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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

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

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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
—

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.

Jun 03, 2021

Difficulty level NEW
by: Anonymous

Eeeeeeeeeeasy

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Math - logic puzzle - ABC + ACB = CBA

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