We are asked to find the smallest possible difference between two 4-digit numbers that can be formed using 8 of the digits from the set of cards (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Steps to Solve:

1. Select the digits: We will select 8 digits from the given set of 10 digits. We leave out two digits, which is a crucial step.

2. Form two numbers: We then arrange these 8 digits to form two 4-digit numbers.

3. Minimize the difference: We try to distribute the digits between the two numbers such that the difference between them is as small as possible.

Approach:

To minimize the difference between the two numbers, we should try to make them as close in value as possible. This means balancing the digits in a way that both numbers are roughly the same magnitude.

Example Calculation:

• Choose the 8 digits with a balance of smaller and larger numbers: Let's select digits $0, 1, 2, 3, 4, 5, 7, 8$, leaving out digits $6$ and $9$.

• Sort the digits to create two 4-digit numbers:

  • For the smallest difference, try to balance the digits by distributing smaller digits evenly.

  One possible arrangement is:

  • Number 1: $1034$
  • Number 2: $2758$

• Calculate the difference: $2758 - 1034 = 1724$

Thus, the smallest possible difference in this case is 1724.

This is one possible way to arrange the numbers, but the key idea is to distribute the digits as evenly as possible.

Would you like more details or further explanations on this approach?

Here are five questions to expand on the topic:

1. How could leaving out different pairs of digits change the outcome?
2. What if we are allowed to form numbers of different lengths?
3. How would adding restrictions (e.g., no leading zeros) affect the solution?
4. Can you generalize the approach for different numbers of digits (e.g., forming three numbers)?
5. What strategies can be used to distribute digits more effectively in optimization problems?

Tip: When minimizing differences between two numbers formed from a set of digits, think about balancing the magnitude of the digits by alternating the largest and smallest available digits between the numbers.