AMC 8 · 2009 · #24
Grade 4 logicPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Cryptarithms are pure place-value puzzles. Tool #3 (Write an Equation) turns each column of the addition and subtraction into a small equation in the unknown digits (with carry/borrow bits). Tool #7 (Identify Subproblems) tells us to work column by column — start with the ones column, which forces B, then move to the tens column, which forces A, C, and finally D — instead of trying to guess all four digits at once.
Ones column of the addition: B + A ends in A, so B = 0 — and no carry into the tens column.
Adding A to something and getting back A in the ones place is exactly the Grade 4 "add multi-digit whole numbers using place value" idea — the only digit that vanishes under addition is 0.
4.NBT.B.4Eliminate PossibilitiesSubtraction ones column: with B = 0, the 0 - A needs a borrow, so 10 - A = A and A = 5.
Borrowing in column subtraction is the standard Grade 4 algorithm; the borrow turns the impossible "0 - A = A" into the solvable "10 - A = A".
4.NBT.B.4Eliminate PossibilitiesSubtraction tens column: the borrow took 1 from the tens, so (A - 1) - C = 0 and C = 4.
Tool #7: once the ones column is solved, the tens column becomes its own small subproblem with one unknown.
4.NBT.B.4Identify SubproblemsAddition tens column: no carry from the ones, so D = A + C = 5 + 4 = 9.
With A and C pinned down, the tens column of the addition reads off D directly.
4.NBT.B.4Eliminate PossibilitiesThis AMC 8 problem only needs the Grade 4 standard column-by-column add/subtract algorithm — once you read each column carefully, the digits fall out one at a time.