Digit selection → perfect square – Using the 1st, 4th, and 6th digits of 531697 (i.e., 5, 6, 7), form a 3-digit perfect square of a two-digit even number. What is the second digit of that two-digit even number?

Introduction / Context:We must permute selected digits to get a 3-digit square whose root is a two-digit even number; then report the second (unit) digit of that even root. The special outputs ‘@’ (no such number) or ‘c’ (more than one) are sentinels.

Given Data / Assumptions:

• Digits available: positions 1,4,6 of 531697 → {5,6,7}.
• Seek a 3-digit perfect square of an even two-digit number.

Concept / Approach:List permutations of {5,6,7} and check against known 3-digit squares. The only candidate matching is 576 = 24^2.

Step-by-Step Solution:Permutations: 567, 576, 657, 675, 756, 765.Perfect square match: 576 = 24^2.Even root = 24 → second digit (units) = 4.

Verification / Alternative check:Other permutations are not perfect squares (quick check against the list of 3-digit squares).

Why Other Options Are Wrong:‘2’ or ‘6’ are not the unit digit of the unique valid even root; ‘@’ (none) and ‘c’ (more than one) do not apply.

Common Pitfalls:Overlooking 576; assuming squares must end with 5 or 6 only for certain roots without checking.

Final Answer:4