If each letter in the equation represents a unique digit, determine the values that represent , and .
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 of is distinct from , (of ) must produce a carry 1 which in turn causes (of ) to produce the carry digit . Thus, and .
Step 1 Suppose
1.1 Let
The table below shows if , then and . 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 will lead to non solutions.
Step 2 Suppose
2.1 Let
The table below shows that if , then which is already taken by .
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 will lead to non solutions.
Step 3 Suppose
This is also a non solution for .
Step 4 Suppose
This is also a non solution for .
Step 5 Suppose
This is a non solution for .
Step 6 Suppose
6.1 Let
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: .