AMC 10 · 2022 · #9
Grade 6 logicPick an answer.
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.
Tabulate the answers
Tabulate each kind's answers.
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.3Introduce A VariableSplit the alternaters
They split by which way they start.
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 ListWrite the three equations
The three counts give three equations.
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 to isolate
Subtracting two equations gives it at once.
One subtraction isolates T — Grade 6 equation-solving avoids touching the other unknowns.
One subtraction isolates the unknown without touching the others.
▸ Why?
The two equations share the same block, so subtracting removes it entirely.
▸ Why?
Subtracting one true equation from another keeps the result true, so the move is legitimate.
Convert to candy
Each truth-teller gets exactly one candy.
Multiply candies-per-child by number of truth-tellers — Grade 3 multiplication.
3.OA.A.3Introduce A VariableMatch the choice
The total is 7.
Final compare-to-options — the smallest, cleanest number on the list.
K.MD.B.3Eliminate PossibilitiesThis AMC 12 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.