In this perfect cube pattern question, select the number which is not a perfect cube among 512, 216, 343 and 719.

Introduction / Context:
This odd one out problem again focuses on perfect cubes, a very common concept in quantitative aptitude. Recognising cube numbers quickly helps in mental math and pattern detection. You are given four numbers, three of which are cubes of integers, and you need to identify the non cube number that does not fit the pattern.

Given Data / Assumptions:

• The numbers are 512, 216, 343 and 719.
• We use the definition that a perfect cube is n^3 for some integer n.
• We recall or compute standard cube values for small integers.

Concept / Approach:
The common technique is to match each number with known cubes: 6^3, 7^3, 8^3, 9^3 and so on. If the number exactly equals one of these cubes, it is a perfect cube. Otherwise, it is not. Three of the given values will fit into cube tables, and one will not, which becomes the odd one out.

Step-by-Step Solution:
Step 1: Check 216. Since 6 * 6 * 6 = 216, we have 6^3 = 216, so 216 is a perfect cube. Step 2: Check 343. Since 7 * 7 * 7 = 343, we have 7^3 = 343, so 343 is also a perfect cube. Step 3: Check 512. Since 8 * 8 * 8 = 512, we have 8^3 = 512, so 512 is another perfect cube. Step 4: Check 719. Nearby cubes are 8^3 = 512 and 9^3 = 729. The number 719 lies between 512 and 729 and does not match any exact cube value. Step 5: Therefore 512, 216 and 343 are perfect cubes, while 719 is not. Hence 719 is the odd one out.

Verification / Alternative check:
We can verify further by trying to factor 719. 719 is not divisible by 2, 3 or 5. Testing 7 gives remainder 5, and 11, 13 or 17 do not divide it evenly. Its prime factorisation cannot be arranged into triples of the same prime, so it cannot be written as n^3. This prime based check aligns with the observation that it lies between 8^3 and 9^3 without being equal to any cube.

Why Other Options Are Wrong:
512 is wrong as the odd one out because it is 8^3 and clearly a perfect cube. 216 is wrong as the odd one out because it is 6^3, a cube number. 343 is wrong as the odd one out because it is 7^3 and also a perfect cube.

Common Pitfalls:
Sometimes candidates do not remember cube values and guess based on size alone. Another error is to treat 719 as special simply because it is not an obvious product of small numbers, without confirming the cube status of the others. Learning standard cubes at least up to 15^3 makes questions like this very straightforward and avoids such confusion.

Final Answer:
The only number that is not a perfect cube, and therefore the odd one out, is 719.