In the following number pair reasoning question, four pairs of numbers are given: (6, 612), (5, 521), (4, 46) and (7, 543). In three pairs, the second number is formed by rearranging the digits of the cube of the first number, while one pair breaks this pattern. Find the odd word or number pair from the given alternatives.

Introduction / Context:
This question belongs to the number reasoning category with a focus on relationships between numbers in ordered pairs. You are given four pairs: (6, 612), (5, 521), (4, 46) and (7, 543). In such questions, the second number often comes from a transformation of the first number, such as squaring, cubing or rearranging digits. Here, the second number in most pairs is obtained by rearranging the digits of the cube of the first number. One pair fails to follow this cube based digit rearrangement pattern and must be selected as the odd one out.

Given Data / Assumptions:

• Number pairs: (6, 612), (5, 521), (4, 46) and (7, 543).
• We are looking for a consistent rule linking the first number to the second.
• The cube of a number is computed as n^3 = n * n * n.
• In three pairs, the second number is a rearrangement of the digits of n^3.
• In one pair, this rule does not hold.

Concept / Approach:
The approach is to compute the cube of each first number and then compare its digits with those of the corresponding second number. If the digits match in some rearranged order, then the pair follows the rule. If the digits differ, then that pair breaks the pattern. Because the numbers involved are small, calculating cubes like 4^3, 5^3, 6^3 and 7^3 is quick, making this an accessible but slightly tricky number puzzle.

Step-by-Step Solution:
Step 1: Analyse the pair (6, 612).Compute 6^3 = 6 * 6 * 6 = 216. The digits of 216 are 2, 1 and 6, which can be rearranged to form 612. So this pair follows the pattern.Step 2: Analyse the pair (5, 521).Compute 5^3 = 125. The digits 1, 2 and 5 can be rearranged to form 521. So this pair also follows the rule.Step 3: Analyse the pair (4, 46).Compute 4^3 = 64. The digits 6 and 4 can be rearranged to form 46. Thus, this pair again follows the pattern.Step 4: Analyse the pair (7, 543).Compute 7^3 = 343. The digits are 3, 4 and 3, but the second number is 543 which uses digits 5, 4 and 3. The digit 5 does not appear in 343, so this pair does not follow the rule.Step 5: Identify the odd pair.The pair (7, 543) is inconsistent with the cube digit rearrangement pattern, so it is the odd one out.

Verification / Alternative check:
An alternative verification is to list quickly the cubes involved: 4^3 = 64, 5^3 = 125, 6^3 = 216 and 7^3 = 343. For the first three pairs, you can see that the second number uses exactly the same digits as the cube, just in a different order. For the fourth pair, the cube 343 has no digit 5, yet the second number 543 includes a 5. This contradiction confirms that the fourth pair breaks the pattern and should be selected.

Why Other Options Are Wrong:
6 , 612: Follows the pattern because 612 is a rearrangement of 6^3 = 216.
5 , 521: Follows the pattern because 521 is a rearrangement of 5^3 = 125.
4 , 46: Follows the pattern because 46 is a rearrangement of 4^3 = 64.

Common Pitfalls:
Candidates sometimes search for direct arithmetic operations, such as multiplication or addition, between the two numbers and miss digit based relationships. Another error is miscalculating cubes, especially if you are not comfortable with mental arithmetic. To avoid mistakes, always compute the small cubes carefully and consciously compare digits. Recognising digit rearrangement patterns is an important skill for many reasoning questions.

Final Answer:
The odd number pair is 7 , 543, because its second number is not formed by rearranging the digits of the cube of the first number, whereas the other three pairs follow this rule.