From the following numbers, which one is the odd one out based on being a perfect cube or not?

Introduction / Context:
This is an odd one out question that focuses on the concept of perfect cubes. Three of the given numbers are exact cubes of integers, while one number does not represent a perfect cube. Your task is to identify the number that breaks this pattern. Such questions test your familiarity with the cubes of common natural numbers and your ability to quickly recognise standard cube values.

Given Data / Assumptions:

• Given numbers: 1728, 2197, 3266, 2744, 3375.
• We consider perfect cubes of small integers such as 10, 11, 12, 13, 14 and 15.
• Three or more numbers will match these cubes and one will not.

Concept / Approach:
A perfect cube is a number that can be written as n^3 for some integer n. We recall that 12^3 = 1728, 13^3 = 2197, and 14^3 = 2744. Also, 15^3 = 3375. These values are standard and often memorised for examinations. The number 3266 does not appear in this list and is not the cube of any integer close to these. Therefore, 3266 is the odd one out because it fails the perfect cube property.

Step-by-Step Solution:

Step 1: Check 1728. Compute 12^3 = 12 * 12 * 12 = 144 * 12 = 1728, so this is a perfect cube. Step 2: Check 2197. Compute 13^3 = 13 * 13 * 13 = 169 * 13 = 2197, so this is also a perfect cube. Step 3: Check 2744. Compute 14^3 = 14 * 14 * 14 = 196 * 14 = 2744, confirming that this is a perfect cube. Step 4: Check 3375. Compute 15^3 = 15 * 15 * 15 = 225 * 15 = 3375, so this is a perfect cube. Step 5: Examine 3266. It lies between 14^3 and 15^3 but is not equal to either of them or to any nearby integer cube. Step 6: Conclude that 3266 is not a perfect cube and therefore is the odd one out.

Verification / Alternative check:
If you approximate cube roots, the cube root of 3266 would lie between 14 and 15 but will not be an integer. Since 14^3 is 2744 and 15^3 is 3375, there is no integer between 14 and 15 that can have a cube equal to 3266. This confirms that 3266 is not of the form n^3 and so does not share the important property that the other numbers share.

Why Other Options Are Wrong:
1728, 2197, 2744, and 3375 are all exact cubes of consecutive integers 12, 13, 14, and 15. They clearly fit into a neat sequence of perfect cubes and therefore cannot be the odd one out. The purpose of the question is to identify the number that does not belong to this group, which is 3266.

Common Pitfalls:
A common error is to misremember cube values or to assume that any number close to a known cube is also a cube. Another pitfall is not checking all numbers systematically and guessing too early. It is helpful to memorise the cubes of integers at least up to 15 for aptitude exams, because such patterns appear frequently.

Final Answer:
The number that does not represent a perfect cube and is therefore the odd one out is 3266.