AMC 8 · 2001 · #20
Grade 6 logicProblem
Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Kaleana shows her test score to Quay, Marty, and Shana, but each of those three keeps their own score hidden. So every one of the three sees only two scores: Kaleana's and their own. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." Using only what each could reasonably conclude from those two scores, order Marty ($M$), Quay ($Q$), and Shana ($S$) from lowest to highest.
Givens: Kaleana's score $K$ is visible to Quay, Marty, and Shana; Quay, Marty, and Shana each know only their own score and $K$; Quay's thought: at least two of the four scores are equal; Marty's thought: $M$ is not the lowest of the four; Shana's thought: $S$ is not the highest of the four; Answer choices: (A) $\text{S,Q,M}$, (B) $\text{Q,M,S}$, (C) $\text{Q,S,M}$, (D) $\text{M,S,Q}$, (E) $\text{S,M,Q}$
Unknowns: The order of $M$, $Q$, $S$ from lowest to highest
Understand
Restated: Kaleana shows her test score to Quay, Marty, and Shana, but each of those three keeps their own score hidden. So every one of the three sees only two scores: Kaleana's and their own. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." Using only what each could reasonably conclude from those two scores, order Marty ($M$), Quay ($Q$), and Shana ($S$) from lowest to highest.
Givens: Kaleana's score $K$ is visible to Quay, Marty, and Shana; Quay, Marty, and Shana each know only their own score and $K$; Quay's thought: at least two of the four scores are equal; Marty's thought: $M$ is not the lowest of the four; Shana's thought: $S$ is not the highest of the four; Answer choices: (A) $\text{S,Q,M}$, (B) $\text{Q,M,S}$, (C) $\text{Q,S,M}$, (D) $\text{M,S,Q}$, (E) $\text{S,M,Q}$
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #2 Make a Systematic List
Each speaker compares only two numbers: their own score and Kaleana's score $K$. So every statement has to be forced by that single comparison. Tool #3 (Eliminate Possibilities) is the right lead: for each of Quay, Marty, and Shana, list the three possibilities ($> K$, $= K$, $< K$) and eliminate the ones that don't make the statement certain. Tool #2 (Make a Systematic List) is the bookkeeping that keeps the three cases straight without overlooking one. We avoid Tool #13 (Algebra) because the whole problem is settled by three short comparisons — no equations needed.
Execute — Answer: A
6.EE.B.5 Step 1 - Pin down Quay.
- Quay sees only his own score $Q$ and Kaleana's $K$.
- He thinks, "At least two of us have the same score." The only equality he can be sure of is between the two scores he can see.
- So $Q = K$.
💡 Quay can't peek at Marty's or Shana's scores, so a guaranteed match must be between his score and Kaleana's.
6.EE.B.8 Step 2 - Pin down Marty.
- Marty sees only $M$ and $K$ and is sure he is not the lowest.
- If $M$ were less than $K$, $M$ might end up the smallest of all four.
- The only way to be sure $M$ isn't the lowest, using only the comparison with $K$, is $M > K$.
💡 "Not the lowest" is a guarantee only if Marty already beats the one score he can see — Kaleana's.
6.EE.B.8 Step 3 - Pin down Shana.
- Shana sees only $S$ and $K$ and is sure she is not the highest.
- By the mirror argument, the only way to be certain is $S < K$.
💡 "Not the highest" is a guarantee only if Shana already loses to the one score she can see.
6.EE.B.8 Step 4 - Chain the three results.
- From $Q = K$, $M > K$, and $S < K$, substitute $K = Q$ to compare $M$, $Q$, $S$ on one number line.
💡 Three short inequalities about the same anchor $K$ line up into one ordering from smallest to largest.
6.EE.B.5 Step 5 - Read off the ordering and match an answer choice.
- Lowest to highest is $S$, then $Q$, then $M$.
💡 Only choice (A) lists the three names in the order $S, Q, M$.
6.EE.B.5 Pin down Quay. Quay sees only his own score $Q$ and Kaleana's $K$. He thinks, "A 6.EE.B.8 Pin down Marty. Marty sees only $M$ and $K$ and is sure he is not the lowest. If 6.EE.B.8 Pin down Shana. Shana sees only $S$ and $K$ and is sure she is not the highest. 6.EE.B.8 Chain the three results. From $Q = K$, $M > K$, and $S < K$, substitute $K = Q$ 6.EE.B.5 Read off the ordering and match an answer choice. Lowest to highest is $S$, then Review
Reasonableness: Plug in numbers to confirm. Let $K = 80$. Then $Q = 80$, $M$ is any score above $80$ (say $90$), and $S$ is any score below $80$ (say $70$). Check each thought: Quay sees $80, 80$ and is sure two scores match — yes, his and Kaleana's. Marty sees $80, 90$ and is sure he isn't the lowest — yes, he beats the one score he can see, and the hidden two can't drag him below his own $90$. Shana sees $80, 70$ and is sure she isn't the highest — yes, she loses to the one score she can see. The ordering $70 < 80 < 90$, i.e. $S < Q < M$, matches answer (A). The other choices fail because they put $M$ behind someone (contradicts $M > K = Q$) or put $S$ ahead of someone (contradicts $S < K = Q$).
Alternative: Tool #2 (Make a Systematic List): list the five answer choices and test each against the three thoughts. For (B) $Q,M,S$ the highest is $S$, contradicting Shana's thought. For (C) $Q,S,M$ the lowest is $Q$, but Quay's certainty about a tie isn't supported — and worse, $S > Q$ would force $S > K$, contradicting Shana. For (D) $M,S,Q$ the lowest is $M$, contradicting Marty. For (E) $S,M,Q$ the highest among the three is $Q$, but with $Q$ above $K$ no equality is forced, contradicting Quay's certainty. Only (A) survives.
CCSS standards used (min grade 6)
6.EE.B.5Understand solving an equation or inequality as answering which values make it true (Testing each of "greater than $K$," "equal to $K$," and "less than $K$" against each thought and keeping only the case that makes the statement certain.)6.EE.B.8Write an inequality of the form $x > c$ or $x < c$ to represent a constraint (Capturing the three conclusions as $Q = K$, $M > K$, $S < K$, then chaining them into the single ordering $S < Q < M$.)
⭐ Each speaker can only compare their score to Kaleana's. So every thought has to be guaranteed by that one comparison. Quay's "tie" forces $Q = K$, Marty's "not lowest" forces $M > K$, and Shana's "not highest" forces $S < K$ — chained together, $S < Q < M$, answer (A).
⭐ Each speaker can only compare their score to Kaleana's. So every thought has to be guaranteed by that one comparison. Quay's "tie" forces $Q = K$, Marty's "not lowest" forces $M > K$, and Shana's "not highest" forces $S < K$ — chained together, $S < Q < M$, answer (A).