Let's solve the problem step by step.

Step 1: Define the number

Let the three-digit number be represented as $100h + 10t + u$, where:

• $h$ = hundreds digit,
• $t$ = tens digit,
• $u$ = units digit.

Step 2: Translate the given conditions

1. The sum of the digits is 19: $h + t + u = 19$

2. When the tens and units digits are interchanged, the number is diminished by 27: The new number becomes $100h + 10u + t$. The equation is: $100h + 10t + u - (100h + 10u + t) = 27$ Simplify: $9t - 9u = 27 \quad \Rightarrow \quad t - u = 3$

3. When the hundreds and tens digits are interchanged, the number is increased by 180: The new number becomes $100t + 10h + u$. The equation is: $100t + 10h + u - (100h + 10t + u) = 180$ Simplify: $90t - 90h = 180 \quad \Rightarrow \quad t - h = 2$

Step 3: Solve the system of equations

We now have three equations:

1. $h + t + u = 19$
2. $t - u = 3$
3. $t - h = 2$

From equation (3): $t = h + 2$

Substitute $t = h + 2$ into equation (2): $h + 2 - u = 3 \quad \Rightarrow \quad u = h - 1$

Substitute $t = h + 2$ and $u = h - 1$ into equation (1):

$h + (h + 2) + (h - 1) = 19$ Simplify: $3h + 1 = 19 \quad \Rightarrow \quad 3h = 18 \quad \Rightarrow \quad h = 6$

Find $t$ and $u$:

$t = h + 2 = 6 + 2 = 8$ $u = h - 1 = 6 - 1 = 5$

Step 4: Verify the solution

The number is $100h + 10t + u = 100(6) + 10(8) + 5 = 685$.

• Sum of digits: $6 + 8 + 5 = 19$ (satisfied).
• Interchanging $t$ and $u$: $685 - 658 = 27$ (satisfied).
• Interchanging $h$ and $t$: $865 - 685 = 180$ (satisfied).

Final Answer:

The number is 685.

Do you have any questions or want further clarification? Here are five related problems to consider:

1. What if the sum of the digits was 21 instead of 19?
2. How would the solution change if the increase when interchanging $h$ and $t$ was 190?
3. Can this problem be solved using matrices for the system of equations?
4. What happens if the digits $t$ and $u$ are equal?
5. How can such problems be represented and solved programmatically?

Tip: Always break multi-condition problems into manageable equations and solve systematically.