From the number 31549786, the second, fifth and eighth digits are taken to form a three-digit number. If it is possible to form a number that is the perfect square of a two-digit even number, what will be the second digit of that two-digit even number?

Introduction / Context:
This question combines digit selection with properties of perfect squares. You are asked to extract certain digits from an eight digit number, rearrange them to form a three digit number, and then determine whether that three digit number can be a perfect square of a two digit even number. Once that two digit even number is identified, you must report its second digit. This tests both numerical reasoning and knowledge of common perfect squares.

Given Data / Assumptions:
- Original number: 31549786. - We take the 2nd, 5th, and 8th digits of this number. - These digits may be rearranged to form a three digit number. - That three digit number must be the perfect square of a two digit even number. - We must find the second digit of that two digit even number, if such a square can be formed.

Concept / Approach:
First, identify the required digits from 31549786 by position. Next, list all distinct three digit numbers that can be formed using those digits. Then compare this set with the list of three digit perfect squares of two digit even numbers. Once a match is found, the corresponding two digit even number is determined, and we can simply read off its second digit. This is a finite search problem rather than an algebraic one.

Step-by-Step Solution:
Step 1: Label the digits of 31549786 by position: Positions: 1 2 3 4 5 6 7 8 Digits: 3 1 5 4 9 7 8 6 Step 2: The second digit is 1, the fifth digit is 9, and the eighth digit is 6. Step 3: Thus the available digits are 1, 9, and 6. Step 4: Possible three digit numbers formed using 1, 9, and 6 are: 196, 169, 916, 961, 619, and 691. Step 5: List the perfect squares of two digit even numbers that are three digit numbers. Even numbers from 10 to 30 give squares like 10^2 = 100, 12^2 = 144, 14^2 = 196, 16^2 = 256, 18^2 = 324, 20^2 = 400, 22^2 = 484, 24^2 = 576, 26^2 = 676, 28^2 = 784, 30^2 = 900. Step 6: Compare our candidate numbers with this list of squares. Step 7: The number 196 appears both in our candidate list and in the perfect square list. Step 8: 196 is the square of 14, that is, 14^2 = 196. Step 9: The two digit even number is 14. Its first digit is 1 and its second digit is 4. Step 10: Therefore, the required second digit is 4.

Verification / Alternative check:
We can confirm no other candidate fits. Numbers like 169 are 13^2, but 13 is not even. Likewise, 961 is 31^2 and 916, 619, and 691 are not squares of any two digit integer. Among the even squares list, only 196 can be made using digits 1, 9, and 6. This guarantees that 14 is the unique valid two digit even number and that 4 is the unique second digit answer.

Why Other Options Are Wrong:
1: This is the first digit of 14, not the second digit. 6: No square of an even two digit number formed from these digits would give a second digit of 6. None of these: This would only be correct if no suitable square existed, but 196 = 14^2 clearly satisfies all conditions.

Common Pitfalls:
One common mistake is to forget the requirement that the two digit number must be even, leading candidates to choose 13 because 169 is a familiar square. Another is to not systematically check all permutations of the digits 1, 9, and 6, causing some learners to overlook 196. Building the full list of permutations and then comparing against known squares is the surest way to avoid missing the correct combination.

Final Answer:
The second digit of the two digit even number whose square is formed from these digits is 4.