If the first and second digits are decreased by 1
and the the third is increased by 3, the digits
will form a geometric progression.
The only possible ascending geometric progression
of digits of a three-digit number are
Case 1. 1,1,1
Case 2. 1,3,9
Case 3. 2,4,8
The digits of a three digit number form an
arithmetic progression. If the first and second
digits are decreased by 1 and the the third is
increased by 3
Case 1 is out because the third number is 1,
and 1 can't be the result of increasing a digit by 3.
Case 3 is out because the previous numbers would have
been 2,5,5, which is not an arithmetic progression.
So it must be case 2.
Checking: the previous digits would have been 2,4,and 6.
which do form an arithmetic progression. We didn't need
the information that the sum of the digits is 12, but it
does turn out that 2+4+6=12.
Answer: The original three-digit number was 246.
Edwin
