1. **State the problem:** We want to find the largest possible sum of the addition problem BEST + OF + LUCK where each letter represents a different digit. 2. **Understand the problem:** Each letter corresponds to a unique digit from 0 to 9. We want to assign digits to letters B, E, S, T, O, F, L, U, C, K to maximize the sum BEST + OF + LUCK. 3. **List the letters:** B, E, S, T, O, F, L, U, C, K (10 letters, so digits 0-9 are all used). 4. **Analyze place values:** - BEST is a 4-digit number: $1000B + 100E + 10S + T$ - OF is a 2-digit number: $10O + F$ - LUCK is a 4-digit number: $1000L + 100U + 10C + K$ 5. **Goal:** Maximize $BEST + OF + LUCK = (1000B + 100E + 10S + T) + (10O + F) + (1000L + 100U + 10C + K)$ 6. **Strategy:** Assign the largest digits to the letters in the highest place values. 7. **Assign digits:** - Highest place values are thousands in BEST and LUCK, so assign 9 and 8 to B and L. - Next highest are hundreds in BEST and LUCK, assign 7 and 6 to E and U. - Then tens in BEST, LUCK, and OF: assign 5, 4, 3 to S, C, O. - Finally, ones place: assign 2, 1, 0 to T, F, K. 8. **Example assignment:** - B=9, L=8 - E=7, U=6 - S=5, C=4, O=3 - T=2, F=1, K=0 9. **Calculate each number:** - BEST = $1000\times9 + 100\times7 + 10\times5 + 2 = 9000 + 700 + 50 + 2 = 9752$ - OF = $10\times3 + 1 = 31$ - LUCK = $1000\times8 + 100\times6 + 10\times4 + 0 = 8000 + 600 + 40 + 0 = 8640$ 10. **Sum:** $9752 + 31 + 8640 = 18423$ 11. **Conclusion:** The largest sum obtainable is **18423** with the digit assignments above.