Using only the second, fourth and seventh digits of the number 739142658, each used exactly once, it is possible to form a number that is the perfect square of a two digit odd number. In that case, which of the following is the second digit of that two digit odd number?

Introduction / Context:
This problem combines digit selection with knowledge of perfect squares and basic number properties. We are told to use three specific digits from a larger number, form a new number using each digit exactly once, and require that this new number be the square of a two digit odd number. The question then asks for the second digit of that two digit odd number, and we must check whether any of the listed options match or whether the correct digit is not among them.

Given Data / Assumptions:

• Original number: 739142658.
• Digits are labelled from left to right as positions 1 to 9.
• We may only use the second, fourth and seventh digits, each exactly once, to form a new number.
• The new number must be a perfect square of a two digit odd number.
• We then need the second digit (units place) of that two digit odd number.

Concept / Approach:
First, identify the second, fourth and seventh digits in the given nine digit number. Then list all permutations of these three digits to form possible three digit numbers. Next, check which of these candidates are perfect squares. A perfect square must equal n * n for some integer n. If one of the candidate numbers is a perfect square of a two digit odd number, we find that two digit number and record its second digit. Finally, we compare that digit with the provided options and decide whether it is present or if the correct response is that none of the listed digits matches.

Step-by-Step Solution:
Step 1: Write the digits of 739142658 with positions: 1:7, 2:3, 3:9, 4:1, 5:4, 6:2, 7:6, 8:5, 9:8. Step 2: The second digit is 3, the fourth digit is 1, and the seventh digit is 6. Step 3: We must form a number using digits 3, 1 and 6, each used exactly once. The possible three digit numbers are: 136, 163, 316, 361, 613 and 631. Step 4: Check which of these are perfect squares. 361 is equal to 19 * 19, so 361 is a perfect square. Step 5: 19 is a two digit odd number, so 361 satisfies all the given conditions. Step 6: The two digit odd number whose square is 361 is 19. The second digit of 19 is 9. Step 7: Now compare the digit 9 with the options given: 4, 7, 3 and none. Step 8: The digit 9 does not appear among 4, 7 or 3, so the correct choice is that none of the listed digits matches the required second digit.

Verification / Alternative check:
We can verify by checking other candidate squares in the three digit range. None of 136, 163, 316, 613 or 631 is a perfect square of an integer, while 361 is clearly 19^2. Therefore there is exactly one suitable three digit number that fits all constraints, and it leads uniquely to the two digit odd number 19 and its second digit 9. Since 9 is absent from the explicit options, the answer must be none of these.

Why Other Options Are Wrong:
Digits 4, 7 and 3 are not the second digit of the required two digit odd number 19. They may appear in the original nine digit number or be among the selected digits, but that does not make them correct for the question asked.

Common Pitfalls:
A common mistake is to assume that the second digit must be one of the digits used to form the square, which is not required by the problem statement. Another error is to overlook 361 as a well known perfect square and assume no such square exists, leading to an incorrect conclusion. Systematically checking all permutations and knowing basic square values such as 19^2 = 361 helps to avoid these pitfalls.

Final Answer:
The correct second digit is 9, which is not among the listed digits, so the answer is none.