In the following list of number pairs, three pairs follow the same numerical rule between the first and the second number. Which number pair is the odd one out that does not follow the common rule?

Introduction / Context:
This question tests pattern recognition in pairs of numbers. In many reasoning examinations, you are given several ordered pairs where three follow a hidden rule and one does not. Your task is to carefully identify the rule that connects the first and second element in most pairs, and then pick the odd pair that breaks this rule. This kind of question strengthens analytical thinking and the ability to work with squares and simple operations like addition.

Given Data / Assumptions:

• The number pairs given are: 4 : 18, 7 : 51, 9 : 83, 6 : 34, and 8 : 66.
• Exactly one pair is odd and does not follow the common pattern.
• We assume the same rule should connect the first and second number in all normal pairs.
• We look for a simple arithmetic rule such as using squares and small additions.

Concept / Approach:
A common approach in such pair questions is to check whether the second number can be derived from the first by applying a standard operation like squaring, doubling, or adding a fixed constant. Since the second number is significantly larger than the first, squaring the first number is a good first guess. After squaring, we check if adding or subtracting a small number such as 1, 2 or 3 converts the squared value into the second number. If the same rule works for most pairs, the one pair that fails this rule will be the odd pair.

Step-by-Step Solution:
Check the pair 4 : 18. Square the first number: 4^2 = 16. Add 2 to it: 16 + 2 = 18, which matches the second number. Check the pair 7 : 51. Square the first number: 7^2 = 49. Add 2 to it: 49 + 2 = 51, which again matches the second number. Check the pair 9 : 83. Square the first number: 9^2 = 81. Add 2 to it: 81 + 2 = 83, which matches perfectly. Check the pair 8 : 66. Square the first number: 8^2 = 64. Add 2 to it: 64 + 2 = 66, which also fits the same rule. Now check the pair 6 : 34. Square the first number: 6^2 = 36. Add 2 to it: 36 + 2 = 38, but the given second number is 34, so this pair does not follow the rule. Thus the common rule is: second number = (first number)^2 + 2, and only one pair fails to satisfy it.

Verification / Alternative check:
We can verify the pattern once more clearly. For 4, 7, 8 and 9, we have second numbers 18, 51, 66 and 83 respectively. When we compute 4^2 + 2, 7^2 + 2, 8^2 + 2 and 9^2 + 2, we get exactly 18, 51, 66 and 83. This shows that the rule holds consistently for four pairs. There is no other simple rule that fits all five pairs simultaneously. Therefore, the pair where the second number deviates from this formula must be the odd one. This additional check confirms that our reasoning is robust.

Why Other Options Are Wrong:
The pair 4 : 18 is correct because 4^2 + 2 = 16 + 2 = 18. The pair 7 : 51 is correct since 7^2 + 2 = 49 + 2 = 51. The pair 9 : 83 is correct because 9^2 + 2 = 81 + 2 = 83. The added option 8 : 66 also fits, as 8^2 + 2 = 64 + 2 = 66. These four pairs all follow the same pattern. Only 6 : 34 does not satisfy this formula, so it is the odd one.

Common Pitfalls:
A common mistake is to look only at simple differences such as second minus first and try to see a pattern in those raw differences, which does not work well here. Another pitfall is to consider unrelated operations like multiplying by 3 or adding random constants without systematically testing squares. Some candidates also stop after checking only one or two pairs and misidentify the pattern. The correct method is to check all pairs, test an operation like squaring, and see if one consistent rule works for most pairs.

Final Answer:
The only pair that does not satisfy the rule second number = (first number)^2 + 2 is 6 : 34, so this is the odd number pair.