AMC 8 · 2001 · #20

Grade 6 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

logical-deductionif-then-reasoning caseworkidentify-subproblems ↑ Prerequisites: logical-deduction

📏 Short solution 💡 3 insights

Problem

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.

Pick an answer.

(A)

S,Q,M

(B)

Q,M,S

(C)

Q,S,M

(D)

M,S,Q

(E)

S,M,Q

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

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.

1STEP 1

Quay sees only his own score and Kaleana's, so the one match he can guarantee is Q = K.

Q = K

2STEP 2

Marty sees only M and K, and "not the lowest" is guaranteed by that single comparison only when M ＞ K.

M ＞ K

3STEP 3

By the mirror argument, Shana's "not the highest" is guaranteed only when S ＜ K.

S ＜ K

4STEP 4

Substitute K = Q into the three inequalities and they line up as S ＜ Q ＜ M.

S ＜ K = Q ＜ M → S ＜ Q ＜ M

5STEP 5

Lowest to highest is S, Q, M, which matches choice (A).

S, Q, M → (A)

Answer

S,Q,M

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).

💡Key takeaway

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).

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