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

Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Cassie and Bridget each see Hannah's score $H$; Cassie does not see Bridget's score $B$; Bridget does not see Cassie's score $C$; Cassie is certain: her score is not the lowest of the three; Bridget is certain: her score is not the highest of the three; Answer choices: (A) Hannah, Cassie, Bridget; (B) Hannah, Bridget, Cassie; (C) Cassie, Bridget, Hannah; (D) Cassie, Hannah, Bridget; (E) Bridget, Cassie, Hannah

Unknowns: The ranking of the three scores from highest to lowest

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #7 Identify Subproblems

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.

Execute — Answer: D

#3 Eliminate Possibilities 4.OA.A.3 Step 1

Subproblem 1: what does Cassie's statement tell us about $C$ vs.
$H$?
The only score Cassie can see besides her own is Hannah's.
If $C < H$, then for all Cassie knows Bridget's hidden score could be anything — possibly higher than both, leaving Cassie at the bottom.
She could *not* be certain she avoided the lowest.
So $C < H$ is ruled out.

Cassie sure she $\ne$ lowest $\;\Longrightarrow\; C > H$

💡 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.

#3 Eliminate Possibilities 4.OA.A.3 Step 2

Subproblem 2: what does Bridget's statement tell us about $B$ vs.
$H$?
The only score Bridget can see besides her own is Hannah's.
If $B > H$, then Cassie's hidden score could be anything — possibly lower than both, leaving Bridget at the top.
She could *not* be certain she avoided the highest.
So $B > H$ is ruled out.

Bridget sure she $\ne$ highest $\;\Longrightarrow\; B < H$

💡 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.

#7 Identify Subproblems 6.EE.B.8 Step 3

Combine the two surviving inequalities through Hannah.
From step 1, $C > H$.
From step 2, $B < H$, i.e.
$H > B$.
Chaining gives $C > H > B$.

$$C > H \text{ and } H > B \;\Longrightarrow\; C > H > B$$

💡 The two subproblems share Hannah, so her score acts as the hinge that links the two pieces into one full ordering.

#3 Eliminate Possibilities 6.EE.B.8 Step 4

Read off the ranking.
Highest to lowest is Cassie, Hannah, Bridget — choice $\textbf{(D)}$.

$$C > H > B \;\Rightarrow\; \text{Cassie, Hannah, Bridget} \;\Rightarrow\; \textbf{(D)}$$

💡 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.

[1] #3 4.OA.A.3 Subproblem 1: what does Cassie's statement tell us about $C$ vs. $H$? The only s

[2] #3 4.OA.A.3 Subproblem 2: what does Bridget's statement tell us about $B$ vs. $H$? The only

[3] #7 6.EE.B.8 Combine the two surviving inequalities through Hannah. From step 1, $C > H$. Fro

[4] #3 6.EE.B.8 Read off the ranking. Highest to lowest is Cassie, Hannah, Bridget — choice $\te

Review

Reasonableness: 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.

Alternative: Tool #2 (Make a Systematic List) on the five answer choices, applying both inequalities $C > H$ and $H > B$ as filters: (A) Hannah, Cassie, Bridget has $H > C$ — fails Cassie's deduction; (B) Hannah, Bridget, Cassie has $H > C$ and $H > B$ but the second is fine, the first kills it; (C) Cassie, Bridget, Hannah has $B > H$ — fails Bridget's deduction; (E) Bridget, Cassie, Hannah has $B > H$ — fails Bridget's deduction. Only (D) Cassie, Hannah, Bridget satisfies both filters.

CCSS standards used (min grade 6)

4.OA.A.3 Solve multistep word problems using the four operations and reasoning (Working through each girl's statement case by case — listing the two possible orderings of the pair she can see and ruling out the one that would leave her certainty unjustified.)
6.EE.B.8 Write and reason about inequalities representing a constraint or condition (Translating "not the lowest" and "not the highest" into the inequalities $C > H$ and $H > B$, then chaining them through the shared term $H$ to get $C > H > B$.)

⭐ 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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