EXAMPLE OF CRYPTARITHMETIC PROBLEM

Solve the cryptarithm below using only the numbers 0, 1, 2, 3, 6, 7, and 9. No letter can represent two different digits. For instance n cannot be 2 and 6 at the same time.

                   sun
               +  fun
                 ______
                 SWIM

Guess and test is the strategy that we are going to use. However, just guess and test is not enough. You may need to make some good observations or use some basic math facts to solve the puzzle in a timely fashion. Otherwise, we may end up guessing all day.        

 Solution:

                    s    u    n
            +      f    u    n
           _______________
              s   w    i    m

Notice that n cannot be 0. When adding n and n, the result is m and m is different than n. However, 0 + 0 = 0, so you are not getting a different number. By the same token, u cannot be 0.

Notice also that n cannot be 2, 7, or 9.  

2 + 2 = 4                     7 + 7 = 14                            9 + 9 = 18

When adding, you will have to write down 4, or 8. However, 4 and 8 are not listed among the numbers we can use.

By the same token u cannot be 2, 7 or 9 and n or u can only be 1, 3, or 6

Notice that the following additions are not possible.

            s     6    1    

         +  f     6    1                            _______________

       s  w     2    2

2 appears twice 

           s     6    3    

     +    f     6    3                          _______________

       s  w    2    6

6 appears twice  

            s      1    6        

       +   f      1    6                            _______________

        s  w     2    2

    2 appears twice

For the one in the middle, you could try to swap 3 and 6 and see what happens.

          s    3    6                                
  +      f    3    6                      
_______________
     s  w    7    2

We have 3 numbers left to use  0, 1, and 9. If you going to get a number with 4 digits as an answer, then 9 must be either s or f

          9    3    6                              
  +      1    3    6                      
  _______________
     1  0    7    2

          1    3    6                              
  +      9    3    6                      
  _______________
     1   0    7    2

The one on the right is of course our answer. The other two additions below are more possibilities that would not have worked if we had tried them.

          9    3    1                              
  +      7    3    1                      
  _______________
     1  6     6    2

          9    1    3                              
  +      7    1    3                      
  _______________
     1  6     2    6