In this odd one out question on number patterns, four numbers are given and three of them are perfect squares while one is different. Choose the number that is different from the rest.

Introduction / Context:
This question asks you to identify the odd number out among four given numbers by observing their hidden mathematical properties. Typically, three numbers follow a common pattern such as being perfect squares, perfect cubes, or prime numbers, while one number does not fit that pattern. Recognising whether numbers are squares or cubes is a very common requirement in quantitative aptitude questions.

Given Data / Assumptions:

• Given numbers: 512, 841, 576, 289. • We suspect that some or all of these may be perfect squares or perfect cubes. • The task is to find the single number that does not share the main property common to the other three.

Concept / Approach:
An effective approach is to test each number to see if it is a perfect square or a perfect cube. If three of the numbers are perfect squares and one is not, then the non square will be the odd one out. Alternatively, if three are cubes and one is not, then the non cube is the answer. We calculate or recall square and cube values around the sizes of the given numbers to determine their nature.

Step-by-Step Solution:
Step 1: Check 512. 8³ = 8 * 8 * 8 = 512, so 512 is a perfect cube, not a perfect square. Step 2: Check 841. 29² = 29 * 29 = 841, so 841 is a perfect square. Step 3: Check 576. 24² = 24 * 24 = 576, so 576 is a perfect square. Step 4: Check 289. 17² = 17 * 17 = 289, so 289 is a perfect square. Step 5: Identify the common property. Three numbers, 841, 576, and 289, are perfect squares. One number, 512, is a perfect cube but not a perfect square.

Verification / Alternative check:
We can further verify that 512 is not a square by checking approximate square roots. 22² = 484 and 23² = 529, so there is no integer n such that n² = 512. Thus, 512 is exclusively a cube, while the other three numbers are exclusively squares. The classification is clear and supports the choice of 512 as the odd one out.

Why Other Options Are Not Odd:
• 841: A perfect square (29²), which matches the pattern of being a square. • 576: A perfect square (24²), again fitting the same pattern. • 289: A perfect square (17²), also consistent with the majority property.

Common Pitfalls:
Some candidates may try to use divisibility tests or look at last digits without checking squares and cubes properly. Others may be misled by the fact that 512 ends with 2, a less common last digit for squares, and guess randomly. The safest method is to recall or quickly compute square and cube values near each number, which makes the difference between 512 and the three squares very clear.

Final Answer:
The odd number out, which is a cube while the others are squares, is 512.