Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #19
Grade 6 logicPick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Each girl's statement only constrains the pair she can actually see — Cassie sees (C, H) and Bridget sees (B, H). Tool #7 (Identify Subproblems) splits the puzzle into two independent two-girl comparisons, which is much easier than reasoning about all three at once. Tool #3 (Eliminate Possibilities) is the engine: in each subproblem we list the two possible orderings of the visible pair and rule out the one that would make the speaker's certainty impossible. Chaining the two surviving inequalities through Hannah pins down the full ranking, so we don't need algebra (Tool #13) at all.
Read Cassie's clue
Cassie sees only Hannah's score. If C ＜ H, a hidden higher Bridget could leave her lowest — she can't be sure, so C ＞ H.
Of the two possible orderings C ＜ H and C ＞ H, only C ＞ H guarantees at least one person (Hannah) is below Cassie no matter what Bridget scored.
Cassie's certainty that she did not get the lowest score forces her own score C to sit above Hannah's score H.
▸ Why?
A statement Cassie can be truly certain of has to hold no matter what the one score she cannot see — Bridget's B — turns out to be, so it can rest only on the single comparison in front of her: her score C against Hannah's score H.
▸ Why?
That one visible comparison rules out being the lowest only when C is above H, because then Hannah is definitely below her; if C did not already top H, the hidden score B could come out higher than her too and leave Cassie at the very bottom.
▸ Why?
A claim about the score she cannot see counts as certain only when it stays true in every case that hidden score could produce, and a single case that breaks it would remove the certainty.
Read Bridget's clue
Bridget sees only Hannah's score. If B ＞ H, a hidden lower Cassie could leave her highest — she can't be sure, so B ＜ H.
Of the two orderings B ＞ H and B ＜ H, only B ＜ H guarantees at least one person (Hannah) is above Bridget no matter what Cassie scored.
4.OA.A.3Eliminate PossibilitiesChain the two inequalities
Chain the survivors through Hannah: C ＞ H from step 1 and H ＞ B from step 2 give C ＞ H ＞ B.
The two subproblems share Hannah, so her score acts as the hinge that links the two pieces into one full ordering.
6.EE.B.8Identify SubproblemsRead off the ranking
Match C ＞ H ＞ B to the choices: Cassie, Hannah, Bridget — choice (D).
Matching the chain C ＞ H ＞ B to the five answer choices leaves only (D); the other four contradict at least one of the two deduced inequalities.
6.EE.B.8Eliminate PossibilitiesThis AMC 8 puzzle only needs the Grade 6 idea that two inequalities sharing a middle term (C ＞ H and H ＞ B) can be chained into one ordering (C ＞ H ＞ B) — no algebra required!
- Read Cassie's clue
- Read Bridget's clue
- Chain the two inequalities
- Read off the ranking
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