What is the smallest three-digit number possible by removing a three-digit number formed using the digits 8, 5, 4, 3, 2, and 1 exactly once from another three-digit number formed using the same digits exactly once?

To solve the problem of finding the smallest three-digit number possible by subtracting one three-digit number from another, both formed using the digits 8, 5, 4, 3, 2, and 1 exactly once, we can proceed as follows:

### Step 1: Identify the Two Three-Digit Numbers
Let's denote the two three-digit numbers as \( A \) and \( B \), where \( A \) is the larger number and \( B \) is the smaller number.

### Step 2: Arrange the Digits
To minimize the difference \( A - B \), we need to make \( A \) as small as possible while still being larger than \( B \). To achieve this, the digits of \( A \) should be arranged in ascending order, and the digits of \( B \) should be arranged in descending order.

### Step 3: Try Different Combinations
Using the digits 8, 5, 4, 3, 2, and 1, let's create the smallest possible \( A \) and largest possible \( B \) with the remaining digits:

1. **Smallest possible \( A \):** Use the smallest available digits.
   - Smallest possible \( A = 123 \)

2. **Largest possible \( B \):** Use the largest available digits.
   - Largest possible \( B = 854 \)

Now, calculate the difference \( A - B \):

\[
854 - 123 = 731
\]

### Step 4: Explore Other Combinations
We need to ensure this is the smallest difference possible. Let's try swapping some digits between \( A \) and \( B \) to check for smaller differences:

- **\( A = 132 \), \( B = 854 \)**: 
  \[
  854 - 132 = 722
  \]
- **\( A = 142 \), \( B = 853 \)**:
  \[
  853 - 142 = 711
  \]
- **\( A = 153 \), \( B = 842 \)**:
  \[
  842 - 153 = 689
  \]
- **\( A = 231 \), \( B = 854 \)**:
  \[
  854 - 231 = 623
  \]
- **\( A = 312 \), \( B = 854 \)**:
  \[
  854 - 312 = 542
  \]
- **\( A = 321 \), \( B = 854 \)**:
  \[
  854 - 321 = 533
  \]

### Step 5: Conclusion
After checking the different possibilities, the smallest difference we can achieve is:

\[
533 \quad (\text{where } A = 321 \text{ and } B = 854)
\]

Thus, the smallest three-digit number possible by subtracting one three-digit number from another, using the digits 8, 5, 4, 3, 2, and 1 exactly once, is **533**.

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Would you like more details on the steps, or do you have any questions?

Here are five related questions you might find interesting:
1. How many different three-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 8 exactly once?
2. What is the largest difference between two three-digit numbers formed using the digits 8, 5, 4, 3, 2, and 1 exactly once?
3. What would the smallest difference be if the digits 0, 1, 2, 3, 4, and 5 were used instead?
4. How would the problem change if you were allowed to use each digit more than once?
5. How many unique pairs of three-digit numbers can be formed with the given digits?

**Tip:** When solving problems involving digit manipulation, systematically testing the extremes (largest and smallest possible values) often leads to the optimal solution.

Discover how to find the smallest three-digit number by subtracting permutations of 854321. Explore the mathematical approach using permutations and combinations, suitable for students in Grades 7-9.

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