The problem:
Find a three-digit number where the digits are in an arithmetic progression (AP), and the sum of the digits is 12. When the digits are reversed, the number is diminished by 396.

Solution:

To solve this problem, we need to consider the given information and use a systematic approach to find the desired number. Let's break down the problem into smaller steps:

Step 1: Setting up the equations
Let the three-digit number be represented by ABC, where A, B, and C are the digits. We know that the digits are in an arithmetic progression, so we can write the following equation:

B = A + d
C = A + 2d

where d is the common difference between the digits.

We also know that the sum of the digits is 12, so we can write the equation:

A + B + C = 12

Step 2: Simplifying the equations
Substituting the values of B and C from the first set of equations into the second equation, we have:

A + (A + d) + (A + 2d) = 12
3A + 3d = 12
A + d = 4

Step 3: Finding the value of d
We can now solve the equation A + d = 4 for d:

d = 4 - A

Step 4: Reversing the digits
When the digits are reversed, the new number becomes CBA. This number is diminished by 396 compared to the original number ABC. Mathematically, we can write:

100C + 10B + A = 100A + 10B + C - 396

Step 5: Solving the equations
Substituting the values of B and C from the first set of equations into the last equation, we have:

100(A + 2d) + 10(A + d) + A = 100A + 10(A + d) + (A + 2d) - 396

Simplifying this equation, we get:

100A + 200d + 10A + 10d + A = 100A + 10A + 10d + A + 2d - 396
290A + 200d = 100A + 10d - 396
190A + 190d = -396

Step 6: Solving for A and d
We now have a system of two equations:

A + d = 4
190A + 190d = -396

Solving these equations simultaneously, we find:

A = 2
d = 2

Step 7: Finding the number
Using the values of A and d, we can determine the digits of the number:

A = 2
B = A + d = 2 + 2 = 4
C = A + 2d = 2 + 2(2) = 6

Therefore, the number is 246.

Conclusion:
The three-digit number where the digits are in an arithmetic progression and the sum of the digits is 12 is 246. When the digits are reversed, the number is diminished by