ABC + DEF = GHI (abc def ghi math problem)

ABC + DEF = GHI and each letter represent number 1-9 and there is no repetition. Can you figure it out? There is how many sets of 3 digit number that satisfy the above condition?  How to find it?

Sum of A+B+C+D+...+F = 45 is an odd number.

Let x + y = z In addition (Without Carrying)

Rule 1: Sum of the digit of number x is odd and sum of the digit of number y is odd then sum of the digit of number z is even.

Rule 2: Sum of the digit of number x is even and sum of the digit of number y is odd then sum of the digit of number z is odd.

Rule 3: Sum of the digit of number x is odd and sum of the digit of number y is even then sum of the digit of number z is odd.

Rule 4: Sum of the digit of number x is even and sum of the digit of number y is even then sum of the digit of number z is even.

In our problem equation never satisfy in addition Without Carrying. Because Sum of A+B+C+D+...+F = 45 is an odd number. It never satisfy the above 4 rules.
For example:
Consider Rule 1:  Let sum of the digit of ABC is odd and sum of the digit of DEF is odd. We need a GHI with sum of the digit is even.
But sum of the digit of ABC + sum of the digit of DEF is even.
Sum of A+B+C+D+...+F = 45 is an odd.
Therefore Sum of GHI is odd(45- even number is an odd number).
Similarly in our problem did not satisfy the other 3 rules also.That means we need addition with carrying  to solve our problem.

Set of number that gives carry

In our problem there is no 0. To get carry sum of single digit number > 10.
2: (2,9)
3: (3,9),(3,8)
4: (4,9),(4,8),(4,7)
5: (5,9),(5,8),(5,7),(5,6)
6: (6,9),(6,8),(6,7)
7: (7,9),(7,8)
8: (8,9)

Consider set (2,9)
2 + 9 = 1 with carry 1. sum of remaining numbers is 33, (3+4+5+6+7+8 = 33) + carry =34. That implies sum of G and H is 17. There is no such number in remaining set.

Consider set (3,9)
3 + 9 = 2 with carry 1. sum of remaining numbers is 31, (1+4+5+6+7+8 = 31) + carry =32. That implies sum of G and H is 16. There is no such number in remaining set.

Consider set (3,8)
3 + 8 = 1 with carry 1. sum of remaining numbers is 33, (2+4+5+6+7+9 = 33) + carry =34. That implies sum of G and H is 17. There is no such number in remaining set.

Consider set (4,9)
4 + 9 = 3 with carry 1. sum of remaining numbers is 29, (1+2+5+6+7+8 = 29) + carry =30. That implies sum of G and H is 15. There is such number in remaining set 8 and 7.

Consider set (4,8)
4 + 8 = 2 with carry 1. sum of remaining numbers is 31, (1+3+5+6+7+9 = 31) + carry =32. That implies sum of G and H is 16. There is such number in remaining set 7 and 9.

Consider set (4,7)
4 + 7 = 1 with carry 1. sum of remaining numbers is 33, (2+3+5+6+8+9 = 33) + carry =34. That implies sum of G and H is 17. There is such number in remaining set 8 and 9.

Consider set (5,9)
5 + 9 = 4 with carry 1. sum of remaining numbers is 27, (1+2+3+6+7+8 = 27) + carry =28. That implies sum of G and H is 14. There is such number in remaining set 6 and 8.

Consider set (5,8)
5 + 8 = 3 with carry 1. sum of remaining numbers is 29, (1+2+4+6+7+9 = 29) + carry =30. That implies sum of G and H is 15. There is such number in remaining set 6 and 9.

Consider set (5,7)
5 + 7 = 2 with carry 1. sum of remaining numbers is 31, (1+3+4+6+8+9 = 31) + carry =32. That implies sum of G and H is 16. There is no such number in remaining set.

Consider set (5,6)
5 + 6 = 1 with carry 1. sum of remaining numbers is 33, (2+3+4+7+8+9 = 33) + carry =34. That implies sum of G and H is 17. There is such number in remaining set 8 and 9.

Consider set (6,9)
6 + 9 = 5 with carry 1. sum of remaining numbers is 25, (1+2+3+4+7+8 = 25) + carry =26. That implies sum of G and H is 13.There is no such number in remaining set.

Consider set (6,8)
6 + 8 = 4 with carry 1. sum of remaining numbers is 27, (1+2+3+5+7+9 = 27) + carry =28. That implies sum of G and H is 14. There is such number in remaining set 5 and 9.

Consider set (6,7)
6 + 7 = 3 with carry 1. sum of remaining numbers is 29, (1+2+4+5+8+9 = 29) + carry =30. That implies sum of G and H is 15. There is no such number in remaining set.

Consider set (7,9)
7 + 9 = 6 with carry 1. sum of remaining numbers is 23, (1+2+3+4+5+8 = 23) + carry =24. That implies sum of G and H is 12.There is such number in remaining set 4 and 8.

Consider set (7,8)
7 + 8 = 5 with carry 1. sum of remaining numbers is 25, (1+2+3+4+6+9 = 25) + carry =26. That implies sum of G and H is 13. There is such number in remaining set 4 and 9.

