AMC 8 · 2013 · #19

Grade 6 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

logical-deductionif-then-reasoning caseworksystematic-enumeration ↑ Prerequisites: logical-deduction

📏 Short solution 💡 2 insights

Problem

Hannah, Bridget, and Cassie just took a test. Hannah showed her score to both of the others, but neither Bridget nor Cassie revealed her own score. Then Cassie says, "I didn't get the lowest score," and Bridget says, "I didn't get the highest score." Each statement must be something the speaker is certain of, given only what she actually sees. Rank the three girls from highest to lowest score.

Pick an answer.

(A)

Hannah, Cassie, Bridget

(B)

Hannah, Bridget, Cassie

(C)

Cassie, Bridget, Hannah

(D)

Cassie, Hannah, Bridget

(E)

Bridget, Cassie, Hannah

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

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.

1STEP 1

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.

Cassie sure she ≠ lowest ⟹ C ＞ H

2STEP 2

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.

Bridget sure she ≠ highest ⟹ B ＜ H

3STEP 3

Chain the survivors through Hannah: C ＞ H from step 1 and H ＞ B from step 2 give C ＞ H ＞ B.

C ＞ H and H ＞ B ⟹ C ＞ H ＞ B

4STEP 4

Match C ＞ H ＞ B to the choices: Cassie, Hannah, Bridget — choice (D).

C ＞ H ＞ B → Cassie, Hannah, Bridget → (D)

Answer

Cassie, Hannah, Bridget

Sanity-check by trying a concrete scoring. Say C = 95, H = 85, B = 70. Cassie sees C = 95 and H = 85; even if Bridget had scored 100, Cassie still beats Hannah and is not lowest — her statement holds with certainty. Bridget sees B = 70 and H = 85; even if Cassie had scored 0, Bridget is still below Hannah and is not highest — her statement holds with certainty. Now flip the inequalities to see the other choices fail: if instead H ＞ C, Cassie cannot rule out Bridget being highest with herself lowest — her certainty breaks. Likewise if B ＞ H, Bridget cannot rule out being the highest. So only (D) survives.

💡Key takeaway

This 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!

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