In the following number relationship question, four number pairs are given: (143, 64), (232, 49), (719, 289) and (462, 169). In three pairs, the second number equals the square of the sum of the digits of the first number, while one pair does not satisfy this rule. Select the odd number group from the given alternatives.

Introduction / Context:
This is a number coding and odd one out question. You are given four ordered pairs: (143, 64), (232, 49), (719, 289) and (462, 169). In three of these pairs, the second number is directly related to the digits of the first number by a simple rule. One pair fails to follow this rule. Your task is to identify that exceptional pair as the odd number group.

Given Data / Assumptions:

• Pairs: (143, 64), (232, 49), (719, 289) and (462, 169).
• We suspect a relation involving the sum of digits of the first number.
• Square of a sum is computed as (sum)^2.
• In three pairs, second number equals the square of the digit sum of the first number.
• Exactly one pair does not match this pattern.

Concept / Approach:
The natural approach when you see a three digit number paired with a two or three digit number is to try operations such as sum of digits, product of digits or difference between digits, then see if the second number is a square or cube of that result. Here, when we take the sum of digits of the first number in each pair and square it, we see an exact match with the second number in three cases. In the remaining pair, the square of the digit sum does not match the second number, revealing it as the odd one out.

Step-by-Step Solution:
Step 1: Analyse (143, 64).Sum of digits of 143: 1 + 4 + 3 = 8. Square of 8 is 8^2 = 64. This equals the second number, so the pair follows the rule.Step 2: Analyse (232, 49).Sum of digits of 232: 2 + 3 + 2 = 7. Square of 7 is 7^2 = 49. This matches the second number, so the pair again follows the rule.Step 3: Analyse (719, 289).Sum of digits of 719: 7 + 1 + 9 = 17. Square of 17 is 17^2 = 289. This equals the second number, so this pair also matches the pattern.Step 4: Analyse (462, 169).Sum of digits of 462: 4 + 6 + 2 = 12. Square of 12 is 12^2 = 144, not 169. The given second number 169 equals 13^2, not 12^2. Therefore this pair does not fit the rule.Step 5: Identify the odd group.Because (462, 169) is the only pair where the second number is not equal to the square of the sum of the digits of the first number, it is the odd one out.

Verification / Alternative check:
You can verify quickly by writing the sum of digits and the corresponding square alongside each pair. For 143 you get 8 and 64, for 232 you get 7 and 49, and for 719 you get 17 and 289. In all these cases the second number matches the square. For 462, however, the sum of digits is 12 but the second number is 169, which is 13^2. This mismatch confirms that the last pair breaks an otherwise perfect pattern, so it must be the odd group.

Why Other Options Are Wrong:
(143, 64): Follows the rule because 64 is equal to (1 + 4 + 3)^2.
(232, 49): Follows the rule because 49 is equal to (2 + 3 + 2)^2.
(719, 289): Follows the rule because 289 is equal to (7 + 1 + 9)^2.

Common Pitfalls:
Some candidates may search for relationships involving multiplication of digits or direct arithmetic between the two numbers, which can be more complicated and lead to confusion. Others may miscalculate the sum of digits or the square, especially under time pressure. To avoid errors, always perform sums and squares carefully and compare them clearly with the second number. Recognising common squares like 64, 49, 144 and 169 can also speed up your reasoning.

Final Answer:
The odd number group is (462, 169), because in this pair the second number is not equal to the square of the sum of the digits of the first number, whereas in all the other pairs it is.