In this perfect power based odd one out question, select the number that is a perfect cube while the others are perfect squares among 256, 289, 343 and 144.

Introduction / Context:
This question belongs to the number system category and focuses on perfect squares and perfect cubes. You must identify the number that is a perfect cube while the rest are perfect squares. Recognizing common squares and cubes of small integers is an essential skill for many competitive examinations.

Given Data / Assumptions:

• The given numbers are 256, 289, 343 and 144.
• We consider perfect squares n^2 and perfect cubes n^3 for integer values of n.
• The pattern contrasts square numbers against a cube number.

Concept / Approach:
The idea is to factor each number or recall standard squares and cubes from memory. If three numbers are perfect squares and one number is a perfect cube but not a perfect square, that cube is the odd one out. Knowledge of squares up to at least 20 and cubes up to at least 10 is very useful here.

Step-by-Step Solution:
Step 1: 256 can be recognized as 16^2, so it is a perfect square.Step 2: 289 is equal to 17^2, so it is also a perfect square.Step 3: 144 is equal to 12^2, which again makes it a perfect square.Step 4: 343 is equal to 7^3. It is not equal to any integer squared value, so it is a perfect cube but not a perfect square.Step 5: Since 256, 289 and 144 are perfect squares and 343 is a perfect cube, 343 does not share the same property and is therefore the odd one out.
Verification / Alternative check:
You can verify by checking nearby squares and cubes. For example, 6^3 = 216, 7^3 = 343 and 8^3 = 512, which confirms that 343 is a cube. Attempting to find an integer n such that n^2 = 343 fails, since 18^2 = 324 and 19^2 = 361. Therefore, 343 is indeed a cube but not a square, whereas the others are confirmed squares.

Why Other Options Are Wrong:
256 is not the odd one out because it is a perfect square and so matches 289 and 144.
289 is also a perfect square and thus part of the majority group of square numbers.
144 is yet another perfect square, aligning with 256 and 289 and not unique in its property.

Common Pitfalls:
Some candidates might think of 256 as special because it is also 2^8, which is a higher power, but that is not the intended comparison. The question clearly focuses on whether numbers are squares or cubes. Another mistake is to forget or misremember key squares and cubes. Regular revision of a table of squares and cubes up to at least 20 helps avoid such errors.

Final Answer:
The only number that is a perfect cube while the others are perfect squares is 343, so 343 is the odd one out.