DONALD + GERALD = ROBERT
Unique letters: D, O, N, A, L, G, E, R, B, T (10 letters)
DONALD + GERALD = ROBERT
This guide walks you through the complete solution step by step, showing how logical deduction eliminates possibilities until the unique answer emerges.
Step-by-Step Walkthrough
Step 1: Start with the ones column
D + D = T (mod 10). T must be even. Since D is a leading digit, D ≠ 0.
Step 2: The hundreds column — the critical constraint
N + R = B (mod 10) with carry. Notice that A + A appears in the tens column (A + A = R or R + 10). Since A + A is always even, R must be even (if no carry from ones to tens) or odd (if there is a carry). This creates an important parity constraint.
Step 3: Ten unique letters = all digits used
This puzzle has exactly 10 unique letters: D, O, N, A, L, G, E, R, B, T. Every digit from 0–9 must be assigned to exactly one letter. This dramatically narrows the search — once 9 letters are determined, the 10th is forced.
Step 4: Leading digit analysis
D + G + carry = R (in the hundred-thousands place, the leftmost column). Since DONALD and GERALD are both 6-digit numbers and ROBERT is also 6-digit, the sum did not gain a digit. This means D + G + carry ≤ 9, so R < D + G + carry. Also D ≠ 0 and G ≠ 0.
Step 5: Use D = 5 as the key starting point
Through systematic constraint analysis starting from the rightmost column, D = 5 can be determined. Then T = 0 (5 + 5 = 10). With T = 0, G ≠ 0 and D ≠ 0 confirmed. From the ten-thousands column: O + E + carry = O (mod 10), which means E + carry = 0 or 10. Since E ≠ T = 0, we need E + carry = 10.
Step 6: Work through the remaining columns
With D=5, T=0 established: From the tens column A + A + carry = R (or R + 10). From thousands: L + L + carry = E (or E + 10). From hundred-thousands: D + G + carry = R → 5 + G + carry = R. Working through all constraints with the remaining digits {1,2,3,4,6,7,8,9} yields: G=1, E=9, A=4, L=8, R=7, O=2, N=6, B=3.
Solution
D=5, O=2, N=6, A=4, L=8, G=1, E=9, R=7, B=3, T=0
526485 + 197485 = 723970
Verification
DONALD = 526485, GERALD = 197485, ROBERT = 723970. Check: 526485 + 197485 = 723970 ✓. All 10 digits 0–9 used exactly once.
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