Reasoning · Grade 3-1 Setting Up the Equation

Problem

Cryptarithm digit deduction

In ABB + ACC = DCD, equal letters are equal digits and different letters differ. Each column is ordinary addition with a carry of 0 or 1. DCD has three digits, so the hundreds column carries nothing further. Find the largest possible DCD.
Your answer
How to solve
Strategy Guess and Check — Add the puzzle one place-value column at a time. The carry rules pin down B at once, turn the units and hundreds columns into a tiny chain of equations, and leave only a few values of A to test, so I list those and keep the largest.
1STEP 1

Look at the tens column

C cancels in the tens, leaving only B and the carry.

B + C + c₁ = C + 10c₂ → B + c₁ = 10c₂
2STEP 2

Pin down B and the units carry

That must make 10, so B = 9.

B + c₁ = 10, B ≤ 9 → B = 9, c₁ = 1
3STEP 3

Use the units column

The ones column gives C = D + 1.

B + C = D + 10 → 9 + C = D + 10 → C = D + 1
4STEP 4

Use the hundreds column

The hundreds column gives D = 2A + 1.

A + A + c₂ = D → 2A + 1 = D, C = 2A + 2
5STEP 5

List the working values of A and pick the largest DCD

Keeping C to one digit caps A at 3, so 787 is largest.

A=3: 399 + 388 = 787, DCD = 787
Answer
787 (399 + 388 = 787)
Check the winning case directly: A=3, B=9, C=8, D=7 gives 399 + 388 = 787, which reads as DCD = 787, and the four digits 3, 9, 8, 7 are all different. DCD is a three-digit number as required, so the answer is sound.
Takeaway

Add the puzzle one column at a time and the carries do the detective work for you - it only needs Grade 3 place-value sense!

  • Look at the tens column
  • Pin down B and the units carry
  • Use the units column
  • Use the hundreds column
  • List the working values of A and pick the largest DCD