AMC 10 · 2022 · #12
Grade 6 arithmeticPick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #4 (Matrix Logic) is the natural fit: build a small grid whose rows are the four kinds of children (truth-teller, liar, alternater-starts-truth, alternater-starts-lie) and whose columns are the three questions. Tool #2 (Systematic List) drives row-by-row Yes/No filling — for each row we just apply the rule mechanically. Tool #7 (Subproblems) breaks the work into (a) build the grid, (b) translate Yes-counts into equations, (c) solve. Tool #13 (Algebra) lets us subtract equations to isolate the truth-teller count without solving the whole system. The final candy count is just (number of truth-tellers) x (Yes answers each gives), and the row for truth-tellers in the grid will read Yes-No-No, so each contributes exactly one candy.
Build the Yes/No grid (Tool #4): rows T, L, A_t, A_ℓ; columns Q1–Q3. Apply each child's rule cell by cell to fill the table.
Kindergarten-level classification: sort children into categories and count Yes per row — the only "hard" part is reading the rules out loud one row at a time.
K.MD.B.3Use Matrix LogicSpot-check the trickiest row A_ℓ (starts with a lie): applying its lie-truth-lie rule across Q1–Q3 gives Yes, Yes, Yes.
Just walk through Yes/No one question at a time using each child's rule — Kindergarten classification, no arithmetic yet.
K.MD.B.3Make A Systematic ListRead each Yes column into an equation: Q1 gives T + L + A_ℓ = 22, Q2 gives L + A_ℓ = 15, Q3 gives A_ℓ = 9.
Each Yes count is just "add up the rows that said Yes in that column" — Grade 6 expression-writing turns the grid into three equations.
6.EE.B.6Identify SubproblemsSubtract equation 2 from equation 1 (Tool #13): L and A_ℓ cancel at once, leaving the truth-teller count T = 7.
One subtraction isolates T — Grade 6 equation-solving avoids touching the other unknowns.
6.EE.B.7Convert To AlgebraThe T-row reads Yes-No-No, so each truth-teller earns just 1 candy (from Q1); with 7 of them the candy total is 7 × 1 = 7.
Multiply candies-per-child by number of truth-tellers — Grade 3 multiplication.
3.OA.A.3Use Matrix LogicMatching the total to the choices gives option (A).
Final compare-to-options — the smallest, cleanest number on the list.
K.MD.B.3Eliminate PossibilitiesThis AMC 10 problem only needs Grade 6 expression-writing you already know — build a tiny Yes/No grid for the four kinds of children, read off the three Yes-counts as equations, and one subtraction (22 - 15) gives the truth-teller count directly. They say Yes only once, so the candy total is exactly 7.