exactly 22, of 529 is exactly 23, and of 625 is exactly 25. These are all integers, confirming that the numbers are perfect squares. For 566, the square root would lie between 23 and 24, but it is not an integer. Only numbers with integer square roots are perfect squares, so 566 fails the test. This second method independently confirms that 566 is not a perfect square.

Why Other Options Are Wrong:
484 is a perfect square (22^2), so it matches the main pattern and is not the odd one out.
529 is a perfect square (23^2), so it also clearly belongs to the same group as 484 and 625.
625 is a perfect square (25^2), which again fits the pattern of perfect squares in the list.
566 is not equal to n^2 for any integer n and therefore breaks the pattern, making it the correct odd-man-out choice.

Common Pitfalls:
One common mistake is to confuse 566 with 576, which actually is 24^2. The similarity of the digits can mislead candidates who are not careful. Another pitfall is to rely solely on a rough sense of magnitude without verifying the exact value. To avoid such errors, it is very helpful to memorize square values at least up to 25^2. Always check each number against these known values instead of trusting intuition alone.

Final Answer:
The number that is not a perfect square and is therefore the odd one out is 566.