## 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: $USSR=9338;\;USA=932;\;PEACE=10270$.