In the pairs 1 : 8, 27 : 64, 125 : 218 and 323 : 512, find the odd pair based on perfect cube patterns.

Introduction / Context:
Odd one out questions using number pairs often hide patterns related to squares, cubes or sequences. Each pair may represent related mathematical values such as consecutive powers or values in a numerical progression. In this problem, each option is a pair of numbers separated by a colon, and we must identify which pair does not fit the pattern that the other pairs follow. The pattern here is based on perfect cubes in the first component of each pair.

Given Data / Assumptions:
The given pairs are 1 : 8, 27 : 64, 125 : 218 and 323 : 512. We use standard cube values of small integers. We focus first on the first number in each pair and check whether it is a perfect cube.

Concept / Approach:
The main concept is the idea of perfect cubes. A number n^3, where n is an integer, is called a perfect cube. We recognize standard cubes such as 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64 and 5^3 = 125. When we examine the pairs, we notice that in three of them, the first number is clearly a perfect cube, but in one pair it is not. That pair is the odd one out. While the second numbers also relate to cubes, the simplest and most consistent pattern here is present in the first number of each pair.

Step-by-Step Solution:
Step 1: For 1 : 8, the first number 1 is 1^3, so it is a perfect cube. Step 2: For 27 : 64, the first number 27 is 3^3, so it is also a perfect cube. Step 3: For 125 : 218, the first number 125 is 5^3, which is again a perfect cube. Step 4: For 323 : 512, the first number 323 is not a perfect cube. 6^3 = 216 and 7^3 = 343, so 323 lies between these cubes and does not match any n^3. Step 5: Therefore, three pairs have a perfect cube as their first element, whereas the pair 323 : 512 does not, making it the odd one out.

Verification / Alternative check:
We can verify by writing out the first numbers in factorized form. 1 = 1 * 1 * 1, 27 = 3 * 3 * 3 and 125 = 5 * 5 * 5 are clear cubes. For 323, we can try nearby cubes: 6^3 = 216 and 7^3 = 343, neither equal to 323, confirming that 323 is not a cube. The second numbers 8, 64 and 512 are themselves cubes of 2, 4 and 8 respectively, which supports the idea that the question is built around cube patterns.

Why Other Options Are Wrong:
1 : 8 is not the odd pair because the first number 1 is a perfect cube and fits the cube based pattern. 27 : 64 is not the odd pair because 27 is also a perfect cube and aligns with the same logic. 125 : 218 is not the odd pair because 125 is again a perfect cube and behaves similarly to 1 and 27 in this context. Only in 323 : 512 is the first number not a perfect cube, which sets it apart.

Common Pitfalls:
Many candidates get distracted by the second numbers or by minor differences like 218 instead of 216, and search for very complex relationships between both members of each pair. Another mistake is to ignore cube values above 5^3 or 6^3, which makes it harder to quickly categorize numbers like 323. Remember that exam setters often use well known cube numbers such as 1, 27 and 125 to signal the underlying pattern, so recognizing these early can save valuable time.

Final Answer:
The pair whose first number is not a perfect cube and is therefore the odd one out is 323 : 512.