Question 3

Step 1 · Eliminate Impossible Digits

From the hint "0 6 4 - Nothing is correct" / "0 3 6 - Nothing is correct":

The digits $0, 3,$ and $6$ are not part of the secret code.

$P_1, P_2, P_3 \notin \{0, 3, 6\}$

Step 2 · Identify Digit 4 and Position Rule

From the hint "0 6 4 - One digit is correct but wrongly placed":

Since $0$ and $6$ are eliminated, the correct digit must be $4$.

Because $4$ is in the second position in this hint, it cannot be in the second position of the code:

$4 \text{ is in the code}$

$P_2 \neq 4$

Step 3 · Deduce Digits 2 and 5

From the hint "5 4 2 - Two digits are correct but wrongly placed":

We know $4$ is one of the correct digits. The other correct digit is either $5$ or $2$.

Possibility A: Correct digits are 4 and 5 (2 is not in the code)

• From hint "2 6 5 - One digit is correct and well placed", since $2$ and $6$ are not in the code, $5$ must be correct and well-placed $\implies P_3 = 5$.
• But from "5 4 2", $5$ must be wrongly placed $\implies P_1 \neq 5, P_2 \neq 5, P_3 \neq 5$.
• This is a contradiction, so Possibility A is false.

Possibility B: Correct digits are 4 and 2 (5 is not in the code)

• From hint "2 6 5 - One digit is correct and well placed", since $6$ and $5$ are not in the code, $2$ must be correct and well-placed $\implies P_1 = 2$.
• In "5 4 2", $2$ is in the third position and wrongly placed ($P_3 \neq 2$), which matches $P_1 = 2$.

Therefore:

• $2$ is in the code with $P_1 = 2$
• $5$ is not in the code

Step 4 · Place the Third Digit

We have:

• $P_1 = 2$
• $4$ is in the code, with $P_2 \neq 4$

Since $P_1 = 2$ and $P_2 \neq 4$, the digit $4$ must be in the third position:

$P_3 = 4$

The code so far is $\mathbf{2 \_ 4}$.

Step 5 · Find the Middle Digit

From the hint "2 7 1 - One digit is correct but wrongly placed":

• $2$ is well-placed ($P_1 = 2$), so $2$ cannot be the "wrongly placed" digit in this hint.
• If $7$ is correct, it is in the second position in this hint, so $P_2 \neq 7$. Since $P_1 = 2$ and $P_3 = 4$, $7$ has no valid position.
• Thus, $1$ must be the correct digit. It is in the third position here, so $P_3 \neq 1$.

Since $P_1 = 2$ and $P_3 = 4$, $1$ must be in the second position:

$P_2 = 1$

Therefore, the complete code is $\mathbf{214}$.