Problem 20 · 2001 AMC 8
Stretch
Logic & Word Problems
deduction
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).
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Answer: A — S, Q, M.
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Hint 1 of 2
The crucial detail is what each person can SEE. Quay only saw Kaleana's score, yet he's CERTAIN two scores match — how could he be certain knowing just one other score?
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Hint 2 of 2
He can only be sure if his own score equals the one he saw: Q = K. Once that's locked, turn Marty's and Shana's thoughts into comparisons with K.
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Approach: turn each statement into an inequality
- Quay is certain two scores are equal but has seen only Kaleana's — the sole way to be sure is if HIS score matches it. So Q = K.
- Marty knows he's not lowest, and the only score he's seen is Kaleana's, so M > K. Shana knows she's not highest, so S < K. Swap K for Q: S < Q < M.
- Lowest to highest: S, Q, M. The transferable move in "who-knows-what" logic: a person's certainty is built only from what they can actually observe — that constraint is what pins everything down.
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