Peace

If each letter in the equation USSR + USA = PEACE represents a unique digit, determine the values that represent USSR, USA, and PEACE.
Source: mathcontest.olemiss.edu 3/24/2008

SOLUTION
This problem is similar to the problem titled “Seven Seas” dated 10/1/2007.

   U  S  S R
        U S A
————–
P E  A  C E

Since U of USSR is distinct from E, S+U (of USA) must produce a carry 1 which in turn causes 1+U (of USSR) to produce the carry digit P=1. Thus, U=9 and E=0.

Step 1 Suppose R=2;\;A=8
1.1 Let S=3
The table below shows if S=3, then A=2 and A=8. Thus, this is not a solution.

Carry

1

  

  

1

  

  

  

  

  

1

  

  

  

  

       
       

9

U

 

3

S

 

3

S

 

2

R

       
  

  

  

  

  

  

  

9

U

  

3

S

  

8

A

       
 

1

P

 

0

E

 

2

A

 

7

C

 

0

E

       

Similarly, we can show that other choices of S=\left \{4,5,6,7\right \} will lead to non solutions.

Step 2 Suppose R=3;\;A=7
2.1 Let S=2
The table below shows that if S=2, then A=1 which is already taken by P.

Carry

1

  

  

1

  

  

  

  

  

1

  

  

  

  

       
       

9

U

 

2

S

 

2

S

 

3

R

       
  

  

  

  

  

  

  

9

U

  

2

S

  

7

A

       
 

1

P

 

0

E

 

1

A

 

5

C

 

0

E

       

Similarly, we can show that other choices of S=\left \{4,5,6,8\right \} will lead to non solutions.

Step 3 Suppose R=4;\;A=6
This is also a non solution for S=\left \{2,3,5,7,8\right \}.

Step 4 Suppose R=6;\;A=4
This is also a non solution for S=\left \{2,3,5,7,8\right \}.

Step 5 Suppose R=7;\;A=3
This is a non solution for S=\left \{2,4,5,6,8\right \}.

Step 6 Suppose R=8;\;A=2
6.1 Let S=3
The table below shows that this is a solution.

Carry

1

  

  

1

  

  


 

  

  

1

  

  

  

  

       
       

9

U

 

3

S

 

3

S

 

8

R

       
  

  

  

  

  


 

  

9

U

  

3

S

  

2

A

       
 

1

P

 

0

E

 

2

A

 

7

C

 

0

E

       

Answer: USSR=9338;\;USA=932;\;PEACE=10270.

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About mvtrinh

Retired high school math teacher.
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