# Math - logic puzzle - ABC + ACB = CBA

by Kerry
(Sacramento, CA)

Logic Puzzle

In the following addition problem, the letters A, B, and C stand for three different digits.

What digit should replace each letter?

A   B   C

+   A   C   B
_________

C   B   A

### Comments for Math - logic puzzle - ABC + ACB = CBA

 Jun 09, 2013 Logic Puzzle by: Staff Answer Part I The one's column: there are only two possibilities 1's column equation 1: C + B = A OR 1's column equation 2: C + B = 10 + A The tens column: there are four possibilities 10's column equation 1: B + C = B OR 10's column equation 2: 1 + B + C = B OR 10's column equation 3: B + C = 10 + B OR 10's column equation 4: 1 + B + C = 10 + B ----------------------------------------------------

 Jun 09, 2013 Logic Puzzle by: Staff ---------------------------------------------------- Part II 10's column equation 1 is not valid because the results will not work for either of the two equations for column 1: if B + C = B, then C = 0 If C = 0, then 1's column equation 1: C + B = A will not work because A, B, and C must stand for three different digits If C = 0, then 1's column equation 2: C + B = 10 + A will not work because B must stand for a single digit Therefore, 10's column equation 1: B + C = B is not valid. 10's column equation 2 is not valid because the results will not work for either of the two equations for column 1: if 1 + B + C = B, then C = -1 C must be a positive integer. Therefore, 10's column equation 2: 1 + B + C = B is not valid. 10's column equation 3 is not valid because the C value is not a single digit integer: if B + C = 10 + B, then C = 10 C must be a single digit positive integer. ----------------------------------------------------

 Jun 09, 2013 Logic Puzzle by: Staff ---------------------------------------------------- Part III 10's column equation 4 is valid: if 1 + B + C = 10 + B, then C = 9 Substitute the value of 9 for C ABC +ACB _____ CBA AB9 +A9B _____ 9BA The hundred's column: there is only one possibility. 100's column equation 1: since C is an odd number, the equation for the hundred's column is: 1 + A + A = C 1 + A + A = 9 therefore, A = 4 ----------------------------------------------------

 Jun 09, 2013 Logic Puzzle by: Staff ---------------------------------------------------- Part IV Substitute the value of 4 for A ABC +ACB _____ CBA AB9 +A9B _____ 9BA 4B9 +49B _____ 9B4 Solve for B using 1's column equation 2: C + B = 10 + A 9 + B = 10 + 4 B = 5 Substitute the value of 5 for B ABC +ACB _____ CBA AB9 +A9B _____ 9BA 459 +495 _____ 954 ----------------------------------------------------

 Jun 09, 2013 Logic Puzzle by: Staff ---------------------------------------------------- Part V The final answer is: 459 +495 _____ 954 Thanks for writing. Staff www.solving-math-problems.com

 Feb 08, 2017 hey NEW by: Anonymous what if the problem is A-B-C and B-D-E and F-E-G are all equal. How gonno solve that?

 Nov 04, 2017 Math prob NEW by: Anonymous A B C + A C B —

 Apr 27, 2018 2 cents NEW by: Anonymous First, let's take a look at the 100 column. There are two possibilities. 1. A + A = C 2. A + A + 1 = C To gather more information, we look at the 10 column, it is B + C = B. There are only two possibilities, either c = 0 or c = 9 and there is a excess in the '1' column. Lastly, we look at the 1 column. As C > A, based on the information we have in the 100 column. We can deduce that C + B = A + 10. With this information, we can deduce that B + C + 1 = B + 10, hence, c = 9. So, the solution is C = 9, A = 4, B = 5.