Classification – Odd one out (three perfect cubes vs one perfect square) Among the following numbers, three are exact perfect cubes while one is a perfect square (but not a cube). Identify the number that does not belong to the “perfect cube” group and mark it as the odd one out.

Introduction / Context:
Odd-one-out questions in number classification frequently exploit familiar lists of powers. If you instantly recognize the standard cube sequence (1, 8, 27, 64, 125, 216, …) and the standard square sequence (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …), you can quickly isolate the single element that does not share the majority property. This item is designed to check whether you can distinguish between “perfect cubes” and “perfect squares” at a glance, and whether you can confirm your first impression with a short calculation.

Given Data / Assumptions:

• Set: 27, 64, 125, 144
• Goal: identify which one is not a perfect cube while the others are.
• No tricks with units or bases; all numbers are in base-10.

Concept / Approach:
Recall small cubes: 3^3 = 27, 4^3 = 64, 5^3 = 125. Also recall that 144 is a well-known square (12^2) that is not a perfect cube. The approach is to match each candidate with a nearby known power and verify equality rather than proximity. If it equals n^3 exactly, it is a cube; if it equals m^2 exactly (but not an exact cube), it is not part of the “cube” group.

Step-by-Step Solution:
27 = 3^3 → perfect cube.64 = 4^3 (also equals 8^2, but crucially it is a cube) → perfect cube.125 = 5^3 → perfect cube.144 = 12^2 → perfect square and not equal to any k^3 for integer k → not a cube.

Verification / Alternative check:
Check nearest cubes around 144: 5^3 = 125 and 6^3 = 216. Since 144 is strictly between these and not equal to either, it cannot be a cube. Independently, 144 = 12 * 12 confirms it as a square. Therefore, the three cubes (27, 64, 125) form the majority pattern, and 144 is the outsider.

Why Other Options Are Wrong:

• 27: Exactly 3^3 → fits the “cube” pattern.
• 64: Exactly 4^3 → fits the “cube” pattern.
• 125: Exactly 5^3 → fits the “cube” pattern.
• None of these: There is a clear odd one (144), so this distractor is not correct.

Common Pitfalls:
A frequent mistake is to rely on “familiarity” without explicit confirmation. While 64 is also a square (8^2), it remains a perfect cube, so it still belongs with 27 and 125 in the “cube” group. The task asks for the element that is not a cube at all, and that is 144.

Final Answer:
144