To solve this problem, we need to understand how an abacus works, especially in terms of representing numbers with beads.

Assuming we are using a typical abacus, where each rod or column represents a place value (thousands, hundreds, tens, and ones), we need to figure out how to place the 16 beads to form the largest and smallest four-digit numbers.

Largest Four-Digit Number:

To make the largest possible four-digit number, you want to place all the beads at the highest possible value for each place.

• Thousands place: The largest value is 9 beads (the maximum value for one place on an abacus).
• Hundreds place: The largest value is also 9 beads.
• Tens place: The largest value is 9 beads.
• Ones place: The largest value is 9 beads.

Thus, the largest four-digit number you can form with the abacus would be 9999.

Smallest Four-Digit Number:

For the smallest four-digit number, you want to use the beads in a way that results in the least value while still maintaining four digits.

• Thousands place: You need at least 1 bead to make the number a four-digit number. So, 1 bead here.
• Hundreds place: The smallest value is 0 beads (because 0 is valid here).
• Tens place: The smallest value is also 0 beads.
• Ones place: The smallest value is 0 beads.

Thus, the smallest four-digit number you can form would be 1000.

Difference Between the Largest and Smallest Four-Digit Numbers:

Now, we can calculate the difference:

$9999 - 1000 = 8999$

Thus, the difference between the largest four-digit number and the smallest four-digit number is 8999.

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Would you like further details or have any additional questions on this topic?

Here are 5 related questions for you to explore:

1. How would the numbers change if we used only 10 beads on the abacus?
2. How does the abacus represent numbers beyond 9 in each place value?
3. What are the historical uses of the abacus in different cultures?
4. How would you represent fractions or decimals using an abacus?
5. What would happen if we used a different number of beads, such as 20?

Tip: To understand the placement of beads clearly, visualize the abacus as a system that groups beads by place value—each rod or column can hold beads representing different powers of 10.