Consider set (8,9)
8 + 9 = 7 with carry 1. sum of remaining numbers is 21, (1+2+3+4+5+6 = 21) + carry =22. That implies sum of G and H is 11.There is such number in remaining set 6 and 5.

Set of number that gives carry and satisfy condition

4: (4,9),(4,8),(4,7)
5: (5,9),(5,8),(5,6)
6: (6,8)
7: (7,9),(7,8)
8: (8,9)

Consider set (4,9):

 The remaining numbers are 1,2,5 and 6. To get 8 we need to add 6 and 2. Therefore one of the solution for set (4,9) is 654 + 219 = 873

Consider set (4,8):

 The remaining numbers are 1,3,5 and 6. To get 9 we need to add 6 and 3. Therefore one of the solution for set (4,8) is 654 + 318 = 972

Consider set (4,7):

 The remaining numbers are 2,3,5 and 6. To get 9 we need to add 6 and 3. Therefore one of the solution for set (4,7) is 654 + 327 = 981

Consider set (5,9):

 The remaining numbers are 1,2,3 and 7. To get 8 we need to add 7 and 1. Therefore one of the solution for set (5,9) is 735 + 127 = 864

Consider set (5,8):

 The remaining numbers are 1,2,4 and 7. To get 9 we need to add 7 and 2. Therefore one of the solution for set (5,8) is 745 + 218 = 963

Consider set (5,6):

 The remaining numbers are 2,3,4 and 7. To get 9 we need to add 7 and 2. Therefore one of the solution for set (5,6) is 745 + 236 = 981

Consider set (6,8):

The remaining numbers are 1,2,3 and 7. To get 9 we need to add 7 and 2. Therefore one of the solution for set (6,8) is 736 + 218 = 954

Consider set (7,9):

The remaining numbers are 1,2,3 and 5. To get 8 we need to add 5 and 3. Therefore one of the solution for set (7,9) is 527 + 319 = 846

Consider set (7,8):

The remaining numbers are 1,2,3 and 6. To get 9 we need to add 6 and 3. Therefore one of the solution for set (7,8) is 627 + 318 = 945

Consider set (8,9):

The remaining numbers are 1,2,3 and 4. To get 6 we need to add 4 and 2. Therefore one of the solution for set (8,9) is 438 + 219 = 657

From the above 10 set each of the set have so-many combination of number that satisfy the equation ABC + DEF = GHI. 

Each set contain 1-3 solution set. The 4 types of solution set are given below.

  A1 = A, B1 = B, D1 = D, E1 = E, H1 = H, G1 = G.

       1)A1+D1= G1 and B1+E1+1 = H1

       2)A1+E1= H1 and B1+D1+1 = G1

       3)B1+D1= H1 and A1+E1+1 = G1

       4)B1+E1= G1 and A1+D1+1 = H1

 Each solution set gives 16 different combination(different values for A-I) of number.

 consider this example : 

                 

First consider the set (4,9). 

      Here there are 3 solution set.

              1) 654 +        2) 654 +     3) 564 +

                  219                129            219

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

                 873                783             783

     set (4,9) gives 16*3 = 48 different combination of number.

First consider the set (4,8). 

      Here there are 2 solution set.

              1) 654 +        2) 654 +

                  318                138   

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

                 972                792    

      set (4,8) gives 16*2 = 32 different combination of number.

First consider the set (4,7). 

      Here there are 3 solution set.

              1) 654 +        2) 654 +    3) 564 +

                  327                237            327

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

                 981                891             891

      set (4,7) gives 16*3 = 48 different combination of number.

First consider the set (5,9). 

      Here there is 1 solution set.

              1) 735 +  

                  129   

                 -------  

                 864      

      set (5,9) gives 16*1 = 16 different combination of number.

First consider the set (5,8). 

      Here there  are 2 solution set.

              1) 745 +     2) 475 +

                  218             218

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

                 963              693 

      set (5,8) gives 16*2 = 32 different combination of number.

First consider the set (5,6). 

      Here there is 1 solution set.

              1) 745 +  

                  236   

                 -------  

                 981    

      set (5,6) gives 16*1 = 16 different combination of number.

First consider the set (6,8). 

      Here there are 2 solution set.

              1) 736 +    2) 376 +

                  218            218

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

                 954             594

      set (6,8) gives 16*2 = 32 different combination of number.

First consider the set (7,8). 

      Here there is 1 solution set.

             1) 627+  

                 318   

                 -------  

                 945    

      set (7,8) gives 16*1 = 16 different combination of number.

First consider the set (7,9). 

      Here there are 2 solution set.

              1) 517 +        2) 157 +   

                  329                329        

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

                 846                486     

      set (7,9) gives 16*2 = 32 different combination of number.

First consider the set (8,9). 

      Here there are 3 solution set.

              1) 438 +        2) 438 +     3) 348 +

                  219                129            219

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

                 657                567             567

      set (8,9) gives 16*3 = 48 different combination of number.

Total 16*21 = 336 combination of number that satisfy the equation ABC + DEF = GHI.