In the number analogy "5 : 124 :: 6 : ____", which number correctly completes the pattern using a cube minus one rule?

Introduction / Context:
This analogy problem involves a transformation from a small integer to a larger number using powers. The pair "5 : 124" indicates that 124 is derived from the number 5 through a specific operation. You must determine this operation and apply it to 6 to find the missing number in the second pair. Recognising patterns involving cubes and simple adjustments is a common requirement in aptitude tests.

Given Data / Assumptions:

• First pair: 5 corresponds to 124.
• Second pair: 6 corresponds to an unknown number.
• Options: 215, 216, 217, 220, 125.
• We suspect use of cube values and addition or subtraction by 1.

Concept / Approach:
Start by checking powers of 5. Compute 5^3 = 125. The given number 124 is exactly one less than 125. Thus, we have 5^3 minus 1 equals 124. This suggests that the rule is to cube the number and then subtract one. To apply the same rule to 6, we compute 6^3 and subtract one from the result.

Step-by-Step Solution:

Step 1: Compute the cube of 5. 5^3 = 5 * 5 * 5 = 125. Step 2: Subtract one from the cube: 125 minus 1 = 124, which matches the given pair. Step 3: Conclude that the transformation is "n maps to n^3 minus 1". Step 4: Apply this to 6. Compute 6^3 = 6 * 6 * 6 = 216. Step 5: Subtract one from the cube of 6. 216 minus 1 = 215. Step 6: Choose 215 from the options as it exactly fits the rule.

Verification / Alternative check:
Verify that using 6^3 directly would give 216, which is an option but does not use the minus 1 adjustment that was applied in the first pair. Similarly, 217 and 220 do not follow from 6^3 by subtracting a simple constant that also works for 5^3. The number 125 is simply 5^3 and would break the analogy since 124 is not equal to 5^3. Thus, 215 is uniquely consistent with the cube minus one transformation.

Why Other Options Are Wrong:
216 equals 6^3 but would only be correct if the first pair used 5^3 without subtraction, which is not the case. 217 and 220 are not 6^3 minus a small integer that matches the pattern of 124. The number 125 represents 5^3 but is not part of the original pair and does not correspond to the mapping of 6. Therefore, only 215 preserves the mapping n to n^3 minus 1 for both given numbers.

Common Pitfalls:
A common mistake is to recognize the cube but forget the minus one operation, leading to the selection of 216. Another pitfall is to look for complicated formulas instead of testing simple adjustments like plus or minus one. In many exam questions, simple patterns with small constants are preferred, so always test cube plus or minus one where it appears relevant.

Final Answer:
Applying the rule "n maps to n^3 minus 1", 5 maps to 124 and 6 maps to 215, completing the analogy as "5 : 124 :: 6 : 215".