USSR + USA = PEACE, what is the value of P + E + A + C + E?

Solution summary: deduce letters by column-wise addition and carries.

Write the columns (right to left) with carry variables:

• Units: R + A = E + 10*c1

• Tens: S + S + c1 = C + 10*c2

• Hundreds: S + U + c2 = A + 10*c3

• Thousands: U + c3 = E + 10*P

Step 1: Determine P, U, E.

Adding a 4-digit number and a 3-digit number yields a 5-digit result, so the final carry is 1 and P = 1.

From the thousands column: U + c3 = E + 10*P = E + 10. Therefore U + c3 >= 10. The only way with single-digit U is c3 = 1 and U = 9, giving E = 0.

So far: P = 1, U = 9, E = 0.

Step 2: Use units and carries.

• Units: R + A = E + 10*c1. Since E = 0, R + A = 10, so c1 = 1 and R = 10 - A.

• Tens: 2S + c1 = C + 10*c2, so 2S + 1 = C + 10*c2.

• Hundreds: S + U + c2 = A + 10*c3. With U = 9 and c3 = 1, this gives S + 9 + c2 = A + 10, so A = S - 1 + c2.

Step 3: Find S, A, R, C by trying feasible carry c2 values (0 or 1).

If c2 = 0: then A = S - 1 and from tens 2S + 1 = C (since c2 = 0). Try small S values that keep digits distinct and unused (P=1, U=9, E=0).

Try S = 3: then A = 2, R = 8 (because R = 10 - A), and C = 2S + 1 = 7. All digits are distinct and unused.

This gives the full assignment: P=1, E=0, A=2, C=7, U=9, S=3, R=8.

Step 4: Verify the addition numerically.

USSR = 9 3 3 8 = 9338

USA = 9 3 2 = 932

Sum = 9338 + 932 = 10270, which corresponds to P E A C E = 1 0 2 7 0.

Final question: P + E + A + C + E = 1 + 0 + 2 + 7 + 0 = 10.