We are tasked with determining a 4-digit code using the given clues. Let's break it down step by step:

Clues:

1. Sum of the first and second digits is a multiple of 6.
2. The difference between the first and second digits is 6.
3. The sum of the third and fourth digits is the same as the sum of the first and second digits.
4. The first digit is the smallest of the four digits.
5. The third and fourth digits are the same.
6. There are no zeros in the code.

Let the digits of the code be represented as $a$, $b$, $c$, and $d$, where:

• $a$ is the first digit,
• $b$ is the second digit,
• $c$ is the third digit,
• $d$ is the fourth digit.

From the clues:

1. The sum of the first and second digits is a multiple of 6: $a + b$ is a multiple of 6.
2. The difference between the first and second digits is 6: $b - a = 6$.
3. The sum of the third and fourth digits equals the sum of the first and second digits: $c + d = a + b$.
4. The first digit is the smallest: $a$ is the smallest.
5. The third and fourth digits are the same: $c = d$.
6. There are no zeros: none of the digits are zero.

Step 1: Use the difference clue

From the second clue, $b - a = 6$, we have: $b = a + 6.$ Since the digits are between 1 and 9 (no zeros allowed), the smallest $a$ can be is 1, because any smaller would make $b$ too large.

Step 2: Use the sum being a multiple of 6

The sum of $a + b$ must be a multiple of 6. Since $b = a + 6$, this gives: $a + (a + 6) = 2a + 6.$ We need $2a + 6$ to be a multiple of 6: $2a + 6 = 6k \quad \text{for some integer } k.$ Simplifying: $2a + 6 = 6 \quad \text{(since } k = 1 \text{ gives a multiple of 6)}.$ $2a = 0 \quad \Rightarrow \quad a = 0,$ which is impossible because there are no zeros in the code. Let’s try the next possible value of $a$.

Step 3: Checking larger values of $a$

We systematically test for larger values:

If $a = 1$:

$b = 1 + 6 = 7,$ so $a + b = 1 + 7 = 8$, which is not a multiple of 6.

If $a = 3$:

$b = 3 + 6 = 9,$ so $a + b = 3 + 9 = 12$, which is a multiple of 6.

Step 4: Determine the third and fourth digits

From the third clue, $c + d = a + b = 12$. Since $c = d$ (from clue 5), we have: $c + c = 12 \quad \Rightarrow \quad 2c = 12 \quad \Rightarrow \quad c = 6.$ Therefore, $c = d = 6$.

Final Answer:

The code is $3966$.

Would you like further details or explanations on any step?

Here are five related questions to expand on the topic:

1. How does the smallest digit restriction affect other possible combinations?
2. Why is it important to check each condition sequentially?
3. How can you generalize the method for codes with different conditions?
4. What other ways can sums and differences be used in puzzle clues?
5. How would the solution change if zeros were allowed?

Tip: When solving multi-step puzzles, verify each condition before finalizing the answer to avoid missing any restrictions.