In the following question on number relationships, four number pairs are given. In each pair, the first number is related to the second number. Carefully examine the pattern in each pair and then select the odd number pair from the given alternatives.

Introduction / Context:
This aptitude question tests your ability to identify hidden numerical relationships between two numbers in a pair. You are given four number pairs where the first number is a perfect square and the second number is close to the square root of the first. Your task is to find the one pair that does not follow the same consistent pattern as the others and mark it as the odd number pair.

Given Data / Assumptions:

• Number pairs: 1444 – 37, 2809 – 54, 2209 – 46 and 6084 – 77.
• The first number in each pair appears to be a large integer, often close to a perfect square.
• The second number seems to be related to the square root of the first number.
• Exactly one pair will break the pattern followed by the other three pairs.

Concept / Approach:
The most natural approach is to check whether the first number in each pair is a perfect square and then compare its square root with the second number. If three pairs satisfy a rule like second number = (square root of first number) minus 1, and one pair differs, that differing pair will be the odd one out. This technique of matching perfect squares and simple adjustments around their roots is very common in number pair reasoning questions.

Step-by-Step Solution:
Step 1: Check the pair 1444 – 37.Compute 38^2 = 38 * 38 = 1444. So the square root of 1444 is 38. The second number is 37, which equals 38 - 1.Step 2: Check the pair 2209 – 46.Compute 47^2 = 47 * 47 = 2209. So the square root is 47 and the second number is 46, that is 47 - 1.Step 3: Check the pair 6084 – 77.Compute 78^2 = 78 * 78 = 6084. The square root is 78 and the second number is 77, again 78 - 1.Step 4: Check the pair 2809 – 54.Compute 53^2 = 53 * 53 = 2809. The square root is 53 but the second number is 54, which equals 53 + 1, not 53 - 1.Step 5: Compare all four results.In three pairs the pattern is second number = (square root of first number) minus 1, while in one pair the second number is one more than the square root. That pair is the odd one out.

Verification / Alternative check:
You can verify your conclusion by writing each pair in the form (n^2, k) and asking whether k equals n - 1 or something else. For 1444 – 37, 2209 – 46 and 6084 – 77, the second number is exactly one less than the square root of the first. Only in 2809 – 54 is the second number one more than the square root. This clean and consistent difference confirms which pair is structurally different.

Why Other Options Are Wrong:
1444 – 37: Here 1444 = 38^2 and 37 = 38 - 1, so it fits the pattern.
2209 – 46: In this case 2209 = 47^2 and 46 = 47 - 1, again matching the rule.
6084 – 77: Similarly, 6084 = 78^2 and 77 = 78 - 1, so this pair is also consistent.

Common Pitfalls:
A common mistake is to attempt complex operations like multiplication or division between the two numbers without first checking for obvious properties such as perfect squares. Another frequent error is miscalculating square values, especially for two digit bases. Memorising squares of integers from 10 to about 30 is extremely helpful for such questions and allows you to quickly see patterns like n^2 and n - 1 or n + 1. Always look for the simplest and most regular relationship first before exploring more complicated possibilities.

Final Answer:
The odd number pair is 2809 – 54, because in this pair the second number is one more than the square root of the first number, whereas in all the other pairs the second number is one less than the square root of the first number.