A number consists of three digits whose sum is 17. The middle one exceeds the sum of other by 1. If the digits are reversed, the number is diminished by 396. Find the number.

Answer

To solve the problem step by step, we will define the digits of the three-digit number and set up equations based on the information given in the question.

### Step 1: Define the digits
Let the three-digit number be represented as:
- \( x \): the digit in the hundreds place
- \( y \): the digit in the tens place
- \( z \): the digit in the units place

### Step 2: Set up the equations
From the problem, we have the following information:
1. The sum of the digits is 17:
\[ x + y + z = 17 \quad \text{(Equation 1)} \]
2. The middle digit exceeds the sum of the other two by 1:
\[ y = x + z + 1 \quad \text{(Equation 2)} \]
3. When the digits are reversed, the number is diminished by 396:
\[ 100x + 10y + z - (100z + 10y + x) = 396 \]
Simplifying this gives:
\[ 99x - 99z = 396 \quad \Rightarrow \quad x - z = 4 \quad \text{(Equation 3)} \]

### Step 3: Solve the equations
Now we have three equations:
1. \( x + y + z = 17 \) (Equation 1)
2. \( y = x + z + 1 \) (Equation 2)
3. \( x - z = 4 \) (Equation 3)

#### Substitute Equation 3 into Equation 1
From Equation 3, we can express \( x \) in terms of \( z \):
\[ x = z + 4 \]
Now substitute \( x \) in Equation 1:
\[ (z + 4) + y + z = 17 \]
This simplifies to:
\[ 2z + y + 4 = 17 \]
\[ 2z + y = 13 \quad \text{(Equation 4)} \]

#### Substitute Equation 3 into Equation 2
Now substitute \( x = z + 4 \) into Equation 2:
\[ y = (z + 4) + z + 1 \]
This simplifies to:
\[ y = 2z + 5 \quad \text{(Equation 5)} \]

### Step 4: Solve Equations 4 and 5
Now we have:
1. \( 2z + y = 13 \) (Equation 4)
2. \( y = 2z + 5 \) (Equation 5)

Substituting Equation 5 into Equation 4:
\[ 2z + (2z + 5) = 13 \]
This simplifies to:
\[ 4z + 5 = 13 \]
\[ 4z = 8 \quad \Rightarrow \quad z = 2 \]

### Step 5: Find \( x \) and \( y \)
Now that we have \( z \), we can find \( x \) and \( y \):
Using Equation 3:
\[ x = z + 4 = 2 + 4 = 6 \]
Using Equation 5:
\[ y = 2z + 5 = 2(2) + 5 = 4 + 5 = 9 \]

### Step 6: Conclusion
The digits of the number are:
- \( x = 6 \)
- \( y = 9 \)
- \( z = 2 \)

Thus, the three-digit number is:
\[ \text{Number} = 100x + 10y + z = 100(6) + 10(9) + 2 = 600 + 90 + 2 = 692 \]

### Final Answer
The number is **692**.