Question 3

We will use the given hints to find the three-digit code for the lock.

Step 1 — Eliminate impossible digits

We first look at the hint that says "Nothing is correct". The hint "0 3 6 - Nothing is correct" means these digits are not in the code. So, the digits 0, 3, and 6 are not part of the secret code. Let the code be $P_1 P_2 P_3$. $P_1, P_2, P_3 \notin \{0, 3, 6\}$

Step 2 — Identify a certain digit and its position rule

Next, we use the hint "0 6 4 - One digit is correct but wrongly placed". We already know that 0 and 6 are not in the code from Step 1. So, the only possible correct digit from "0 6 4" must be 4. This digit 4 is in the second position in "0 6 4". The hint says this digit is wrongly placed. So, 4 is in the code, but it is not in the second position. $4 \text{ is in the code}$ $P_2 \neq 4$

Step 3 — Deduce more digits and their positions

Now we use the hint "5 4 2 - Two digits are correct but wrongly placed". We know from Step 2 that 4 is in the code. So, 4 is one of the two correct digits in this hint. The other correct digit must be either 5 or 2. Also, all correct digits in this hint are wrongly placed.

Let us consider two possibilities for the second correct digit:

Possibility A: The correct digits are 4 and 5. This means that 2 is NOT in the code. Let us check Hint 1: "2 6 5 - One digit is correct and well placed". We know 2 is not in the code, and 6 is not in the code (from Step 1). So, 5 must be the correct and well-placed digit. This means $P_3 = 5$. However, from Hint 3, if 5 is a correct digit, it must be wrongly placed. This means $P_1 \neq 5$, $P_2 \neq 5$, and $P_3 \neq 5$. This contradicts our finding that $P_3 = 5$. So, Possibility A is incorrect.

Possibility B: The correct digits are 4 and 2. This means that 5 is NOT in the code. Let us check Hint 1: "2 6 5 - One digit is correct and well placed". We know 6 is not in the code (from Step 1), and 5 is not in the code (from this possibility). So, 2 must be the correct and well-placed digit. This means $P_1 = 2$. Now, let us check this with Hint 3 ("5 4 2 - Two digits are correct but wrongly placed"). The correct digits are 4 and 2. They are both wrongly placed. For 4: $P_2 \neq 4$. This is consistent with Step 2. For 2: $P_3 \neq 2$. We have $P_1 = 2$. This does not contradict $P_3 \neq 2$. So, Possibility B is consistent.

From this step, we have found:

• 5 is NOT in the code.
• 2 IS in the code.
• $P_1 = 2$.
• $P_3 \neq 2$.

Step 4 — Place the third digit

We know the first digit of the code is 2 ($P_1 = 2$). We also know that 4 is in the code, and $P_2 \neq 4$ (from Step 2). Since $P_1 = 2$, 4 cannot be $P_1$. Since $P_2 \neq 4$, 4 cannot be $P_2$. So, 4 must be in the third position. $P_3 = 4$ We now have the first and third digits of the code: 2 _ 4. We also confirmed $P_3 \neq 2$ because $4 \neq 2$.

Step 5 — Find the middle digit

We need to find the digit for $P_2$. The digits not in the code are 0, 3, 5, 6. The digits we have placed are $P_1 = 2$ and $P_3 = 4$. So, $P_2$ must be one of the remaining digits: {1, 7, 8, 9}.

Let us use the hint "2 7 1 - One digit is correct but wrongly placed". We know 2 is in the code ($P_1 = 2$). If 2 was the correct digit from "2 7 1", it would have to be wrongly placed. But 2 is in the first position in "2 7 1", and $P_1 = 2$. So 2 is well placed. Therefore, 2 is not the "one digit correct but wrongly placed" from this hint. This means the correct digit from "2 7 1" must be either 7 or 1. And it must be wrongly placed.

Let us check if 7 is the correct digit. If 7 is correct, it must be wrongly placed. 7 is in the second position in "2 7 1". So, $P_2 \neq 7$. If 7 is in the code, it must be $P_1=7$ or $P_3=7$. But we know $P_1 = 2$ and $P_3 = 4$. So, 7 cannot be in the code.

Let us check if 1 is the correct digit. If 1 is correct, it must be wrongly placed. 1 is in the third position in "2 7 1". So, $P_3 \neq 1$. We know $P_3 = 4$, so $P_3 \neq 1$ is true. Since 1 is in the code, and $P_1 = 2$ and $P_3 = 4$, 1 must be $P_2$. So, the middle digit is 1. $P_2 = 1$

The code is 2 1 4.

Answer

The 3-digit code is 214.