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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

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Jun 09, 2013
Logic Puzzle
by: Staff


Answer

Part I


Math - logic puzzle - ABC + ACB = CB




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


ABC + ACB = CBA, the one's column




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


ABC + ACB = CBA, the ten's column




----------------------------------------------------

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.


ABC + ACB = CBA, the ten's column equation 1 is invalid





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.


ABC + ACB = CBA, the ten's column equation 2 is invalid




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


ABC + ACB = CBA, the ten's column equation 3 is invalid




10's column equation 4 is valid:

if 1 + B + C = 10 + B, then C = 9


ABC + ACB = CBA, the ten's column equation 4 is valid



Substitute the value of 9 for C


ABC
+ACB
_____
CBA

AB9
+A9B
_____
9BA


ABC + ACB = CBA, C = 9



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


ABC + ACB = CBA, A = 4





Substitute the value of 4 for A

ABC
+ACB
_____
CBA

AB9
+A9B
_____
9BA


4B9
+49B
_____
9B4

ABC + ACB = CBA, substitute A = 4




Solve for B using 1's column equation 2:


C + B = 10 + A

9 + B = 10 + 4

B = 5



ABC + ACB = CBA, solve for 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


ABC + ACB = CBA, final answer








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

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