Let's solve the problem.

Let the three-digit number be represented as $\overline{abc}$, where:

• $a$ is the hundreds digit,
• $b$ is the tens digit,
• $c$ is the units digit.

The problem gives us two conditions:

1. The units digit $c$ is the sum of the hundreds and tens digits: $c = a + b$
2. The units digit $c$ exceeds the tens digit $b$ by as much as the tens digit exceeds the hundreds digit: $c - b = b - a$ This simplifies to: $c = 2b - a$

Now, we have the system of equations:

1. $c = a + b$
2. $c = 2b - a$

Step 1: Set the two expressions for $c$ equal to each other:

$a + b = 2b - a$

Step 2: Solve for $a$:

$a + b = 2b - a$ Add $a$ to both sides: $2a + b = 2b$ Subtract $b$ from both sides: $2a = b$ Thus, the tens digit $b = 2a$.

Step 3: Substitute $b = 2a$ into the first equation for $c$:

$c = a + b = a + 2a = 3a$

Step 4: Now, since $a$, $b$, and $c$ are digits, they must be integers between 0 and 9. Let's check possible values of $a$:

• $a = 1$: $b = 2a = 2$, $c = 3a = 3$ → The number is $123$.
• $a = 2$: $b = 2a = 4$, $c = 3a = 6$ → The number is $246$.
• $a = 3$: $b = 2a = 6$, $c = 3a = 9$ → The number is $369$.

For $a = 4$ or higher, $b = 2a$ would be 8 or higher, and $c = 3a$ would be greater than 9, which is not allowed since digits must be between 0 and 9.

Conclusion:

There are 3 such three-digit numbers: $123$, $246$, and $369$.

Thus, the answer is $\boxed{3}$.

Would you like more details on any step? Here are 5 related questions to further explore this topic:

1. How would the solution change if the problem specified a different relationship between the digits?
2. Can this method be applied to other multi-digit numbers? If so, how?
3. What if the sum of the digits had to follow a different rule, such as being divisible by a certain number?
4. How could the concept of modular arithmetic help in solving this type of digit problem?
5. Can you find other problems where the difference between digits plays a crucial role in the solution?

Tip: When working with digit problems, always pay attention to the constraints on the values of the digits (i.e., they must be between 0 and 9).